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... 1523 A Atomic Masses.................................................... 1531 B Selected Radioactive Isotopes............................................ 1537 C Useful Information.................................................. 1541 D Glossary of Key Symbols and Notation....................................... 1545 Index........................................................... 1659 This content is available for free at http://cnx.org/content/col11844/1.13 Preface 1 PREFACE The OpenStax College Physics: AP® Edition program has been developed with several goals in mind: accessibility, customization, and student engagement—all while encouraging science students toward high levels of academic scholarship. Instructors and students alike will find that this program offers a strong foundation in physics in an accessible format. Welcome! About OpenStax College OpenStax College is a nonprofit organization committed to improving student access to quality learning materials. Our free textbooks are developed and peer-reviewed by educators to ensure they are readable, accurate, and meet the scope and sequence requirements of today’s high school courses. Unlike traditional textbooks, OpenStax College resources live online and are owned by the community of educators using them. Through our partnerships with companies and foundations committed to reducing costs for students, OpenStax College is working to improve access to education for all. OpenStax College is an initiative of Rice University and is made possible through the generous support of several philanthropic foundations. OpenStax College

resources provide quality academic instruction. Three key features set our materials apart from others: they can be customized by instructors for each class, they are a “living” resource that grows online through contributions from science educators, and they are available for free or at minimal cost. Customization OpenStax College learning resources are designed to be customized for each course. Our textbooks provide a solid foundation on which instructors can build, and our resources are conceived and written with flexibility in mind. Instructors can simply select the sections most relevant to their curricula and create a textbook that speaks directly to the needs of their classes and students. Teachers are encouraged to expand on existing examples by adding unique context via geographically localized applications and topical connections. This customization feature will help bring physics to life for students and will ensure that your textbook truly reflects the goals of your course. Curation To broaden access and encourage community curation, OpenStax College Physics: AP® Edition is “open source” licensed under a Creative Commons Attribution (CC-BY) license. The scientific community is invited to submit examples, emerging research, and other feedback to enhance and strengthen the material and keep it current and relevant for today’s students. Submit your suggestions to info@openstaxcollege.org, and find information on edition status, alternate versions, errata, and news on the StaxDash at http://openstaxcollege.org (http://openstaxcollege.org). Cost Our textbooks are available for free online and in low-cost print and e-book editions. About OpenStax College Physics: AP® Edition In 2012, OpenStax College published College Physics as part of a series that offers free and open college textbooks for higher education. College Physics was quickly adopted for science courses all around the country, and as word about this valuable resource spread, advanced placement teachers around the country started utilizing the book in AP® courses too. Physics: AP® Edition is the result of an effort to better serve these teachers and students. Based on College Physics—a program based on the teaching and research experience of numerous physicists—Physics: AP® Edition focuses on and emphasizes the new AP® curriculum's concepts and practices. Alignment to the AP® curriculum The new AP® Physics curriculum framework outlines the two full-year physics courses AP® Physics 1: Algebra-Based and AP® Physics 2: Algebra-Based. These two courses replaced the one-year AP® Physics B course, which over the years had become a

fast-paced survey of physics facts and formulas that did not provide in-depth conceptual understanding of major physics ideas and the connections between them. The new AP® Physics 1 and 2 courses focus on the big ideas typically included in the first and second semesters of an algebrabased, introductory college-level physics course, providing students with the essential knowledge and skills required to support future advanced course work in physics. The AP® Physics 1 curriculum includes mechanics, mechanical waves, sound, and electrostatics. The AP® Physics 2 curriculum focuses on thermodynamics, fluid statics, dynamics, electromagnetism, geometric and physical optics, quantum physics, atomic physics, and nuclear physics. Seven unifying themes of physics called the Big Ideas each include three to seven Enduring Understandings (EU), which are themselves composed of Essential Knowledge (EK) that provides details and context for students as they explore physics. AP® Science Practices emphasize inquiry-based learning and development of critical thinking and reasoning skills. Inquiry usually uses a series of steps to gain new knowledge, beginning with an observation and following with a hypothesis to explain the observation; then experiments are conducted to test the hypothesis, gather results, and draw conclusions from data. The 2 Preface AP® framework has identified seven major science practices, which can be described by short phrases: using representations and models to communicate information and solve problems; using mathematics appropriately; engaging in questioning; planning and implementing data collection strategies; analyzing and evaluating data; justifying scientific explanations; and connecting concepts. The framework’s Learning Objectives merge content (EU and EK) with one or more of the seven science practices that students should develop as they prepare for the AP® Physics exam. Each chapter of OpenStax College Physics: AP® Edition begins with a Connection for AP® Courses introduction that explains how the content in the chapter sections align to the Big Ideas, Enduring Understandings, and Essential Knowledge in the AP® framework. Physics: AP® Edition contains a wealth of information and the Connection for AP® Courses sections will help you distill the required AP® content from material that, although interesting, exceeds the scope of an introductory-level course. Each section opens with the program’s learning objectives as well as the AP® learning objectives and science practices addressed. We have also developed Real World Connections features and Applying the Science Practices features that highlight concepts, examples, and practices in the framework. Pedagogical Foundation and Features OpenStax College Physics: AP® Edition

is organized such that topics are introduced conceptually with a steady progression to precise definitions and analytical applications. The analytical aspect (problem solving) is tied back to the conceptual before moving on to another topic. Each introductory chapter, for example, opens with an engaging photograph relevant to the subject of the chapter and interesting applications that are easy for most students to visualize. Our features include: • Connections for AP® Courses introduce each chapter and explain how its content addresses the AP® curriculum. • Worked examples promote both analytical and conceptual skills. They are introduced using an application of interest followed by a strategy that emphasizes the concepts involved, a mathematical solution, and a discussion. • Problem-solving strategies are presented independently and subsequently appear at crucial points in the text where students can benefit most from them. • Misconception Alerts address common misconceptions that students may bring to class. • Take Home Investigations provide the opportunity for students to apply or explore what they have learned with a hands- on activity. • Real World Connections highlight important concepts and examples in the AP® framework. • Applying the Science Practices includes activities and challenging questions that engage students while they apply the AP® science practices. • Things Great and Small explain macroscopic phenomena (such as air pressure) with submicroscopic phenomena (such as atoms bouncing off walls). • Simulations direct students to further explore the physics concepts they have learned about in the module through the interactive PHeT physics simulations developed by the University of Colorado. Assessment Physics: AP® Edition offers a wealth of assessment options that include: • End-of-Module Problems include conceptual questions that challenge students’ ability to explain what they have learned conceptually, independent of the mathematical details, and problems and exercises that challenge students to apply both concepts and skills to solve mathematical physics problems. Integrated Concept Problems challenge students to apply concepts and skills to solve a problem. • • Unreasonable Results encourage students to analyze the answer with respect to how likely or realistic it really is. • Construct Your Own Problem requires students to construct the details of a problem, justify their starting assumptions, show specific steps in the problem’s solution, and finally discuss the meaning of the result. • Test Prep for AP® Courses consists of end-of-module problems that include assessment items with the format and rigor found in the AP® exam to help prepare students. About Our Team Physics: AP® Edition would not be possible if not for the tremendous contributions of the authors and community reviewing team. Contributors

to OpenStax College Physics: AP® Edition This content is available for free at http://cnx.org/content/col11844/1.13 Preface Senior Contributors 3 Irna Lyublinskaya CUNY College of Staten Island, Staten Island, NY Gregg Wolfe Avonworth High School, Pittsburgh, PA Douglas Ingram TCU Department of Physics and Astronomy, Fort Worth, TX Liza Pujji Manukau Institute of Technology (MIT), New Zealand Sudhi Oberoi Visiting Research Student, QuIC Lab, Raman Research Institute, India Nathan Czuba Sabio Academy, Chicago, IL Julie Kretchman Science Writer, BS, University of Toronto, Canada John Stoke Science Writer, MS, University of Chicago, IL David Anderson Science Writer, PhD, College of William and Mary, Williamsburg, VA Erika Gasper Science Writer, MA, University of California, Santa Cruz, CA Advanced Placement Teacher Reviewers Faculty Reviewers Michelle Burgess Avon Lake High School, Avon Lake, OH Alexander Lavy Xavier High School, New York, NY Brian Hastings Spring Grove Area School District, York, PA John Boehringer Prosper High School, Prosper, TX Victor Brazil Petaluma High School, Petaluma, CA Jerome Mass Glastonbury Public Schools, Glastonbury, CT Bryan Callow Lindenwold High School, Lindenwold, NJ Anand Batra Howard University, Washington, DC John Aiken Georgia Institute of Technology, Atlanta, GA Robert Arts University of Pikeville, Pikeville, KY Ulrich Zurcher Cleveland State University, Cleveland, OH Michael Ottinger Missouri Western State University, Kansas City, MO James Smith Caldwell University, Caldwell, NJ Additional Resources Preparing for the AP® Physics 1 Exam Rice Online’s dynamic new course, available on edX, is fully integrated with Physics for AP® Courses for free. Developed by nationally recognized Rice Professor Dr. Jason Hafner and AP® Physics teachers Gigi Nevils-Noe and Matt Wilson the course combines innovative learning technologies with engaging, professionally-produced Concept Trailers™, inquiry based labs, practice problems, lectures, demonstrations, assessments, and other compelling resources to promote engagement and longterm retention of AP® Physics 1 concepts and application. Learn more at online.rice.edu. Other learning resources (powerpoint slides, testbanks, online homework etc) are updated frequently and can be viewed by going to https://openstaxcollege.org. To the AP

® Physics Student The fundamental goal of physics is to discover and understand the “laws” that govern observed phenomena in the world around us. Why study physics? If you plan to become a physicist, the answer is obvious—introductory physics provides the foundation for your career; or if you want to become an engineer, physics provides the basis for the engineering principles used to solve applied and practical problems. For example, after the discovery of the photoelectric effect by physicists, engineers developed photocells that are used in solar panels to convert sunlight to electricity. What if you are an aspiring medical doctor? Although the applications of the laws of physics may not be obvious, their understanding is tremendously valuable. Physics is involved in medical diagnostics, such as x-rays, magnetic resonance imaging (MRI), and ultrasonic blood flow measurements. Medical therapy sometimes directly involves physics; cancer radiotherapy uses ionizing radiation. What if you are planning a nonscience career? Learning physics provides you with a well-rounded education and the ability to make important decisions, such as evaluating the pros and cons of energy production sources or voting on decisions about nuclear waste disposal. 4 Preface This AP® Physics 1 course begins with kinematics, the study of motion without considering its causes. Motion is everywhere: from the vibration of atoms to the planetary revolutions around the Sun. Understanding motion is key to understanding other concepts in physics. You will then study dynamics, which considers the forces that affect the motion of moving objects and systems. Newton’s laws of motion are the foundation of dynamics. These laws provide an example of the breadth and simplicity of the principles under which nature functions. One of the most remarkable simplifications in physics is that only four distinct forces account for all known phenomena. Your journey will continue as you learn about energy. Energy plays an essential role both in everyday events and in scientific phenomena. You can likely name many forms of energy, from that provided by our foods, to the energy we use to run our cars, to the sunlight that warms us on the beach. The next stop is learning about oscillatory motion and waves. All oscillations involve force and energy: you push a child in a swing to get the motion started and you put energy into a guitar string when you pluck it. Some oscillations create waves. For example, a guitar creates sound waves. You will conclude this first physics course with the study of static electricity and electric currents. Many of the characteristics of static electricity can be explored by rubbing things together. Rubbing creates the

spark you get from walking across a wool carpet, for example. Similarly, lightning results from air movements under certain weather conditions. In AP® Physics 2 course you will continue your journey by studying fluid dynamics, which explains why rising smoke curls and twists and how the body regulates blood flow. The next stop is thermodynamics, the study of heat transfer—energy in transit—that can be used to do work. Basic physical laws govern how heat transfers and its efficiency. Then you will learn more about electric phenomena as you delve into electromagnetism. An electric current produces a magnetic field; similarly, a magnetic field produces a current. This phenomenon, known as magnetic induction, is essential to our technological society. The generators in cars and nuclear plants use magnetism to generate a current. Other devices that use magnetism to induce currents include pickup coils in electric guitars, transformers of every size, certain microphones, airport security gates, and damping mechanisms on sensitive chemical balances. From electromagnetism you will continue your journey to optics, the study of light. You already know that visible light is the type of electromagnetic waves to which our eyes respond. Through vision, light can evoke deep emotions, such as when we view a magnificent sunset or glimpse a rainbow breaking through the clouds. Optics is concerned with the generation and propagation of light. The quantum mechanics, atomic physics, and nuclear physics are at the end of your journey. These areas of physics have been developed at the end of the 19th and early 20th centuries and deal with submicroscopic objects. Because these objects are smaller than we can observe directly with our senses and generally must be observed with the aid of instruments, parts of these physics areas may seem foreign and bizarre to you at first. However, we have experimentally confirmed most of the ideas in these areas of physics. AP® Physics is a challenging course. After all, you are taking physics at the introductory college level. You will discover that some concepts are more difficult to understand than others; most students, for example, struggle to understand rotational motion and angular momentum or particle-wave duality. The AP® curriculum promotes depth of understanding over breadth of content, and to make your exploration of topics more manageable, concepts are organized around seven major themes called the Big Ideas that apply to all levels of physical systems and interactions between them (see web diagram below). Each Big Idea identifies Enduring Understandings (EU), Essential Knowledge (EK), and illustrative examples that support key concepts and content. Simple descriptions define

the focus of each Big Idea. • Big Idea 1: Objects and systems have properties. • Big Idea 2: Fields explain interactions. • Big Idea 3: The interactions are described by forces. • Big Idea 4: Interactions result in changes. • Big Idea 5: Changes are constrained by conservation laws. • Big Idea 6: Waves can transfer energy and momentum. • Big Idea 7: The mathematics of probability can to describe the behavior of complex and quantum mechanical systems. Doing college work is not easy, but completion of AP® classes is a reliable predictor of college success and prepares you for subsequent courses. The more you engage in the subject, the easier your journey through the curriculum will be. Bring your enthusiasm to class every day along with your notebook, pencil, and calculator. Prepare for class the day before, and review concepts daily. Form a peer study group and ask your teacher for extra help if necessary. The AP® lab program focuses on more open-ended, student-directed, and inquiry-based lab investigations designed to make you think, ask questions, and analyze data like scientists. You will develop critical thinking and reasoning skills and apply different means of communicating information. By the time you sit for the AP® exam in May, you will be fluent in the language of physics; because you have been doing real science, you will be ready to show what you have learned. Along the way, you will find the study of the world around us to be one of the most relevant and enjoyable experiences of your high school career. Irina Lyublinskaya, PhD Professor of Science Education To the AP® Physics Teacher The AP® curriculum was designed to allow instructors flexibility in their approach to teaching the physics courses. OpenStax College Physics: AP® Edition helps you orient students as they delve deeper into the world of physics. Each chapter includes a Connection for AP® Courses introduction that describes the AP® Physics Big Ideas, Enduring Understandings, and Essential Knowledge addressed in that chapter. Each section starts with specific AP® learning objectives and includes essential concepts, illustrative examples, and science practices, along with suggestions for applying the learning objectives through take home experiments, virtual lab investigations, and activities and questions for preparation and review. At the end of each section, students will find the Test Prep for AP® courses with multiple-choice and open-response questions addressing AP® learning objectives to help them prepare for the AP® exam. This content is available for free at http://cnx.org/content/col118

44/1.13 Preface 5 OpenStax College Physics: AP® Edition has been written to engage students in their exploration of physics and help them relate what they learn in the classroom to their lives outside of it. Physics underlies much of what is happening today in other sciences and in technology. Thus, the book content includes interesting facts and ideas that go beyond the scope of the AP® course. The AP® Connection in each chapter directs students to the material they should focus on for the AP® exam, and what content—although interesting—is not part of the AP® curriculum. Physics is a beautiful and fascinating science. It is in your hands to engage and inspire your students to dive into an amazing world of physics, so they can enjoy it beyond just preparation for the AP® exam. Irina Lyublinskaya, PhD Professor of Science Education The concept map showing major links between Big Ideas and Enduring Understandings is provided below for visual reference. 6 Preface This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 7 INTRODUCTION: THE NATURE OF 1 SCIENCE AND PHYSICS Figure 1.1 Galaxies are as immense as atoms are small. Yet the same laws of physics describe both, and all the rest of nature—an indication of the underlying unity in the universe. The laws of physics are surprisingly few in number, implying an underlying simplicity to nature's apparent complexity. (credit: NASA, JPL-Caltech, P. Barmby, Harvard-Smithsonian Center for Astrophysics) Chapter Outline 1.1. Physics: An Introduction 1.2. Physical Quantities and Units 1.3. Accuracy, Precision, and Significant Figures 1.4. Approximation Connection for AP® Courses What is your first reaction when you hear the word “physics”? Did you imagine working through difficult equations or memorizing formulas that seem to have no real use in life outside the physics classroom? Many people come to the subject of physics with a bit of fear. But as you begin your exploration of this broad-ranging subject, you may soon come to realize that physics plays a much larger role in your life than you first thought, no matter your life goals or career choice. For example, take a look at the image above. This image is of the Andromeda Galaxy, which contains billions of individual stars, huge clouds of gas, and

dust. Two smaller galaxies are also visible as bright blue spots in the background. At a staggering 2.5 million light years from Earth, this galaxy is the nearest one to our own galaxy (which is called the Milky Way). The stars and planets that make up Andromeda might seem to be the furthest thing from most people's regular, everyday lives. But Andromeda is a great starting point to think about the forces that hold together the universe. The forces that cause Andromeda to act as it does are the same forces we contend with here on Earth, whether we are planning to send a rocket into space or simply raise the walls for a new home. The same gravity that causes the stars of Andromeda to rotate and revolve also causes water to flow over hydroelectric dams here on Earth. Tonight, take a moment to look up at the stars. The forces out there are the same as the ones here on Earth. Through a study of physics, you may gain a greater understanding of the interconnectedness of everything we can see and know in this universe. Think now about all of the technological devices that you use on a regular basis. Computers, smart phones, GPS systems, MP3 players, and satellite radio might come to mind. Next, think about the most exciting modern technologies that you have heard about in the news, such as trains that levitate above tracks, “invisibility cloaks” that bend light around them, and microscopic robots that fight cancer cells in our bodies. All of these groundbreaking advancements, commonplace or unbelievable, rely on the principles of physics. Aside from playing a significant role in technology, professionals such as engineers, pilots, physicians, physical therapists, electricians, and computer programmers apply physics concepts in their daily work. For example, a pilot must understand how wind forces affect a flight path and a physical therapist must understand how the muscles in the body experience forces as they move and bend. As you will learn in this text, physics principles are propelling new, exciting technologies, and these principles are applied in a wide range of careers. In this text, you will begin to explore the history of the formal study of physics, beginning with natural philosophy and the ancient Greeks, and leading up through a review of Sir Isaac Newton and the laws of physics that bear his name. You will also be introduced to the standards scientists use when they study physical quantities and the interrelated system of measurements most of the scientific community uses to communicate in a single mathematical language. Finally, you will study the limits of our ability to be

accurate and precise, and the reasons scientists go to painstaking lengths to be as clear as possible regarding their own limitations. 8 Chapter 1 | Introduction: The Nature of Science and Physics Chapter 1 introduces many fundamental skills and understandings needed for success with the AP® Learning Objectives. While this chapter does not directly address any Big Ideas, its content will allow for a more meaningful understanding when these Big Ideas are addressed in future chapters. For instance, the discussion of models, theories, and laws will assist you in understanding the concept of fields as addressed in Big Idea 2, and the section titled ‘The Evolution of Natural Philosophy into Modern Physics' will help prepare you for the statistical topics addressed in Big Idea 7. This chapter will also prepare you to understand the Science Practices. In explicitly addressing the role of models in representing and communicating scientific phenomena, Section 1.1 supports Science Practice 1. Additionally, anecdotes about historical investigations and the inset on the scientific method will help you to engage in the scientific questioning referenced in Science Practice 3. The appropriate use of mathematics, as called for in Science Practice 2, is a major focus throughout sections 1.2, 1.3, and 1.4. 1.1 Physics: An Introduction Figure 1.2 The flight formations of migratory birds such as Canada geese are governed by the laws of physics. (credit: David Merrett) Learning Objectives By the end of this section, you will be able to: • Explain the difference between a principle and a law. • Explain the difference between a model and a theory. The physical universe is enormously complex in its detail. Every day, each of us observes a great variety of objects and phenomena. Over the centuries, the curiosity of the human race has led us collectively to explore and catalog a tremendous wealth of information. From the flight of birds to the colors of flowers, from lightning to gravity, from quarks to clusters of galaxies, from the flow of time to the mystery of the creation of the universe, we have asked questions and assembled huge arrays of facts. In the face of all these details, we have discovered that a surprisingly small and unified set of physical laws can explain what we observe. As humans, we make generalizations and seek order. We have found that nature is remarkably cooperative—it exhibits the underlying order and simplicity we so value. It is the underlying order of nature that makes science in general, and physics in particular, so enjoyable to study. For example, what do a bag of chips and a car battery have in common

? Both contain energy that can be converted to other forms. The law of conservation of energy (which says that energy can change form but is never lost) ties together such topics as food calories, batteries, heat, light, and watch springs. Understanding this law makes it easier to learn about the various forms energy takes and how they relate to one another. Apparently unrelated topics are connected through broadly applicable physical laws, permitting an understanding beyond just the memorization of lists of facts. The unifying aspect of physical laws and the basic simplicity of nature form the underlying themes of this text. In learning to apply these laws, you will, of course, study the most important topics in physics. More importantly, you will gain analytical abilities that will enable you to apply these laws far beyond the scope of what can be included in a single book. These analytical skills will help you to excel academically, and they will also help you to think critically in any professional career you choose to pursue. This module discusses the realm of physics (to define what physics is), some applications of physics (to illustrate its relevance to other disciplines), and more precisely what constitutes a physical law (to illuminate the importance of experimentation to theory). Science and the Realm of Physics Science consists of the theories and laws that are the general truths of nature as well as the body of knowledge they encompass. Scientists are continually trying to expand this body of knowledge and to perfect the expression of the laws that describe it. Physics is concerned with describing the interactions of energy, matter, space, and time, and it is especially interested in what fundamental mechanisms underlie every phenomenon. The concern for describing the basic phenomena in nature essentially defines the realm of physics. Physics aims to describe the function of everything around us, from the movement of tiny charged particles to the motion of people, cars, and spaceships. In fact, almost everything around you can be described quite accurately by the laws of physics. Consider a smart phone (Figure 1.3). Physics describes how electricity interacts with the various circuits inside the device. This This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 9 knowledge helps engineers select the appropriate materials and circuit layout when building the smart phone. Next, consider a GPS system. Physics describes the relationship between the speed of an object, the distance over which it travels, and the time it takes to travel that distance. When you use a GPS device in a vehicle, it

utilizes these physics equations to determine the travel time from one location to another. Figure 1.3 The Apple “iPhone” is a common smart phone with a GPS function. Physics describes the way that electricity flows through the circuits of this device. Engineers use their knowledge of physics to construct an iPhone with features that consumers will enjoy. One specific feature of an iPhone is the GPS function. GPS uses physics equations to determine the driving time between two locations on a map. (credit: @gletham GIS, Social, Mobile Tech Images) Applications of Physics You need not be a scientist to use physics. On the contrary, knowledge of physics is useful in everyday situations as well as in nonscientific professions. It can help you understand how microwave ovens work, why metals should not be put into them, and why they might affect pacemakers. (See Figure 1.4 and Figure 1.5.) Physics allows you to understand the hazards of radiation and rationally evaluate these hazards more easily. Physics also explains the reason why a black car radiator helps remove heat in a car engine, and it explains why a white roof helps keep the inside of a house cool. Similarly, the operation of a car's ignition system as well as the transmission of electrical signals through our body's nervous system are much easier to understand when you think about them in terms of basic physics. Physics is the foundation of many important disciplines and contributes directly to others. Chemistry, for example—since it deals with the interactions of atoms and molecules—is rooted in atomic and molecular physics. Most branches of engineering are applied physics. In architecture, physics is at the heart of structural stability, and is involved in the acoustics, heating, lighting, and cooling of buildings. Parts of geology rely heavily on physics, such as radioactive dating of rocks, earthquake analysis, and heat transfer in the Earth. Some disciplines, such as biophysics and geophysics, are hybrids of physics and other disciplines. Physics has many applications in the biological sciences. On the microscopic level, it helps describe the properties of cell walls and cell membranes (Figure 1.6 and Figure 1.7). On the macroscopic level, it can explain the heat, work, and power associated with the human body. Physics is involved in medical diagnostics, such as x-rays, magnetic resonance imaging (MRI), and ultrasonic blood flow measurements. Medical therapy sometimes directly involves physics; for example, cancer radiotherapy uses ionizing radiation. Physics can also explain sensory phenomena

, such as how musical instruments make sound, how the eye detects color, and how lasers can transmit information. It is not necessary to formally study all applications of physics. What is most useful is knowledge of the basic laws of physics and a skill in the analytical methods for applying them. The study of physics also can improve your problem-solving skills. Furthermore, physics has retained the most basic aspects of science, so it is used by all of the sciences, and the study of physics makes other sciences easier to understand. Figure 1.4 The laws of physics help us understand how common appliances work. For example, the laws of physics can help explain how microwave ovens heat up food, and they also help us understand why it is dangerous to place metal objects in a microwave oven. (credit: MoneyBlogNewz) 10 Chapter 1 | Introduction: The Nature of Science and Physics Figure 1.5 These two applications of physics have more in common than meets the eye. Microwave ovens use electromagnetic waves to heat food. Magnetic resonance imaging (MRI) also uses electromagnetic waves to yield an image of the brain, from which the exact location of tumors can be determined. (credit: Rashmi Chawla, Daniel Smith, and Paul E. Marik) Figure 1.6 Physics, chemistry, and biology help describe the properties of cell walls in plant cells, such as the onion cells seen here. (credit: Umberto Salvagnin) Figure 1.7 An artist's rendition of the the structure of a cell membrane. Membranes form the boundaries of animal cells and are complex in structure and function. Many of the most fundamental properties of life, such as the firing of nerve cells, are related to membranes. The disciplines of biology, chemistry, and physics all help us understand the membranes of animal cells. (credit: Mariana Ruiz) Models, Theories, and Laws; The Role of Experimentation The laws of nature are concise descriptions of the universe around us; they are human statements of the underlying laws or rules that all natural processes follow. Such laws are intrinsic to the universe; humans did not create them and so cannot change them. We can only discover and understand them. Their discovery is a very human endeavor, with all the elements of mystery, imagination, struggle, triumph, and disappointment inherent in any creative effort. (See Figure 1.8 and Figure 1.9.) The cornerstone of discovering natural laws is observation; science must describe the universe as it is, not as we may imagine

it to be. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 11 Figure 1.8 Isaac Newton (1642–1727) was very reluctant to publish his revolutionary work and had to be convinced to do so. In his later years, he stepped down from his academic post and became exchequer of the Royal Mint. He took this post seriously, inventing reeding (or creating ridges) on the edge of coins to prevent unscrupulous people from trimming the silver off of them before using them as currency. (credit: Arthur Shuster and Arthur E. Shipley: Britain's Heritage of Science. London, 1917.) Figure 1.9 Marie Curie (1867–1934) sacrificed monetary assets to help finance her early research and damaged her physical well-being with radiation exposure. She is the only person to win Nobel prizes in both physics and chemistry. One of her daughters also won a Nobel Prize. (credit: Wikimedia Commons) We all are curious to some extent. We look around, make generalizations, and try to understand what we see—for example, we look up and wonder whether one type of cloud signals an oncoming storm. As we become serious about exploring nature, we become more organized and formal in collecting and analyzing data. We attempt greater precision, perform controlled experiments (if we can), and write down ideas about how the data may be organized and unified. We then formulate models, theories, and laws based on the data we have collected and analyzed to generalize and communicate the results of these experiments. A model is a representation of something that is often too difficult (or impossible) to display directly. While a model is justified with experimental proof, it is only accurate under limited situations. An example is the planetary model of the atom in which electrons are pictured as orbiting the nucleus, analogous to the way planets orbit the Sun. (See Figure 1.10.) We cannot observe electron orbits directly, but the mental image helps explain the observations we can make, such as the emission of light from hot gases (atomic spectra). Physicists use models for a variety of purposes. For example, models can help physicists analyze a scenario and perform a calculation, or they can be used to represent a situation in the form of a computer simulation. A theory is an explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers.

Some theories include models to help visualize phenomena, whereas others do not. Newton's theory of gravity, for example, does not require a model or mental image, because we can observe the objects directly with our own senses. The kinetic theory of gases, on the other hand, is a model in which a gas is viewed as being composed of atoms and molecules. Atoms and molecules are too small to be observed directly with our senses—thus, we picture them mentally to understand what our instruments tell us about the behavior of gases. A law uses concise language to describe a generalized pattern in nature that is supported by scientific evidence and repeated experiments. Often, a law can be expressed in the form of a single mathematical equation. Laws and theories are similar in that they are both scientific statements that result from a tested hypothesis and are supported by scientific evidence. However, the designation law is reserved for a concise and very general statement that describes phenomena in nature, such as the law that 12 Chapter 1 | Introduction: The Nature of Science and Physics energy is conserved during any process, or Newton's second law of motion, which relates force, mass, and acceleration by the simple equation F = a. A theory, in contrast, is a less concise statement of observed phenomena. For example, the Theory of Evolution and the Theory of Relativity cannot be expressed concisely enough to be considered a law. The biggest difference between a law and a theory is that a theory is much more complex and dynamic. A law describes a single action, whereas a theory explains an entire group of related phenomena. And, whereas a law is a postulate that forms the foundation of the scientific method, a theory is the end result of that process. Less broadly applicable statements are usually called principles (such as Pascal's principle, which is applicable only in fluids), but the distinction between laws and principles often is not carefully made. Figure 1.10 What is a model? This planetary model of the atom shows electrons orbiting the nucleus. It is a drawing that we use to form a mental image of the atom that we cannot see directly with our eyes because it is too small. Models, Theories, and Laws Models, theories, and laws are used to help scientists analyze the data they have already collected. However, often after a model, theory, or law has been developed, it points scientists toward new discoveries they would not otherwise have made. The models, theories, and laws we devise sometimes imply the existence of objects or phenomena as yet unobserved. These predictions are remarkable triumphs

and tributes to the power of science. It is the underlying order in the universe that enables scientists to make such spectacular predictions. However, if experiment does not verify our predictions, then the theory or law is wrong, no matter how elegant or convenient it is. Laws can never be known with absolute certainty because it is impossible to perform every imaginable experiment in order to confirm a law in every possible scenario. Physicists operate under the assumption that all scientific laws and theories are valid until a counterexample is observed. If a good-quality, verifiable experiment contradicts a well-established law, then the law must be modified or overthrown completely. The study of science in general and physics in particular is an adventure much like the exploration of uncharted ocean. Discoveries are made; models, theories, and laws are formulated; and the beauty of the physical universe is made more sublime for the insights gained. The Scientific Method As scientists inquire and gather information about the world, they follow a process called the scientific method. This process typically begins with an observation and question that the scientist will research. Next, the scientist typically performs some research about the topic and then devises a hypothesis. Then, the scientist will test the hypothesis by performing an experiment. Finally, the scientist analyzes the results of the experiment and draws a conclusion. Note that the scientific method can be applied to many situations that are not limited to science, and this method can be modified to suit the situation. Consider an example. Let us say that you try to turn on your car, but it will not start. You undoubtedly wonder: Why will the car not start? You can follow a scientific method to answer this question. First off, you may perform some research to determine a variety of reasons why the car will not start. Next, you will state a hypothesis. For example, you may believe that the car is not starting because it has no engine oil. To test this, you open the hood of the car and examine the oil level. You observe that the oil is at an acceptable level, and you thus conclude that the oil level is not contributing to your car issue. To troubleshoot the issue further, you may devise a new hypothesis to test and then repeat the process again. The Evolution of Natural Philosophy into Modern Physics Physics was not always a separate and distinct discipline. It remains connected to other sciences to this day. The word physics comes from Greek, meaning nature. The study of nature came to be called “natural philosophy.” From ancient

times through the Renaissance, natural philosophy encompassed many fields, including astronomy, biology, chemistry, physics, mathematics, and medicine. Over the last few centuries, the growth of knowledge has resulted in ever-increasing specialization and branching of natural philosophy into separate fields, with physics retaining the most basic facets. (See Figure 1.11, Figure 1.12, and Figure 1.13.) Physics as it developed from the Renaissance to the end of the 19th century is called classical physics. It was transformed into modern physics by revolutionary discoveries made starting at the beginning of the 20th century. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 13 Figure 1.11 Over the centuries, natural philosophy has evolved into more specialized disciplines, as illustrated by the contributions of some of the greatest minds in history. The Greek philosopher Aristotle (384–322 B.C.) wrote on a broad range of topics including physics, animals, the soul, politics, and poetry. (credit: Jastrow (2006)/Ludovisi Collection) Figure 1.12 Galileo Galilei (1564–1642) laid the foundation of modern experimentation and made contributions in mathematics, physics, and astronomy. (credit: Domenico Tintoretto) Figure 1.13 Niels Bohr (1885–1962) made fundamental contributions to the development of quantum mechanics, one part of modern physics. (credit: United States Library of Congress Prints and Photographs Division) Classical physics is not an exact description of the universe, but it is an excellent approximation under the following conditions: Matter must be moving at speeds less than about 1% of the speed of light, the objects dealt with must be large enough to be seen with a microscope, and only weak gravitational fields, such as the field generated by the Earth, can be involved. Because humans live under such circumstances, classical physics seems intuitively reasonable, while many aspects of modern physics seem bizarre. This is why models are so useful in modern physics—they let us conceptualize phenomena we do not ordinarily experience. We can relate to models in human terms and visualize what happens when objects move at high speeds or imagine what objects too small to observe with our senses might be like. For example, we can understand an atom's properties because we can picture it in our minds, although we have never seen an atom with our eyes. New tools, of course, allow us to better

picture phenomena we cannot see. In fact, new instrumentation has allowed us in recent years to actually “picture” the atom. 14 Chapter 1 | Introduction: The Nature of Science and Physics Limits on the Laws of Classical Physics For the laws of classical physics to apply, the following criteria must be met: Matter must be moving at speeds less than about 1% of the speed of light, the objects dealt with must be large enough to be seen with a microscope, and only weak gravitational fields (such as the field generated by the Earth) can be involved. Figure 1.14 Using a scanning tunneling microscope (STM), scientists can see the individual atoms that compose this sheet of gold. (credit: Erwinrossen) Some of the most spectacular advances in science have been made in modern physics. Many of the laws of classical physics have been modified or rejected, and revolutionary changes in technology, society, and our view of the universe have resulted. Like science fiction, modern physics is filled with fascinating objects beyond our normal experiences, but it has the advantage over science fiction of being very real. Why, then, is the majority of this text devoted to topics of classical physics? There are two main reasons: Classical physics gives an extremely accurate description of the universe under a wide range of everyday circumstances, and knowledge of classical physics is necessary to understand modern physics. Modern physics itself consists of the two revolutionary theories, relativity and quantum mechanics. These theories deal with the very fast and the very small, respectively. Relativity must be used whenever an object is traveling at greater than about 1% of the speed of light or experiences a strong gravitational field such as that near the Sun. Quantum mechanics must be used for objects smaller than can be seen with a microscope. The combination of these two theories is relativistic quantum mechanics, and it describes the behavior of small objects traveling at high speeds or experiencing a strong gravitational field. Relativistic quantum mechanics is the best universally applicable theory we have. Because of its mathematical complexity, it is used only when necessary, and the other theories are used whenever they will produce sufficiently accurate results. We will find, however, that we can do a great deal of modern physics with the algebra and trigonometry used in this text. Check Your Understanding A friend tells you he has learned about a new law of nature. What can you know about the information even before your friend describes the law? How would the information be different if your friend told you he had learned about a scientific theory rather than a law

? Solution Without knowing the details of the law, you can still infer that the information your friend has learned conforms to the requirements of all laws of nature: it will be a concise description of the universe around us; a statement of the underlying rules that all natural processes follow. If the information had been a theory, you would be able to infer that the information will be a large-scale, broadly applicable generalization. PhET Explorations: Equation Grapher Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. = ) to see how they add to generate the polynomial curve. Figure 1.15 Equation Grapher (http://cnx.org/content/m54764/1.2/equation-grapher_en.jar) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 15 1.2 Physical Quantities and Units Figure 1.16 The distance from Earth to the Moon may seem immense, but it is just a tiny fraction of the distances from Earth to other celestial bodies. (credit: NASA) By the end of this section, you will be able to: Learning Objectives • Perform unit conversions both in the SI and English units. • Explain the most common prefixes in the SI units and be able to write them in scientific notation. The range of objects and phenomena studied in physics is immense. From the incredibly short lifetime of a nucleus to the age of Earth, from the tiny sizes of sub-nuclear particles to the vast distance to the edges of the known universe, from the force exerted by a jumping flea to the force between Earth and the Sun, there are enough factors of 10 to challenge the imagination of even the most experienced scientist. Giving numerical values for physical quantities and equations for physical principles allows us to understand nature much more deeply than does qualitative description alone. To comprehend these vast ranges, we must also have accepted units in which to express them. And we shall find that (even in the potentially mundane discussion of meters, kilograms, and seconds) a profound simplicity of nature appears—all physical quantities can be expressed as combinations of only four fundamental physical quantities: length, mass, time, and electric current. We define a physical quantity either by specifying how it is measured or by stating how it is calculated from other

measurements. For example, we define distance and time by specifying methods for measuring them, whereas we define average speed by stating that it is calculated as distance traveled divided by time of travel. Measurements of physical quantities are expressed in terms of units, which are standardized values. For example, the length of a race, which is a physical quantity, can be expressed in units of meters (for sprinters) or kilometers (for distance runners). Without standardized units, it would be extremely difficult for scientists to express and compare measured values in a meaningful way. (See Figure 1.17.) Figure 1.17 Distances given in unknown units are maddeningly useless. There are two major systems of units used in the world: SI units (also known as the metric system) and English units (also known as the customary or imperial system). English units were historically used in nations once ruled by the British Empire and are still widely used in the United States. Virtually every other country in the world now uses SI units as the standard; the metric system is also the standard system agreed upon by scientists and mathematicians. The acronym “SI” is derived from the French Système International. 16 Chapter 1 | Introduction: The Nature of Science and Physics SI Units: Fundamental and Derived Units Table 1.1 gives the fundamental SI units that are used throughout this textbook. This text uses non-SI units in a few applications where they are in very common use, such as the measurement of blood pressure in millimeters of mercury (mm Hg). Whenever non-SI units are discussed, they will be tied to SI units through conversions. Table 1.1 Fundamental SI Units Length Mass Time Electric Charge meter (m) kilogram (kg) second (s) coulomb (c) It is an intriguing fact that some physical quantities are more fundamental than others and that the most fundamental physical quantities can be defined only in terms of the procedure used to measure them. The units in which they are measured are thus called fundamental units. In this textbook, the fundamental physical quantities are taken to be length, mass, time, and electric charge. (Note that electric current will not be introduced until much later in this text.) All other physical quantities, such as force and electric current, can be expressed as algebraic combinations of length, mass, time, and current (for example, speed is length divided by time); these units are called derived units. Units of Time, Length, and Mass: The Second, Meter, and Kil

ogram The Second The SI unit for time, the second(abbreviated s), has a long history. For many years it was defined as 1/86,400 of a mean solar day. More recently, a new standard was adopted to gain greater accuracy and to define the second in terms of a non-varying, or constant, physical phenomenon (because the solar day is getting longer due to very gradual slowing of the Earth's rotation). Cesium atoms can be made to vibrate in a very steady way, and these vibrations can be readily observed and counted. In 1967 the second was redefined as the time required for 9,192,631,770 of these vibrations. (See Figure 1.18.) Accuracy in the fundamental units is essential, because all measurements are ultimately expressed in terms of fundamental units and can be no more accurate than are the fundamental units themselves. Figure 1.18 An atomic clock such as this one uses the vibrations of cesium atoms to keep time to a precision of better than a microsecond per year. The fundamental unit of time, the second, is based on such clocks. This image is looking down from the top of an atomic fountain nearly 30 feet tall! (credit: Steve Jurvetson/Flickr) The Meter The SI unit for length is the meter (abbreviated m); its definition has also changed over time to become more accurate and precise. The meter was first defined in 1791 as 1/10,000,000 of the distance from the equator to the North Pole. This measurement was improved in 1889 by redefining the meter to be the distance between two engraved lines on a platinum-iridium bar now kept near Paris. By 1960, it had become possible to define the meter even more accurately in terms of the wavelength of light, so it was again redefined as 1,650,763.73 wavelengths of orange light emitted by krypton atoms. In 1983, the meter was given its present definition (partly for greater accuracy) as the distance light travels in a vacuum in 1/299,792,458 of a second. (See Figure 1.19.) This change defines the speed of light to be exactly 299,792,458 meters per second. The length of the meter will change if the speed of light is someday measured with greater accuracy. The Kilogram The SI unit for mass is the kilogram (abbreviated kg); it is defined to be the mass of a platinum-iridium cylinder kept with

the old meter standard at the International Bureau of Weights and Measures near Paris. Exact replicas of the standard kilogram are also kept at the United States' National Institute of Standards and Technology, or NIST, located in Gaithersburg, Maryland outside of Washington D.C., and at other locations around the world. The determination of all other masses can be ultimately traced to a comparison with the standard mass. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 17 Figure 1.19 The meter is defined to be the distance light travels in 1/299,792,458 of a second in a vacuum. Distance traveled is speed multiplied by time. Electric current and its accompanying unit, the ampere, will be introduced in Introduction to Electric Current, Resistance, and Ohm's Law when electricity and magnetism are covered. The initial modules in this textbook are concerned with mechanics, fluids, heat, and waves. In these subjects all pertinent physical quantities can be expressed in terms of the fundamental units of length, mass, and time. Metric Prefixes SI units are part of the metric system. The metric system is convenient for scientific and engineering calculations because the units are categorized by factors of 10. Table 1.2 gives metric prefixes and symbols used to denote various factors of 10. Metric systems have the advantage that conversions of units involve only powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as the system of U.S. customary units, the relationships are not as simple—there are 12 inches in a foot, 5280 feet in a mile, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by using an appropriate metric prefix. For example, distances in meters are suitable in construction, while distances in kilometers are appropriate for air travel, and the tiny measure of nanometers are convenient in optical design. With the metric system there is no need to invent new units for particular applications. The term order of magnitude refers to the scale of a value expressed in the metric system. Each power of 10 in the metric system represents a different order of magnitude. For example, 101 102 103 magnitude. All quantities that can be expressed as a product of a specific power of 10 are said to be of the same order of magnitude. For example

, the number 800 can be written as 8×102, and the number 450 can be written as 4.5×102. Thus, the numbers 800 and 450 are of the same order of magnitude: 102. Order of magnitude can be thought of as a ballpark estimate for the scale of a value. The diameter of an atom is on the order of 10−9 m, while the diameter of the Sun is on the order of 109 m., and so forth are all different orders of The Quest for Microscopic Standards for Basic Units The fundamental units described in this chapter are those that produce the greatest accuracy and precision in measurement. There is a sense among physicists that, because there is an underlying microscopic substructure to matter, it would be most satisfying to base our standards of measurement on microscopic objects and fundamental physical phenomena such as the speed of light. A microscopic standard has been accomplished for the standard of time, which is based on the oscillations of the cesium atom. The standard for length was once based on the wavelength of light (a small-scale length) emitted by a certain type of atom, but it has been supplanted by the more precise measurement of the speed of light. If it becomes possible to measure the mass of atoms or a particular arrangement of atoms such as a silicon sphere to greater precision than the kilogram standard, it may become possible to base mass measurements on the small scale. There are also possibilities that electrical phenomena on the small scale may someday allow us to base a unit of charge on the charge of electrons and protons, but at present current and charge are related to large-scale currents and forces between wires. 18 Chapter 1 | Introduction: The Nature of Science and Physics Table 1.2 Metric Prefixes for Powers of 10 and their Symbols Value[1] Symbol Prefix Example (some are approximate) exa peta tera giga mega kilo hecto deka — deci centi milli micro nano pico femto atto E P T G M k h da — d c m µ n p f a 1018 1015 1012 109 106 103 102 101 100 (=1) 10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18 exameter Em 1018 m distance light travels in a century petasecond Ps 1015 s 30 million years terawatt TW 1012 W powerful laser output gigahertz GHz 109 Hz a microwave frequency megacurie MCi 106

Ci high radioactivity kilometer km 103 m about 6/10 mile hectoliter hL 102 L 26 gallons dekagram dag 101 g teaspoon of butter deciliter dL 10−1 L less than half a soda centimeter cm 10−2 m fingertip thickness millimeter mm 10−3 m flea at its shoulders micrometer µm 10−6 m detail in microscope nanogram ng 10−9 g small speck of dust picofarad pF 10−12 F small capacitor in radio femtometer fm 10−15 m size of a proton attosecond as 10−18 s time light crosses an atom Known Ranges of Length, Mass, and Time The vastness of the universe and the breadth over which physics applies are illustrated by the wide range of examples of known lengths, masses, and times in Table 1.3. Examination of this table will give you some feeling for the range of possible topics and numerical values. (See Figure 1.20 and Figure 1.21.) Figure 1.20 Tiny phytoplankton swims among crystals of ice in the Antarctic Sea. They range from a few micrometers to as much as 2 millimeters in length. (credit: Prof. Gordon T. Taylor, Stony Brook University; NOAA Corps Collections) 1. See Appendix A for a discussion of powers of 10. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 19 Figure 1.21 Galaxies collide 2.4 billion light years away from Earth. The tremendous range of observable phenomena in nature challenges the imagination. (credit: NASA/CXC/UVic./A. Mahdavi et al. Optical/lensing: CFHT/UVic./H. Hoekstra et al.) Unit Conversion and Dimensional Analysis It is often necessary to convert from one type of unit to another. For example, if you are reading a European cookbook, some quantities may be expressed in units of liters and you need to convert them to cups. Or, perhaps you are reading walking directions from one location to another and you are interested in how many miles you will be walking. In this case, you will need to convert units of feet to miles. Let us consider a simple example of how to convert units. Let us say that we want to convert 80 meters (m) to kilometers (km). The first thing to do

is to list the units that you have and the units that you want to convert to. In this case, we have units in meters and we want to convert to kilometers. Next, we need to determine a conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit are equal to another unit. For example, there are 12 inches in 1 foot, 100 centimeters in 1 meter, 60 seconds in 1 minute, and so on. In this case, we know that there are 1,000 meters in 1 kilometer. Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor so that the units cancel out, as shown: 80m× 1 km 1000m = 0.080 km. (1.1) Note that the unwanted m unit cancels, leaving only the desired km unit. You can use this method to convert between any types of unit. Click Appendix C for a more complete list of conversion factors. 20 Chapter 1 | Introduction: The Nature of Science and Physics Table 1.3 Approximate Values of Length, Mass, and Time Lengths in meters Masses in kilograms (more precise values in parentheses) Times in seconds (more precise values in parentheses) 10−18 Present experimental limit to smallest observable detail 10−30 Mass of an electron 9.11×10−31 kg 10−23 Time for light to cross a proton 10−15 Diameter of a proton 10−27 Mass of a hydrogen atom 1.67×10−27 kg 10−22 Mean life of an extremely unstable nucleus 10−14 Diameter of a uranium nucleus 10−15 Mass of a bacterium 10−10 Diameter of a hydrogen atom 10−5 Mass of a mosquito 10−15 Time for one oscillation of visible light 10−13 Time for one vibration of an atom in a solid 10−8 Thickness of membranes in cells of living organisms 10−2 Mass of a hummingbird 10−8 Time for one oscillation of an FM radio wave Mass of a liter of water (about a quart) 10−3 Duration of a nerve impulse 10−6 Wavelength of visible light 10−3 Size of a grain of sand Height of a 4-year-old child Length of a football field 1 102 103 108 Mass of a person Mass of a car Mass of a large ship Greatest ocean depth 1012 Mass of a large iceberg Diameter of the Earth 1015 Mass of the nucleus of a comet 101

1 Recorded history 1 105 107 109 Time for one heartbeat One day 8.64×104 s One year (y) 3.16×107 s About half the life expectancy of a human 1017 1018 Age of the Earth Age of the universe 1 102 104 107 1011 Distance from the Earth to the Sun 1023 Mass of the Moon 7.35×1022 kg 1016 Distance traveled by light in 1 year (a light year) 1025 Mass of the Earth 5.97×1024 kg 1021 1022 1026 Diameter of the Milky Way galaxy 1030 Mass of the Sun 1.99×1030 kg Distance from the Earth to the nearest large galaxy (Andromeda) Distance from the Earth to the edges of the known universe 1042 Mass of the Milky Way galaxy (current upper limit) 1053 Mass of the known universe (current upper limit) Example 1.1 Unit Conversions: A Short Drive Home Suppose that you drive the 10.0 km from your university to home in 20.0 min. Calculate your average speed (a) in kilometers per hour (km/h) and (b) in meters per second (m/s). (Note: Average speed is distance traveled divided by time of travel.) Strategy First we calculate the average speed using the given units. Then we can get the average speed into the desired units by picking the correct conversion factor and multiplying by it. The correct conversion factor is the one that cancels the unwanted unit and leaves the desired unit in its place. Solution for (a) (1) Calculate average speed. Average speed is distance traveled divided by time of travel. (Take this definition as a given for now—average speed and other motion concepts will be covered in a later module.) In equation form, average speed =distance time. (1.2) (2) Substitute the given values for distance and time. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics average speed = 10.0 km 20.0 min = 0.500 km min. (3) Convert km/min to km/h: multiply by the conversion factor that will cancel minutes and leave hours. That conversion factor is 60 min/hr. Thus, average speed =0.500 km min × 60 min 1 h = 30.0 km h. Discussion for (a) To check your answer, consider the following: 21 (1.

3) (1.4) (1) Be sure that you have properly cancelled the units in the unit conversion. If you have written the unit conversion factor upside down, the units will not cancel properly in the equation. If you accidentally get the ratio upside down, then the units will not cancel; rather, they will give you the wrong units as follows: = 1 60 × 1 hr 60 min km min, km ⋅ hr min2 (1.5) which are obviously not the desired units of km/h. (2) Check that the units of the final answer are the desired units. The problem asked us to solve for average speed in units of km/h and we have indeed obtained these units. (3) Check the significant figures. Because each of the values given in the problem has three significant figures, the answer should also have three significant figures. The answer 30.0 km/hr does indeed have three significant figures, so this is appropriate. Note that the significant figures in the conversion factor are not relevant because an hour is defined to be 60 minutes, so the precision of the conversion factor is perfect. (4) Next, check whether the answer is reasonable. Let us consider some information from the problem—if you travel 10 km in a third of an hour (20 min), you would travel three times that far in an hour. The answer does seem reasonable. Solution for (b) There are several ways to convert the average speed into meters per second. (1) Start with the answer to (a) and convert km/h to m/s. Two conversion factors are needed—one to convert hours to seconds, and another to convert kilometers to meters. (2) Multiplying by these yields Average speed = 30.0km h 3,600 s Average speed = 8.33m s. × 1 h × 1,000 m 1 km, (1.6) (1.7) Discussion for (b) If we had started with 0.500 km/min, we would have needed different conversion factors, but the answer would have been the same: 8.33 m/s. You may have noted that the answers in the worked example just covered were given to three digits. Why? When do you need to be concerned about the number of digits in something you calculate? Why not write down all the digits your calculator produces? The module Accuracy, Precision, and Significant Figures will help you answer these questions. Nonstandard Units While there are numerous types of units

that we are all familiar with, there are others that are much more obscure. For example, a firkin is a unit of volume that was once used to measure beer. One firkin equals about 34 liters. To learn more about nonstandard units, use a dictionary or encyclopedia to research different “weights and measures.” Take note of any unusual units, such as a barleycorn, that are not listed in the text. Think about how the unit is defined and state its relationship to SI units. Check Your Understanding Some hummingbirds beat their wings more than 50 times per second. A scientist is measuring the time it takes for a hummingbird to beat its wings once. Which fundamental unit should the scientist use to describe the measurement? Which factor of 10 is the scientist likely to use to describe the motion precisely? Identify the metric prefix that corresponds to this factor of 10. Solution 22 Chapter 1 | Introduction: The Nature of Science and Physics The scientist will measure the time between each movement using the fundamental unit of seconds. Because the wings beat so fast, the scientist will probably need to measure in milliseconds, or 10−3 20 milliseconds per beat.) seconds. (50 beats per second corresponds to Check Your Understanding One cubic centimeter is equal to one milliliter. What does this tell you about the different units in the SI metric system? Solution The fundamental unit of length (meter) is probably used to create the derived unit of volume (liter). The measure of a milliliter is dependent on the measure of a centimeter. 1.3 Accuracy, Precision, and Significant Figures Figure 1.22 A double-pan mechanical balance is used to compare different masses. Usually an object with unknown mass is placed in one pan and objects of known mass are placed in the other pan. When the bar that connects the two pans is horizontal, then the masses in both pans are equal. The “known masses” are typically metal cylinders of standard mass such as 1 gram, 10 grams, and 100 grams. (credit: Serge Melki) Figure 1.23 Many mechanical balances, such as double-pan balances, have been replaced by digital scales, which can typically measure the mass of an object more precisely. Whereas a mechanical balance may only read the mass of an object to the nearest tenth of a gram, many digital scales can measure the mass of an object up to the nearest thousandth of a gram. (credit: Karel Jakubec) By the end of this section, you will be able

to: Learning Objectives • Determine the appropriate number of significant figures in both addition and subtraction, as well as multiplication and division calculations. • Calculate the percent uncertainty of a measurement. Accuracy and Precision of a Measurement Science is based on observation and experiment—that is, on measurements. Accuracy is how close a measurement is to the correct value for that measurement. For example, let us say that you are measuring the length of standard computer paper. The This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 23 packaging in which you purchased the paper states that it is 11.0 inches long. You measure the length of the paper three times and obtain the following measurements: 11.1 in., 11.2 in., and 10.9 in. These measurements are quite accurate because they are very close to the correct value of 11.0 inches. In contrast, if you had obtained a measurement of 12 inches, your measurement would not be very accurate. The precision of a measurement system is refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). Consider the example of the paper measurements. The precision of the measurements refers to the spread of the measured values. One way to analyze the precision of the measurements would be to determine the range, or difference, between the lowest and the highest measured values. In that case, the lowest value was 10.9 in. and the highest value was 11.2 in. Thus, the measured values deviated from each other by at most 0.3 in. These measurements were relatively precise because they did not vary too much in value. However, if the measured values had been 10.9, 11.1, and 11.9, then the measurements would not be very precise because there would be significant variation from one measurement to another. The measurements in the paper example are both accurate and precise, but in some cases, measurements are accurate but not precise, or they are precise but not accurate. Let us consider an example of a GPS system that is attempting to locate the position of a restaurant in a city. Think of the restaurant location as existing at the center of a bull's-eye target, and think of each GPS attempt to locate the restaurant as a black dot. In Figure 1.24, you can see that the GPS measurements are spread out far apart from each other, but they are all relatively close to the actual location of

the restaurant at the center of the target. This indicates a low precision, high accuracy measuring system. However, in Figure 1.25, the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high precision, low accuracy measuring system. Figure 1.24 A GPS system attempts to locate a restaurant at the center of the bull's-eye. The black dots represent each attempt to pinpoint the location of the restaurant. The dots are spread out quite far apart from one another, indicating low precision, but they are each rather close to the actual location of the restaurant, indicating high accuracy. (credit: Dark Evil) Figure 1.25 In this figure, the dots are concentrated rather closely to one another, indicating high precision, but they are rather far away from the actual location of the restaurant, indicating low accuracy. (credit: Dark Evil) Accuracy, Precision, and Uncertainty The degree of accuracy and precision of a measuring system are related to the uncertainty in the measurements. Uncertainty is a quantitative measure of how much your measured values deviate from a standard or expected value. If your measurements are not very accurate or precise, then the uncertainty of your values will be very high. In more general terms, uncertainty can be thought of as a disclaimer for your measured values. For example, if someone asked you to provide the mileage on your car, you might say that it is 45,000 miles, plus or minus 500 miles. The plus or minus amount is the uncertainty in your value. That is, you are indicating that the actual mileage of your car might be as low as 44,500 miles or as high as 45,500 miles, or anywhere in between. All measurements contain some amount of uncertainty. In our example of measuring the length of the paper, we might say that the length of the paper is 11 in., plus or minus 0.2 in. The uncertainty in a measurement,, is often denoted as (“delta ”), so the measurement result would be recorded as ±. In our paper example, the length of the paper could be expressed as 11 in. ± 0.2. The factors contributing to uncertainty in a measurement include: 24 Chapter 1 | Introduction: The Nature of Science and Physics 1. Limitations of the measuring device, 2. The skill of the person making the measurement, 3. Irregularities in the object being measured, 4. Any other factors that affect the outcome (highly dependent on the situation).

In our example, such factors contributing to the uncertainty could be the following: the smallest division on the ruler is 0.1 in., the person using the ruler has bad eyesight, or one side of the paper is slightly longer than the other. At any rate, the uncertainty in a measurement must be based on a careful consideration of all the factors that might contribute and their possible effects. Making Connections: Real-World Connections—Fevers or Chills? Uncertainty is a critical piece of information, both in physics and in many other real-world applications. Imagine you are caring for a sick child. You suspect the child has a fever, so you check his or her temperature with a thermometer. What if the uncertainty of the thermometer were 3.0ºC? If the child's temperature reading was 37.0ºC (which is normal body temperature), the “true” temperature could be anywhere from a hypothermic 34.0ºC to a dangerously high 40.0ºC. A thermometer with an uncertainty of 3.0ºC would be useless. Percent Uncertainty One method of expressing uncertainty is as a percent of the measured value. If a measurement is expressed with uncertainty,, the percent uncertainty (%unc) is defined to be % unc = ×100%. (1.8) Example 1.2 Calculating Percent Uncertainty: A Bag of Apples A grocery store sells 5 lb bags of apples. You purchase four bags over the course of a month and weigh the apples each time. You obtain the following measurements: • Week 1 weight: 4.8 lb • Week 2 weight: 5.3 lb • Week 3 weight: 4.9 lb • Week 4 weight: 5.4 lb You determine that the weight of the 5 lb bag has an uncertainty of ±0.4 lb. What is the percent uncertainty of the bag's weight? Strategy First, observe that the expected value of the bag's weight,, is 5 lb. The uncertainty in this value,, is 0.4 lb. We can use the following equation to determine the percent uncertainty of the weight: Solution Plug the known values into the equation: Discussion % unc = ×100%. % unc =0.4 lb 5 lb ×100% = 8%. (1.9) (1.10) We can conclude that the weight of the apple bag is 5 lb ± 8%. Consider how this percent uncertainty would change if the bag of apples were half as heavy,

but the uncertainty in the weight remained the same. Hint for future calculations: when calculating percent uncertainty, always remember that you must multiply the fraction by 100%. If you do not do this, you will have a decimal quantity, not a percent value. Uncertainties in Calculations There is an uncertainty in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties (a few percent or less), then the method of adding percents can be used for multiplication or division. This method says that the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 25 uncertainties in the items used to make the calculation. For example, if a floor has a length of 4.00 m and a width of 3.00 m, with uncertainties of 2% and 1%, respectively, then the area of the floor is 12.0 m2 and has an uncertainty of 3%. (Expressed as an area this is 0.36 m2, which we round to 0.4 m2 since the area of the floor is given to a tenth of a square meter.) Check Your Understanding A high school track coach has just purchased a new stopwatch. The stopwatch manual states that the stopwatch has an uncertainty of ±0.05 s. Runners on the track coach's team regularly clock 100 m sprints of 11.49 s to 15.01 s. At the school's last track meet, the first-place sprinter came in at 12.04 s and the second-place sprinter came in at 12.07 s. Will the coach's new stopwatch be helpful in timing the sprint team? Why or why not? Solution No, the uncertainty in the stopwatch is too great to effectively differentiate between the sprint times. Precision of Measuring Tools and Significant Figures An important factor in the accuracy and precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter, while a caliper can measure length to the nearest 0.01 mill

imeter. The caliper is a more precise measuring tool because it can measure extremely small differences in length. The more precise the measuring tool, the more precise and accurate the measurements can be. When we express measured values, we can only list as many digits as we initially measured with our measuring tool. For example, if you use a standard ruler to measure the length of a stick, you may measure it to be 36.7 cm. You could not express this value as 36.71 cm because your measuring tool was not precise enough to measure a hundredth of a centimeter. It should be noted that the last digit in a measured value has been estimated in some way by the person performing the measurement. For example, the person measuring the length of a stick with a ruler notices that the stick length seems to be somewhere in between 36.6 cm and 36.7 cm, and he or she must estimate the value of the last digit. Using the method of significant figures, the rule is that the last digit written down in a measurement is the first digit with some uncertainty. In order to determine the number of significant digits in a value, start with the first measured value at the left and count the number of digits through the last digit written on the right. For example, the measured value 36.7 cm has three digits, or significant figures. Significant figures indicate the precision of a measuring tool that was used to measure a value. Zeros Special consideration is given to zeros when counting significant figures. The zeros in 0.053 are not significant, because they are only placekeepers that locate the decimal point. There are two significant figures in 0.053. The zeros in 10.053 are not placekeepers but are significant—this number has five significant figures. The zeros in 1300 may or may not be significant depending on the style of writing numbers. They could mean the number is known to the last digit, or they could be placekeepers. So 1300 could have two, three, or four significant figures. (To avoid this ambiguity, write 1300 in scientific notation.) Zeros are significant except when they serve only as placekeepers. Check Your Understanding Determine the number of significant figures in the following measurements: a. 0.0009 b. 15,450.0 c. 6×103 d. 87.990 e. 30.42 Solution (a) 1; the zeros in this number are placekeepers that indicate the decimal point (b) 6; here, the zeros indicate

that a measurement was made to the 0.1 decimal point, so the zeros are significant (c) 1; the value 103 signifies the decimal place, not the number of measured values (d) 5; the final zero indicates that a measurement was made to the 0.001 decimal point, so it is significant (e) 4; any zeros located in between significant figures in a number are also significant 26 Chapter 1 | Introduction: The Nature of Science and Physics Significant Figures in Calculations When combining measurements with different degrees of accuracy and precision, the number of significant digits in the final answer can be no greater than the number of significant digits in the least precise measured value. There are two different rules, one for multiplication and division and the other for addition and subtraction, as discussed below. 1. For multiplication and division: The result should have the same number of significant figures as the quantity having the least significant figures entering into the calculation. For example, the area of a circle can be calculated from its radius using = 2. Let us see how many significant figures the area has if the radius has only two—say, = 1.2 m. Then, = 2 = (3.1415927...)×(1.2 m)2 = 4.5238934 m2 (1.11) is what you would get using a calculator that has an eight-digit output. But because the radius has only two significant figures, it limits the calculated quantity to two significant figures or =4.5 m2, (1.12) even though is good to at least eight digits. 2. For addition and subtraction: The answer can contain no more decimal places than the least precise measurement. Suppose that you buy 7.56 kg of potatoes in a grocery store as measured with a scale with precision 0.01 kg. Then you drop off 6.052 kg of potatoes at your laboratory as measured by a scale with precision 0.001 kg. Finally, you go home and add 13.7 kg of potatoes as measured by a bathroom scale with precision 0.1 kg. How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer? The mass is found by simple addition and subtraction: 7.56 kg - 6.052 kg kg +13.7 15.208 kg = 15.2 kg. (1.13) Next, we identify the least precise measurement: 13.7 kg. This measurement is expressed to the 0.1 decimal place,

so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer is rounded to the tenths place, giving us 15.2 kg. Significant Figures in this Text In this text, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits, for example. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, particularly in optics, more accurate numbers are needed and more than three significant figures will be used. Finally, if a number is exact, such as the two in the formula for the circumference of a circle, = 2π, it does not affect the number of significant figures in a calculation. Check Your Understanding Perform the following calculations and express your answer using the correct number of significant digits. (a) A woman has two bags weighing 13.5 pounds and one bag with a weight of 10.2 pounds. What is the total weight of the bags? (b) The force on an object is equal to its mass multiplied by its acceleration. If a wagon with mass 55 kg accelerates at a rate of 0.0255 m/s2, what is the force on the wagon? (The unit of force is called the newton, and it is expressed with the symbol N.) Solution (a) 37.2 pounds; Because the number of bags is an exact value, it is not considered in the significant figures. (b) 1.4 N; Because the value 55 kg has only two significant figures, the final value must also contain two significant figures. PhET Explorations: Estimation Explore size estimation in one, two, and three dimensions! Multiple levels of difficulty allow for progressive skill improvement. Figure 1.26 Estimation (http://cnx.org/content/m54766/1.7/estimation_en.jar) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 27 1.4 Approximation Learning Objectives By the end of this section, you will be able to: • Make reasonable approximations based on given data. On many occasions, physicists, other scientists, and engineers need to make approx

imations or “guesstimates” for a particular quantity. What is the distance to a certain destination? What is the approximate density of a given item? About how large a current will there be in a circuit? Many approximate numbers are based on formulae in which the input quantities are known only to a limited accuracy. As you develop problem-solving skills (that can be applied to a variety of fields through a study of physics), you will also develop skills at approximating. You will develop these skills through thinking more quantitatively, and by being willing to take risks. As with any endeavor, experience helps, as well as familiarity with units. These approximations allow us to rule out certain scenarios or unrealistic numbers. Approximations also allow us to challenge others and guide us in our approaches to our scientific world. Let us do two examples to illustrate this concept. Example 1.3 Approximate the Height of a Building Can you approximate the height of one of the buildings on your campus, or in your neighborhood? Let us make an approximation based upon the height of a person. In this example, we will calculate the height of a 39-story building. Strategy Think about the average height of an adult male. We can approximate the height of the building by scaling up from the height of a person. Solution Based on information in the example, we know there are 39 stories in the building. If we use the fact that the height of one story is approximately equal to about the length of two adult humans (each human is about 2 m tall), then we can estimate the total height of the building to be 2 m 1 person × 2 person 1 story ×39 stories = 156 m. (1.14) Discussion You can use known quantities to determine an approximate measurement of unknown quantities. If your hand measures 10 cm across, how many hand lengths equal the width of your desk? What other measurements can you approximate besides length? Example 1.4 Approximating Vast Numbers: a Trillion Dollars Figure 1.27 A bank stack contains one-hundred $100 bills, and is worth $10,000. How many bank stacks make up a trillion dollars? (credit: Andrew Magill) The U.S. federal deficit in the 2008 fiscal year was a little greater than $10 trillion. Most of us do not have any concept of how much even one trillion actually is. Suppose that you were given a trillion dollars in $100 bills. If you made 100-bill stacks and

used them to evenly cover a football field (between the end zones), make an approximation of how high the money pile 28 Chapter 1 | Introduction: The Nature of Science and Physics would become. (We will use feet/inches rather than meters here because football fields are measured in yards.) One of your friends says 3 in., while another says 10 ft. What do you think? Strategy When you imagine the situation, you probably envision thousands of small stacks of 100 wrapped $100 bills, such as you might see in movies or at a bank. Since this is an easy-to-approximate quantity, let us start there. We can find the volume of a stack of 100 bills, find out how many stacks make up one trillion dollars, and then set this volume equal to the area of the football field multiplied by the unknown height. Solution (1) Calculate the volume of a stack of 100 bills. The dimensions of a single bill are approximately 3 in. by 6 in. A stack of 100 of these is about 0.5 in. thick. So the total volume of a stack of 100 bills is: volume of stack = length×width×height, volume of stack = 6 in.×3 in.×0.5 in. volume of stack = 9 in.3. (1.15) (2) Calculate the number of stacks. Note that a trillion dollars is equal to $1×1012, and a stack of one-hundred $100 bills is equal to $10000 or $1×104. The number of stacks you will have is: $1×1012(a trillion dollars)/ $1×104 per stack = 1×108 stacks. (3) Calculate the area of a football field in square inches. The area of a football field is 100 yd×50 yd, which gives 5,000 yd2. Because we are working in inches, we need to convert square yards to square inches: Area = 5,000 yd2× 3 ft 1 yd × 3 ft 1 yd × 12 in. 1 ft × 12 in. 1 ft = 6,480000 in.2 Area ≈ 6×106 in.2. (1.16) (1.17) This conversion gives us 6×106 in.2 for the area of the field. (Note that we are using only one significant figure in these calculations.) (4) Calculate the total volume of the bills. The volume of all the $100 -

bill stacks is 9 in.3 / stack×108 stacks = 9×108 in.3. (5) Calculate the height. To determine the height of the bills, use the equation: = area of field×height of money: volume of bills Height of money = volume of bills area of field Height of money = 9×108in.3 6×106in.2 = 1.33×102 in., Height of money ≈ 1×102 in. = 100 in. The height of the money will be about 100 in. high. Converting this value to feet gives 100 in.× 1 ft 12 in. = 8.33 ft ≈ 8 ft. Discussion (1.18) (1.19) The final approximate value is much higher than the early estimate of 3 in., but the other early estimate of 10 ft (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise tell you in terms of rough “guesstimates” versus carefully calculated approximations? Check Your Understanding Using mental math and your understanding of fundamental units, approximate the area of a regulation basketball court. Describe the process you used to arrive at your final approximation. Solution An average male is about two meters tall. It would take approximately 15 men laid out end to end to cover the length, and about 7 to cover the width. That gives an approximate area of 420 m2. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 29 Glossary accuracy: the degree to which a measured value agrees with correct value for that measurement approximation: an estimated value based on prior experience and reasoning classical physics: physics that was developed from the Renaissance to the end of the 19th century conversion factor: a ratio expressing how many of one unit are equal to another unit derived units: units that can be calculated using algebraic combinations of the fundamental units English units: system of measurement used in the United States; includes units of measurement such as feet, gallons, and pounds fundamental units: units that can only be expressed relative to the procedure used to measure them kilogram: the SI unit for mass, abbreviated (kg) law: a description, using concise language or a mathematical formula, a generalized pattern in nature that is supported by scientific evidence and repeated experiments meter: the SI unit for length, abbreviated (m) method of adding percents

: the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation metric system: a system in which values can be calculated in factors of 10 model: representation of something that is often too difficult (or impossible) to display directly modern physics: the study of relativity, quantum mechanics, or both order of magnitude: refers to the size of a quantity as it relates to a power of 10 percent uncertainty: the ratio of the uncertainty of a measurement to the measured value, expressed as a percentage physical quantity : a characteristic or property of an object that can be measured or calculated from other measurements physics: the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon precision: the degree to which repeated measurements agree with each other quantum mechanics: the study of objects smaller than can be seen with a microscope relativity: the study of objects moving at speeds greater than about 1% of the speed of light, or of objects being affected by a strong gravitational field scientific method: a method that typically begins with an observation and question that the scientist will research; next, the scientist typically performs some research about the topic and then devises a hypothesis; then, the scientist will test the hypothesis by performing an experiment; finally, the scientist analyzes the results of the experiment and draws a conclusion second: the SI unit for time, abbreviated (s) SI units : the international system of units that scientists in most countries have agreed to use; includes units such as meters, liters, and grams significant figures: express the precision of a measuring tool used to measure a value theory: an explanation for patterns in nature that is supported by scientific evidence and verified multiple times by various groups of researchers uncertainty: a quantitative measure of how much your measured values deviate from a standard or expected value units : a standard used for expressing and comparing measurements Section Summary 1.1 Physics: An Introduction • Science seeks to discover and describe the underlying order and simplicity in nature. 30 Chapter 1 | Introduction: The Nature of Science and Physics • Physics is the most basic of the sciences, concerning itself with energy, matter, space and time, and their interactions. • Scientific laws and theories express the general truths of nature and the body of knowledge they encompass. These laws of nature are rules that all natural processes appear to follow. 1.2 Physical Quantities and Units • Physical quantities are a characteristic or property of an object that can be measured or calculated

from other measurements. • Units are standards for expressing and comparing the measurement of physical quantities. All units can be expressed as combinations of four fundamental units. • The four fundamental units we will use in this text are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature. • The four fundamental units are abbreviated as follows: meter, m; kilogram, kg; second, s; and ampere, A. The metric system also uses a standard set of prefixes to denote each order of magnitude greater than or lesser than the fundamental unit itself. • Unit conversions involve changing a value expressed in one type of unit to another type of unit. This is done by using conversion factors, which are ratios relating equal quantities of different units. 1.3 Accuracy, Precision, and Significant Figures • Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value. • Precision of measured values refers to how close the agreement is between repeated measurements. • The precision of a measuring tool is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool. • Significant figures express the precision of a measuring tool. • When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value. • When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value. 1.4 Approximation Scientists often approximate the values of quantities to perform calculations and analyze systems. Conceptual Questions 1.1 Physics: An Introduction 1. Models are particularly useful in relativity and quantum mechanics, where conditions are outside those normally encountered by humans. What is a model? 2. How does a model differ from a theory? 3. If two different theories describe experimental observations equally well, can one be said to be more valid than the other (assuming both use accepted rules of logic)? 4. What determines the validity of a theory? 5. Certain criteria must be satisfied if a measurement or observation is to be believed. Will the criteria necessarily be as strict for an expected result as for an unexpected result? 6. Can the validity of a model be limited, or must it be universally valid? How does this

compare to the required validity of a theory or a law? 7. Classical physics is a good approximation to modern physics under certain circumstances. What are they? 8. When is it necessary to use relativistic quantum mechanics? 9. Can classical physics be used to accurately describe a satellite moving at a speed of 7500 m/s? Explain why or why not. 1.2 Physical Quantities and Units 10. Identify some advantages of metric units. 1.3 Accuracy, Precision, and Significant Figures 11. What is the relationship between the accuracy and uncertainty of a measurement? 12. Prescriptions for vision correction are given in units called diopters (D). Determine the meaning of that unit. Obtain information (perhaps by calling an optometrist or performing an internet search) on the minimum uncertainty with which corrections in diopters are determined and the accuracy with which corrective lenses can be produced. Discuss the sources of uncertainties in both the prescription and accuracy in the manufacture of lenses. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 1 | Introduction: The Nature of Science and Physics 31 Problems & Exercises 1.2 Physical Quantities and Units 1. The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this? 2. A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit? 3. Show that 1.0 m/s = 3.6 km/h. Hint: Show the explicit steps involved in converting 1.0 m/s = 3.6 km/h. 4. American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1 meter equals 3.281 feet.) 5. Soccer fields vary in size. A large soccer field is 115 m long and 85 m wide. What are its dimensions in feet and inches? (Assume that 1 meter equals 3.281 feet.) 6. What is the height in meters of a person who is 6 ft 1.0 in. tall? (Assume that 1 meter equals 39.37 in.) 7. Mount Everest, at 29,028 feet, is the tallest mountain on the Earth. What is its height in kilometers

? (Assume that 1 kilometer equals 3,281 feet.) 8. The speed of sound is measured to be 342 m/s on a certain day. What is this in km/h? 9. Tectonic plates are large segments of the Earth's crust that move slowly. Suppose that one such plate has an average speed of 4.0 cm/year. (a) What distance does it move in 1 s at this speed? (b) What is its speed in kilometers per million years? 10. (a) Refer to Table 1.3 to determine the average distance between the Earth and the Sun. Then calculate the average speed of the Earth in its orbit in kilometers per second. (b) What is this in meters per second? 1.3 Accuracy, Precision, and Significant Figures Express your answers to problems in this section to the correct number of significant figures and proper units. 11. Suppose that your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (in kilograms)? 12. A good-quality measuring tape can be off by 0.50 cm over a distance of 20 m. What is its percent uncertainty? 13. (a) A car speedometer has a 5.0% uncertainty. What is the range of possible speeds when it reads 90 km/h? (b) Convert this range to miles per hour. (1 km = 0.6214 mi) 14. An infant's pulse rate is measured to be 130 ± 5 beats/ min. What is the percent uncertainty in this measurement? 15. (a) Suppose that a person has an average heart rate of 72.0 beats/min. How many beats does he or she have in 2.0 y? (b) In 2.00 y? (c) In 2.000 y? (106.7)(98.2) / (46.210)(1.01) (b) (18.7)2 (c) 1.60×10−19 (3712). 18. (a) How many significant figures are in the numbers 99 and 100? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers, significant figures or percent uncertainties? 19. (a) If your speedometer has an uncertainty of 2.0 km/h at a speed of 90 km/h, what is the

percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/h, what is the range of speeds you could be going? 20. (a) A person's blood pressure is measured to be 120 ± 2 mm Hg. What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm Hg? 21. A person measures his or her heart rate by counting the number of beats in 30 s. If 40 ± 1 beats are counted in 30.0 ± 0.5 s, what is the heart rate and its uncertainty in beats per minute? 22. What is the area of a circle 3.102 cm in diameter? 23. If a marathon runner averages 9.5 mi/h, how long does it take him or her to run a 26.22 mi marathon? 24. A marathon runner completes a 42.188 km course in 2 h, 30 min, and 12 s. There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the uncertainty in the elapsed time. (c) What is the average speed in meters per second? (d) What is the uncertainty in the average speed? 25. The sides of a small rectangular box are measured to be 1.80 ± 0.01 cm, 2.05 ± 0.02 cm, and 3.1 ± 0.1 cm long. Calculate its volume and uncertainty in cubic centimeters. 26. When non-metric units were used in the United Kingdom, a unit of mass called the pound-mass (lbm) was employed, where 1 lbm = 0.4539 kg. (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms? 27. The length and width of a rectangular room are measured to be 3.955 ± 0.005 m and 3.050 ± 0.005 m. Calculate the area of the room and its uncertainty in square meters. 28. A car engine moves a piston with a circular cross section of 7.500 ± 0.002 cm diameter a distance of 3.250 ± 0.001 cm to compress the gas in the cylinder. (a

) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume. 16. A can contains 375 mL of soda. How much is left after 308 mL is removed? 1.4 Approximation 17. State how many significant figures are proper in the results of the following calculations: (a) 29. How many heartbeats are there in a lifetime? 32 Chapter 1 | Introduction: The Nature of Science and Physics 30. A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD? 31. How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10−22 s.) 32. Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of 10−27 kg and the mass of a bacterium is on the order of 10−15 kg. ) Figure 1.28 This color-enhanced photo shows Salmonella typhimurium (red) attacking human cells. These bacteria are commonly known for causing foodborne illness. Can you estimate the number of atoms in each bacterium? (credit: Rocky Mountain Laboratories, NIAID, NIH) 33. Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom? 34. (a) What fraction of Earth's diameter is the greatest ocean depth? (b) The greatest mountain height? 35. (a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human? 36. Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 33 2 KINEMATICS Figure 2.1 The motion of an American kestrel through the air can be described by the bird's displacement, speed, velocity, and acceleration. When it flies in a straight line without any change in direction, its motion is said to

be one dimensional. (credit: Vince Maidens, Wikimedia Commons) Chapter Outline 2.1. Displacement 2.2. Vectors, Scalars, and Coordinate Systems 2.3. Time, Velocity, and Speed 2.4. Acceleration 2.5. Motion Equations for Constant Acceleration in One Dimension 2.6. Problem-Solving Basics for One Dimensional Kinematics 2.7. Falling Objects 2.8. Graphical Analysis of One Dimensional Motion Connection for AP® Courses Objects are in motion everywhere we look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. Even in inanimate objects, there is a continuous motion in the vibrations of atoms and molecules. Questions about motion are interesting in and of themselves: How long will it take for a space probe to get to Mars? Where will a football land if it is thrown at a certain angle? Understanding motion will not only provide answers to these questions, but will be key to understanding more advanced concepts in physics. For example, the discussion of force in Chapter 4 will not fully make sense until you understand acceleration. This relationship between force and acceleration is also critical to understanding Big Idea 3. Additionally, this unit will explore the topic of reference frames, a critical component to quantifying how things move. If you have ever waved to a departing friend at a train station, you are likely familiar with this idea. While you see your friend move away from you at a considerable rate, those sitting with her will likely see her as not moving. The effect that the chosen reference frame has on your observations is substantial, and an understanding of this is needed to grasp both Enduring Understanding 3.A and Essential Knowledge 3.A.1. Our formal study of physics begins with kinematics, which is defined as the study of motion without considering its causes. In one- and two-dimensional kinematics we will study only the motion of a football, for example, without worrying about what forces cause or change its motion. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one-dimensional motion. Later, in two-dimensional kinematics, we apply concepts developed here to study motion along curved paths (two- and three-dimensional motion), for example, that of a car rounding a curve. The content in this chapter supports: Big Idea

3 The interactions of an object with other objects can be described by forces. 34 Chapter 2 | Kinematics Enduring Understanding 3.A All forces share certain common characteristics when considered by observers in inertial reference frames. Essential Knowledge 3.A.1 An observer in a particular reference frame can describe the motion of an object using such quantities as position, displacement, distance, velocity, speed, and acceleration. 2.1 Displacement Figure 2.2 These cyclists in Vietnam can be described by their position relative to buildings and a canal. Their motion can be described by their change in position, or displacement, in the frame of reference. (credit: Suzan Black, Fotopedia) Learning Objectives By the end of this section, you will be able to: • Define position, displacement, distance, and distance traveled in a particular frame of reference. • Explain the relationship between position and displacement. • Distinguish between displacement and distance traveled. • Calculate displacement and distance given initial position, final position, and the path between the two. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) Position In order to describe the motion of an object, you must first be able to describe its position—where it is at any particular time. More precisely, you need to specify its position relative to a convenient reference frame. Earth is often used as a reference frame, and we often describe the position of an object as it relates to stationary objects in that reference frame. For example, a rocket launch would be described in terms of the position of the rocket with respect to the Earth as a whole, while a professor's position could be described in terms of where she is in relation to the nearby white board. (See Figure 2.3.) In other cases, we use reference frames that are not stationary but are in motion relative to the Earth. To describe the position of a person in an airplane, for example, we use the airplane, not the Earth, as the reference frame. (See Figure 2.4

.) Displacement If an object moves relative to a reference frame (for example, if a professor moves to the right relative to a white board or a passenger moves toward the rear of an airplane), then the object's position changes. This change in position is known as displacement. The word “displacement” implies that an object has moved, or has been displaced. Displacement Displacement is the change in position of an object: where Δ is displacement, f is the final position, and 0 is the initial position. Δ = f − 0, (2.1) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 35 In this text the upper case Greek letter Δ (delta) always means “change in” whatever quantity follows it; thus, Δ means change in position. Always solve for displacement by subtracting initial position 0 from final position f. Note that the SI unit for displacement is the meter (m) (see Physical Quantities and Units), but sometimes kilometers, miles, feet, and other units of length are used. Keep in mind that when units other than the meter are used in a problem, you may need to convert them into meters to complete the calculation. Figure 2.3 A professor paces left and right while lecturing. Her position relative to the blackboard is given by. The +2.0 m displacement of the professor relative to the blackboard is represented by an arrow pointing to the right. Figure 2.4 A passenger moves from his seat to the back of the plane. His location relative to the airplane is given by. The −4 m displacement of the passenger relative to the plane is represented by an arrow toward the rear of the plane. Notice that the arrow representing his displacement is twice as long as the arrow representing the displacement of the professor (he moves twice as far) in Figure 2.3. Note that displacement has a direction as well as a magnitude. The professor's displacement is 2.0 m to the right, and the airline passenger's displacement is 4.0 m toward the rear. In one-dimensional motion, direction can be specified with a plus or minus sign. When you begin a problem, you should select which direction is positive (usually that will be to the right or up, but you are free to select positive as being any direction). The professor's initial position is 0 = 1.5 m and her final position is

f = 3.5 m. Thus her displacement is Δ = f −0 = 3.5 m − 1.5 m = + 2.0 m. (2.2) In this coordinate system, motion to the right is positive, whereas motion to the left is negative. Similarly, the airplane passenger's initial position is 0 = 6.0 m and his final position is f = 2.0 m, so his displacement is 36 Chapter 2 | Kinematics His displacement is negative because his motion is toward the rear of the plane, or in the negative direction in our coordinate system. Δ = f −0 = 2.0 m − 6.0 m = −4.0 m. (2.3) Distance Although displacement is described in terms of direction, distance is not. Distance is defined to be the magnitude or size of displacement between two positions. Note that the distance between two positions is not the same as the distance traveled between them. Distance traveled is the total length of the path traveled between two positions. Distance has no direction and, thus, no sign. For example, the distance the professor walks is 2.0 m. The distance the airplane passenger walks is 4.0 m. Misconception Alert: Distance Traveled vs. Magnitude of Displacement It is important to note that the distance traveled, however, can be greater than the magnitude of the displacement (by magnitude, we mean just the size of the displacement without regard to its direction; that is, just a number with a unit). For example, the professor could pace back and forth many times, perhaps walking a distance of 150 m during a lecture, yet still end up only 2.0 m to the right of her starting point. In this case her displacement would be +2.0 m, the magnitude of her displacement would be 2.0 m, but the distance she traveled would be 150 m. In kinematics we nearly always deal with displacement and magnitude of displacement, and almost never with distance traveled. One way to think about this is to assume you marked the start of the motion and the end of the motion. The displacement is simply the difference in the position of the two marks and is independent of the path taken in traveling between the two marks. The distance traveled, however, is the total length of the path taken between the two marks. Check Your Understanding A cyclist rides 3 km west and then turns around and rides 2 km east. (a) What is her displacement? (b) What distance does

she ride? (c) What is the magnitude of her displacement? Solution Figure 2.5 (a) The rider's displacement is Δ = f − 0 = −1 km. (The displacement is negative because we take east to be positive and west to be negative.) (b) The distance traveled is 3 km + 2 km = 5 km. (c) The magnitude of the displacement is 1 km. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 37 2.2 Vectors, Scalars, and Coordinate Systems Figure 2.6 The motion of this Eclipse Concept jet can be described in terms of the distance it has traveled (a scalar quantity) or its displacement in a specific direction (a vector quantity). In order to specify the direction of motion, its displacement must be described based on a coordinate system. In this case, it may be convenient to choose motion toward the left as positive motion (it is the forward direction for the plane), although in many cases, the -coordinate runs from left to right, with motion to the right as positive and motion to the left as negative. (credit: Armchair Aviator, Flickr) Learning Objectives By the end of this section, you will be able to: • Define and distinguish between scalar and vector quantities. • Assign a coordinate system for a scenario involving one-dimensional motion. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.2 The student is able to design an experimental investigation of the motion of an object. What is the difference between distance and displacement? Whereas displacement is defined by both direction and magnitude, distance is defined only by magnitude. Displacement is an example of a vector quantity. Distance is an example of a scalar quantity. A vector is any quantity with both magnitude and direction. Other examples of vectors include a velocity of 90 km/h east and a force of 500 newtons straight down. The direction of a vector in one-dimensional motion is given simply by a plus ( + ) or minus ( − ) sign. Vectors are represented graphically by arrows. An arrow used to represent a vector has a length proportional to the vector's magnitude (e.g., the larger the magnitude, the longer the length of the vector) and points in the same direction as the vector. Some physical quantities, like distance, either have no

direction or none is specified. A scalar is any quantity that has a magnitude, but no direction. For example, a 20ºC temperature, the 250 kilocalories (250 Calories) of energy in a candy bar, a 90 km/h speed limit, a person's 1.8 m height, and a distance of 2.0 m are all scalars—quantities with no specified direction. Note, however, that a scalar can be negative, such as a −20ºC temperature. In this case, the minus sign indicates a point on a scale rather than a direction. Scalars are never represented by arrows. Coordinate Systems for One-Dimensional Motion In order to describe the direction of a vector quantity, you must designate a coordinate system within the reference frame. For one-dimensional motion, this is a simple coordinate system consisting of a one-dimensional coordinate line. In general, when describing horizontal motion, motion to the right is usually considered positive, and motion to the left is considered negative. With vertical motion, motion up is usually positive and motion down is negative. In some cases, however, as with the jet in Figure 2.6, it can be more convenient to switch the positive and negative directions. For example, if you are analyzing the motion of falling objects, it can be useful to define downwards as the positive direction. If people in a race are running to the left, it is useful to define left as the positive direction. It does not matter as long as the system is clear and consistent. Once you assign a positive direction and start solving a problem, you cannot change it. 38 Chapter 2 | Kinematics Figure 2.7 It is usually convenient to consider motion upward or to the right as positive ( + ) and motion downward or to the left as negative ( − ). Check Your Understanding A person's speed can stay the same as he or she rounds a corner and changes direction. Given this information, is speed a scalar or a vector quantity? Explain. Solution Speed is a scalar quantity. It does not change at all with direction changes; therefore, it has magnitude only. If it were a vector quantity, it would change as direction changes (even if its magnitude remained constant). Switching Reference Frames A fundamental tenet of physics is that information about an event can be gathered from a variety of reference frames. For example, imagine that you are a passenger walking toward the front of a bus. As you walk, your motion is observed by a fellow bus passenger

and by an observer standing on the sidewalk. Both the bus passenger and sidewalk observer will be able to collect information about you. They can determine how far you moved and how much time it took you to do so. However, while you moved at a consistent pace, both observers will get different results. To the passenger sitting on the bus, you moved forward at what one would consider a normal pace, something similar to how quickly you would walk outside on a sunny day. To the sidewalk observer though, you will have moved much quicker. Because the bus is also moving forward, the distance you move forward against the sidewalk each second increases, and the sidewalk observer must conclude that you are moving at a greater pace. To show that you understand this concept, you will need to create an event and think of a way to view this event from two different frames of reference. In order to ensure that the event is being observed simultaneously from both frames, you will need an assistant to help out. An example of a possible event is to have a friend ride on a skateboard while tossing a ball. How will your friend observe the ball toss, and how will those observations be different from your own? Your task is to describe your event and the observations of your event from both frames of reference. Answer the following questions below to demonstrate your understanding. For assistance, you can review the information given in the ‘Position' paragraph at the start of Section 2.1. 1. What is your event? What object are both you and your assistant observing? 2. What do you see as the event takes place? 3. What does your assistant see as the event takes place? 4. How do your reference frames cause you and your assistant to have two different sets of observations? This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 39 2.3 Time, Velocity, and Speed Figure 2.8 The motion of these racing snails can be described by their speeds and their velocities. (credit: tobitasflickr, Flickr) By the end of this section, you will be able to: Learning Objectives • Explain the relationships between instantaneous velocity, average velocity, instantaneous speed, average speed, displacement, and time. • Calculate velocity and speed given initial position, initial time, final position, and final time. • Derive a graph of velocity vs. time given a graph of position vs. time. • Interpret a graph of velocity vs

. time. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) There is more to motion than distance and displacement. Questions such as, “How long does a foot race take?” and “What was the runner's speed?” cannot be answered without an understanding of other concepts. In this section we add definitions of time, velocity, and speed to expand our description of motion. Time As discussed in Physical Quantities and Units, the most fundamental physical quantities are defined by how they are measured. This is the case with time. Every measurement of time involves measuring a change in some physical quantity. It may be a number on a digital clock, a heartbeat, or the position of the Sun in the sky. In physics, the definition of time is simple— time is change, or the interval over which change occurs. It is impossible to know that time has passed unless something changes. The amount of time or change is calibrated by comparison with a standard. The SI unit for time is the second, abbreviated s. We might, for example, observe that a certain pendulum makes one full swing every 0.75 s. We could then use the pendulum to measure time by counting its swings or, of course, by connecting the pendulum to a clock mechanism that registers time on a dial. This allows us to not only measure the amount of time, but also to determine a sequence of events. How does time relate to motion? We are usually interested in elapsed time for a particular motion, such as how long it takes an airplane passenger to get from his seat to the back of the plane. To find elapsed time, we note the time at the beginning and end of the motion and subtract the two. For example, a lecture may start at 11:00 A.M. and end at 11:50 A.M., so that the elapsed time would be 50 min. Elapsed time Δt is the difference between the ending time and beginning time, Δ = f − 0, (2.

4) where Δ is the change in time or elapsed time, f is the time at the end of the motion, and 0 is the time at the beginning of the motion. (As usual, the delta symbol, Δ, means the change in the quantity that follows it.) Life is simpler if the beginning time 0 is taken to be zero, as when we use a stopwatch. If we were using a stopwatch, it would simply read zero at the start of the lecture and 50 min at the end. If 0 = 0, then Δ = f ≡. In this text, for simplicity's sake, • motion starts at time equal to zero (0 = 0) • the symbol is used for elapsed time unless otherwise specified (Δ = f ≡ ) 40 Velocity Chapter 2 | Kinematics Your notion of velocity is probably the same as its scientific definition. You know that if you have a large displacement in a small amount of time you have a large velocity, and that velocity has units of distance divided by time, such as miles per hour or kilometers per hour. Average Velocity Average velocity is displacement (change in position) divided by the time of travel, f − 0 f − 0 - is the average (indicated by the bar over the ) velocity, Δ is the change in position (or displacement), and f where and 0 are the final and beginning positions at times f and 0, respectively. If the starting time 0 is taken to be zero, then the average velocity is simply - = Δ Δ = (2.5) - = Δ. (2.6) Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The SI unit for velocity is meters per second or m/s, but many other units, such as km/h, mi/h (also written as mph), and cm/s, are in common use. Suppose, for example, an airplane passenger took 5 seconds to move −4 m (the minus sign indicates that displacement is toward the back of the plane). His average velocity would be - = Δ = −4 m 5 s = − 0.8 m/s. (2.7) The minus sign indicates the average velocity is also toward the rear of the plane. The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he goes

to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals. Figure 2.9 A more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip. The smaller the time intervals considered in a motion, the more detailed the information. When we carry this process to its logical conclusion, we are left with an infinitesimally small interval. Over such an interval, the average velocity becomes the instantaneous velocity or the velocity at a specific instant. A car's speedometer, for example, shows the magnitude (but not the direction) of the instantaneous velocity of the car. (Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to use average velocity.) Instantaneous velocity is the average velocity at a specific instant in time (or over an infinitesimally small time interval). Mathematically, finding instantaneous velocity,, at a precise instant can involve taking a limit, a calculus operation beyond the scope of this text. However, under many circumstances, we can find precise values for instantaneous velocity without calculus. Speed In everyday language, most people use the terms “speed” and “velocity” interchangeably. In physics, however, they do not have the same meaning and they are distinct concepts. One major difference is that speed has no direction. Thus speed is a scalar. Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 41 Instantaneous speed is the magnitude of instantaneous velocity. For example, suppose the airplane passenger at one instant had an instantaneous velocity of −3.0 m/s (the minus meaning toward the rear of the plane). At that same time his instantaneous speed was 3.0 m/s. Or suppose that at one time during a shopping trip your instantaneous velocity is 40 km/h due north. Your instantaneous speed at that instant would be 40 km/h—the same magnitude but without a direction. Average speed, however, is very different from average velocity. Average speed is the distance traveled divided by elapsed time. We have noted that distance traveled can be greater than displacement. So average speed can be greater than average velocity, which is displacement divided by time

. For example, if you drive to a store and return home in half an hour, and your car's odometer shows the total distance traveled was 6 km, then your average speed was 12 km/h. Your average velocity, however, was zero, because your displacement for the round trip is zero. (Displacement is change in position and, thus, is zero for a round trip.) Thus average speed is not simply the magnitude of average velocity. Figure 2.10 During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, since there was no net change in position. Thus the average velocity is zero. Another way of visualizing the motion of an object is to use a graph. A plot of position or of velocity as a function of time can be very useful. For example, for this trip to the store, the position, velocity, and speed-vs.-time graphs are displayed in Figure 2.11. (Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we'll probably stop at the store. But for simplicity's sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.) 42 Chapter 2 | Kinematics Figure 2.11 Position vs. time, velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative. Making Connections: Take-Home Investigation—Getting a Sense of Speed If you have spent much time driving, you probably have a good sense of speeds between about 10 and 70 miles per hour. But what are these in meters per second? What do we mean when we say that something is moving at 10 m/s? To get a better sense of what these values really mean, do some observations and calculations on your own: • calculate typical car speeds in meters per second • estimate jogging and walking speed by timing yourself; convert the measurements into both m/s and mi/h • determine the speed of an ant, snail, or falling leaf This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 43 Check Your Understanding A commuter train travels from Baltimore to Washington, DC, and back in 1 hour and 45

minutes. The distance between the two stations is approximately 40 miles. What is (a) the average velocity of the train, and (b) the average speed of the train in m/s? Solution (a) The average velocity of the train is zero because f = 0 ; the train ends up at the same place it starts. (b) The average speed of the train is calculated below. Note that the train travels 40 miles one way and 40 miles back, for a total distance of 80 miles. = 80 miles 105 minutes distance time × 5280 feet 1 mile × 1 meter 3.28 feet × 1 minute 60 seconds = 20 m/s (2.8) (2.9) 80 miles 105 minutes 2.4 Acceleration Figure 2.12 A plane decelerates, or slows down, as it comes in for landing in St. Maarten. Its acceleration is opposite in direction to its velocity. (credit: Steve Conry, Flickr) Learning Objectives By the end of this section, you will be able to: • Define and distinguish between instantaneous acceleration and average acceleration. • Calculate acceleration given initial time, initial velocity, final time, and final velocity. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, and graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time. The formal definition of acceleration is consistent with these notions, but more inclusive. Average Acceleration Average Acceleration is the rate at which velocity changes, where is average acceleration, is velocity, and is time. (The bar over the means average acceleration.) - = 2.10) Because acceleration is velocity in m/s divided by time in s, the SI units for acceleration are m/s2, meters per second squared or meters per second per second, which literally means by how many meters per second the velocity changes every second. 44 Chapter 2 | Kinematics Recall

that velocity is a vector—it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both. Acceleration as a Vector Acceleration is a vector in the same direction as the change in velocity, Δ. Since velocity is a vector, it can change either in magnitude or in direction. Acceleration is therefore a change in either speed or direction, or both. Keep in mind that although acceleration is in the direction of the change in velocity, it is not always in the direction of motion. When an object's acceleration is in the same direction of its motion, the object will speed up. However, when an object's acceleration is opposite to the direction of its motion, the object will slow down. Speeding up and slowing down should not be confused with a positive and negative acceleration. The next two examples should help to make this distinction clear. Figure 2.13 A subway train in Sao Paulo, Brazil, decelerates as it comes into a station. It is accelerating in a direction opposite to its direction of motion. (credit: Yusuke Kawasaki, Flickr) Making Connections: Car Motion Figure 2.14 Above are arrows representing the motion of five cars (A–E). In all five cases, the positive direction should be considered to the right of the page. Consider the acceleration and velocity of each car in terms of its direction of travel. Figure 2.15 Car A is speeding up. Because the positive direction is considered to the right of the paper, Car A is moving with a positive velocity. Because it is speeding up while moving with a positive velocity, its acceleration is also considered positive. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 45 Figure 2.16 Car B is slowing down. Because the positive direction is considered to the right of the paper, Car B is also moving with a positive velocity. However, because it is slowing down while moving with a positive velocity, its acceleration is considered negative. (This can be viewed in a mathematical manner as well. If the car was originally moving with a velocity of +25 m

/s, it is finishing with a speed less than that, like +5 m/s. Because the change in velocity is negative, the acceleration will be as well.) Figure 2.17 Car C has a constant speed. Because the positive direction is considered to the right of the paper, Car C is moving with a positive velocity. Because all arrows are of the same length, this car is not changing its speed. As a result, its change in velocity is zero, and its acceleration must be zero as well. Figure 2.18 Car D is speeding up in the opposite direction of Cars A, B, C. Because the car is moving opposite to the positive direction, Car D is moving with a negative velocity. Because it is speeding up while moving in a negative direction, its acceleration is negative as well. Figure 2.19 Car E is slowing down in the same direction as Car D and opposite of Cars A, B, C. Because it is moving opposite to the positive direction, Car E is moving with a negative velocity as well. However, because it is slowing down while moving in a negative direction, its acceleration is actually positive. As in example B, this may be more easily understood in a mathematical sense. The car is originally moving with a large negative velocity (−25 m/s) but slows to a final velocity that is less negative (−5 m/s). This change in velocity, from −25 m/s to −5 m/s, is actually a positive change ( − = − 5 m/s − − 25 m/s of 20 m/s. Because the change in velocity is positive, the acceleration must also be positive. Making Connection - Illustrative Example The three graphs below are labeled A, B, and C. Each one represents the position of a moving object plotted against time. 46 Chapter 2 | Kinematics Figure 2.20 Three position and time graphs: A, B, and C. As we did in the previous example, let's consider the acceleration and velocity of each object in terms of its direction of travel. Figure 2.21 Graph A of Position (y axis) vs. Time (x axis). Object A is continually increasing its position in the positive direction. As a result, its velocity is considered positive. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 47 Figure 2.22 Breakdown of Graph A into two separate sections. During the first portion of

time (shaded grey) the position of the object does not change much, resulting in a small positive velocity. During a later portion of time (shaded green) the position of the object changes more, resulting in a larger positive velocity. Because this positive velocity is increasing over time, the acceleration of the object is considered positive. Figure 2.23 Graph B of Position (y axis) vs. Time (x axis). As in case A, Object B is continually increasing its position in the positive direction. As a result, its velocity is considered positive. Figure 2.24 Breakdown of Graph B into two separate sections. During the first portion of time (shaded grey) the position of the object changes a large amount, resulting in a large positive velocity. During a later portion of time (shaded green) the position of the object does not change as much, resulting in a smaller positive velocity. Because this positive velocity is decreasing over time, the acceleration of the object is considered negative. 48 Chapter 2 | Kinematics Figure 2.25 Graph C of Position (y axis) vs. Time (x axis). Object C is continually decreasing its position in the positive direction. As a result, its velocity is considered negative. Figure 2.26 Breakdown of Graph C into two separate sections. During the first portion of time (shaded grey) the position of the object does not change a large amount, resulting in a small negative velocity. During a later portion of time (shaded green) the position of the object changes a much larger amount, resulting in a larger negative velocity. Because the velocity of the object is becoming more negative during the time period, the change in velocity is negative. As a result, the object experiences a negative acceleration. Example 2.1 Calculating Acceleration: A Racehorse Leaves the Gate A racehorse coming out of the gate accelerates from rest to a velocity of 15.0 m/s due west in 1.80 s. What is its average acceleration? Figure 2.27 (credit: Jon Sullivan, PD Photo.org) Strategy This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 49 First we draw a sketch and assign a coordinate system to the problem. This is a simple problem, but it always helps to visualize it. Notice that we assign east as positive and west as negative. Thus, in this case, we have negative velocity. Figure 2.28 We can solve

this problem by identifying Δ and Δ from the given information and then calculating the average f − 0 f − 0 acceleration directly from the equation - = Δ Δ =. Solution 1. Identify the knowns. 0 = 0, f = −15.0 m/s (the minus sign indicates direction toward the west), Δ = 1.80 s. 2. Find the change in velocity. Since the horse is going from zero to − 15.0 m/s, its change in velocity equals its final velocity: Δ = f = −15.0 m/s. -. 3. Plug in the known values ( Δ and Δ ) and solve for the unknown - = Δ Δ = −15.0 m/s 1.80 s = −8.33 m/s2. (2.11) Discussion The minus sign for acceleration indicates that acceleration is toward the west. An acceleration of 8.33 m/s2 due west means that the horse increases its velocity by 8.33 m/s due west each second, that is, 8.33 meters per second per second, which we write as 8.33 m/s2. This is truly an average acceleration, because the ride is not smooth. We shall see later that an acceleration of this magnitude would require the rider to hang on with a force nearly equal to his weight. Instantaneous Acceleration Instantaneous acceleration, or the acceleration at a specific instant in time, is obtained by the same process as discussed for instantaneous velocity in Time, Velocity, and Speed—that is, by considering an infinitesimally small interval of time. How do we find instantaneous acceleration using only algebra? The answer is that we choose an average acceleration that is representative of the motion. Figure 2.29 shows graphs of instantaneous acceleration versus time for two very different motions. In Figure 2.29(a), the acceleration varies slightly and the average over the entire interval is nearly the same as the instantaneous acceleration at any time. In this case, we should treat this motion as if it had a constant acceleration equal to the average (in this case about 1.8 m/s2 ). In Figure 2.29(b), the acceleration varies drastically over time. In such situations it is best to consider smaller time intervals and choose an average acceleration for each. For example, we could consider motion over the time intervals from 0 to 1.0 s and from 1.0 to 3.0 s as separate motions with accelerations of +3.0 m/

s2 and –2.0 m/s2, respectively. 50 Chapter 2 | Kinematics Figure 2.29 Graphs of instantaneous acceleration versus time for two different one-dimensional motions. (a) Here acceleration varies only slightly and is always in the same direction, since it is positive. The average over the interval is nearly the same as the acceleration at any given time. (b) Here the acceleration varies greatly, perhaps representing a package on a post office conveyor belt that is accelerated forward and backward as it bumps along. It is necessary to consider small time intervals (such as from 0 to 1.0 s) with constant or nearly constant acceleration in such a situation. The next several examples consider the motion of the subway train shown in Figure 2.30. In (a) the shuttle moves to the right, and in (b) it moves to the left. The examples are designed to further illustrate aspects of motion and to illustrate some of the reasoning that goes into solving problems. Figure 2.30 One-dimensional motion of a subway train considered in Example 2.2, Example 2.3, Example 2.4, Example 2.5, Example 2.6, and Example 2.7. Here we have chosen the -axis so that + means to the right and − means to the left for displacements, velocities, and accelerations. (a) The subway train moves to the right from 0 to f. Its displacement Δ is +2.0 km. (b) The train moves to the left from ′0 to ′f. Its displacement Δ′ is −1.5 km. (Note that the prime symbol (′) is used simply to distinguish between displacement in the two different situations. The distances of travel and the size of the cars are on different scales to fit everything into the diagram.) Example 2.2 Calculating Displacement: A Subway Train What are the magnitude and sign of displacements for the motions of the subway train shown in parts (a) and (b) of Figure 2.30? Strategy This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 51 A drawing with a coordinate system is already provided, so we don't need to make a sketch, but we should analyze it to make sure we understand what it is showing. Pay particular attention to the coordinate system. To find displacement, we use the equation Δ = f − 0. This is

straightforward since the initial and final positions are given. Solution 1. Identify the knowns. In the figure we see that f = 6.70 km and 0 = 4.70 km for part (a), and ′f = 3.75 km and ′0 = 5.25 km for part (b). 2. Solve for displacement in part (a). Δ = f − 0 = 6.70 km − 4.70 km= +2.00 km 3. Solve for displacement in part (b). Δ′ = ′f − ′0 = 3.75 km − 5.25 km = − 1.50 km Discussion (2.12) (2.13) The direction of the motion in (a) is to the right and therefore its displacement has a positive sign, whereas motion in (b) is to the left and thus has a minus sign. Example 2.3 Comparing Distance Traveled with Displacement: A Subway Train What are the distances traveled for the motions shown in parts (a) and (b) of the subway train in Figure 2.30? Strategy To answer this question, think about the definitions of distance and distance traveled, and how they are related to displacement. Distance between two positions is defined to be the magnitude of displacement, which was found in Example 2.2. Distance traveled is the total length of the path traveled between the two positions. (See Displacement.) In the case of the subway train shown in Figure 2.30, the distance traveled is the same as the distance between the initial and final positions of the train. Solution 1. The displacement for part (a) was +2.00 km. Therefore, the distance between the initial and final positions was 2.00 km, and the distance traveled was 2.00 km. 2. The displacement for part (b) was −1.5 km. Therefore, the distance between the initial and final positions was 1.50 km, and the distance traveled was 1.50 km. Discussion Distance is a scalar. It has magnitude but no sign to indicate direction. Example 2.4 Calculating Acceleration: A Subway Train Speeding Up Suppose the train in Figure 2.30(a) accelerates from rest to 30.0 km/h in the first 20.0 s of its motion. What is its average acceleration during that time interval? Strategy It is worth it at this point to make a simple sketch: Figure 2.31 This problem

involves three steps. First we must determine the change in velocity, then we must determine the change in time, and finally we use these values to calculate the acceleration. Solution 52 Chapter 2 | Kinematics 1. Identify the knowns. 0 = 0 (the trains starts at rest), f = 30.0 km/h, and Δ = 20.0 s. 2. Calculate Δ. Since the train starts from rest, its change in velocity is Δ= +30.0 km/h, where the plus sign means velocity to the right. -. 3. Plug in known values and solve for the unknown, - = Δ Δ = +30.0 km/h 20.0 s 4. Since the units are mixed (we have both hours and seconds for time), we need to convert everything into SI units of meters and seconds. (See Physical Quantities and Units for more guidance.) - = +30 km/h 20.0 s 103 m 1 km 1 h 3600 s = 0.417 m/s2 (2.14) (2.15) Discussion The plus sign means that acceleration is to the right. This is reasonable because the train starts from rest and ends up with a velocity to the right (also positive). So acceleration is in the same direction as the change in velocity, as is always the case. Example 2.5 Calculate Acceleration: A Subway Train Slowing Down Now suppose that at the end of its trip, the train in Figure 2.30(a) slows to a stop from a speed of 30.0 km/h in 8.00 s. What is its average acceleration while stopping? Strategy Figure 2.32 In this case, the train is decelerating and its acceleration is negative because it is toward the left. As in the previous example, we must find the change in velocity and the change in time and then solve for acceleration. Solution 1. Identify the knowns. 0 = 30.0 km/h, f = 0 km/h (the train is stopped, so its velocity is 0), and Δ = 8.00 s. 2. Solve for the change in velocity − 30.0 km/h = −30.0 km/h -. 3. Plug in the knowns, Δ and Δ, and solve for - = Δ Δ = −30.0 km/h 8.00 s 4. Convert the units to meters and seconds. - = Δ Δ = −30.0 km

/h 8.00 s 103 m 1 km 1 h 3600 s = −1.04 m/s2. (2.16) (2.17) (2.18) Discussion The minus sign indicates that acceleration is to the left. This sign is reasonable because the train initially has a positive velocity in this problem, and a negative acceleration would oppose the motion. Again, acceleration is in the same direction as the change in velocity, which is negative here. This acceleration can be called a deceleration because it has a direction opposite to the velocity. The graphs of position, velocity, and acceleration vs. time for the trains in Example 2.4 and Example 2.5 are displayed in Figure 2.33. (We have taken the velocity to remain constant from 20 to 40 s, after which the train decelerates.) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 53 Figure 2.33 (a) Position of the train over time. Notice that the train's position changes slowly at the beginning of the journey, then more and more quickly as it picks up speed. Its position then changes more slowly as it slows down at the end of the journey. In the middle of the journey, while the velocity remains constant, the position changes at a constant rate. (b) Velocity of the train over time. The train's velocity increases as it accelerates at the beginning of the journey. It remains the same in the middle of the journey (where there is no acceleration). It decreases as the train decelerates at the end of the journey. (c) The acceleration of the train over time. The train has positive acceleration as it speeds up at the beginning of the journey. It has no acceleration as it travels at constant velocity in the middle of the journey. Its acceleration is negative as it slows down at the end of the journey. Example 2.6 Calculating Average Velocity: The Subway Train What is the average velocity of the train in part b of Example 2.2, and shown again below, if it takes 5.00 min to make its trip? 54 Chapter 2 | Kinematics Figure 2.34 Strategy Average velocity is displacement divided by time. It will be negative here, since the train moves to the left and has a negative displacement. Solution 1. Identify the knowns. ′f = 3.75 km, ′0 = 5.25 km, Δ = 5

.00 min. 2. Determine displacement, Δ′. We found Δ′ to be − 1.5 km in Example 2.2. 3. Solve for average velocity. 4. Convert units. - = Δ′ Δ = −1.50 km 5.00 min - = Δ′ Δ = −1.50 km 5.00 min 60 min 1 h = −18.0 km/h Discussion The negative velocity indicates motion to the left. Example 2.7 Calculating Deceleration: The Subway Train Finally, suppose the train in Figure 2.34 slows to a stop from a velocity of 20.0 km/h in 10.0 s. What is its average acceleration? Strategy Once again, let's draw a sketch: Figure 2.35 As before, we must find the change in velocity and the change in time to calculate average acceleration. Solution 1. Identify the knowns. 0 = −20 km/h, f = 0 km/h, Δ = 10.0 s. 2. Calculate Δ. The change in velocity here is actually positive, since Δ = f − 0 = 0 − (−20 km/h)=+20 km/h. -. 3. Solve for 4. Convert units. - = Δ Δ = +20.0 km/h 10.0 s This content is available for free at http://cnx.org/content/col11844/1.13 (2.19) (2.20) (2.21) (2.22) Chapter 2 | Kinematics Discussion - = +20.0 km/h 10.0 s 103 m 1 km 1 h 3600 s = +0.556 m/s2 55 (2.23) The plus sign means that acceleration is to the right. This is reasonable because the train initially has a negative velocity (to the left) in this problem and a positive acceleration opposes the motion (and so it is to the right). Again, acceleration is in the same direction as the change in velocity, which is positive here. As in Example 2.5, this acceleration can be called a deceleration since it is in the direction opposite to the velocity. Sign and Direction Perhaps the most important thing to note about these examples is the signs of the answers. In our chosen coordinate system, plus means the quantity is to the right and minus means it is to the left. This is easy to imagine for displacement and velocity. But it is a little less

obvious for acceleration. Most people interpret negative acceleration as the slowing of an object. This was not the case in Example 2.7, where a positive acceleration slowed a negative velocity. The crucial distinction was that the acceleration was in the opposite direction from the velocity. In fact, a negative acceleration will increase a negative velocity. For example, the train moving to the left in Figure 2.34 is sped up by an acceleration to the left. In that case, both and are negative. The plus and minus signs give the directions of the accelerations. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down. Check Your Understanding An airplane lands on a runway traveling east. Describe its acceleration. Solution If we take east to be positive, then the airplane has negative acceleration, as it is accelerating toward the west. It is also decelerating: its acceleration is opposite in direction to its velocity. PhET Explorations: Moving Man Simulation Learn about position, velocity, and acceleration graphs. Move the little man back and forth with the mouse and plot his motion. Set the position, velocity, or acceleration and let the simulation move the man for you. Figure 2.36 Moving Man (http://cnx.org/content/m54772/1.3/moving-man_en.jar) 2.5 Motion Equations for Constant Acceleration in One Dimension Figure 2.37 Kinematic equations can help us describe and predict the motion of moving objects such as these kayaks racing in Newbury, England. (credit: Barry Skeates, Flickr) By the end of this section, you will be able to: Learning Objectives 56 Chapter 2 | Kinematics • Calculate displacement of an object that is not accelerating, given initial position and velocity. • Calculate final velocity of an accelerating object, given initial velocity, acceleration, and time. • Calculate displacement and final position of an accelerating object, given initial position, initial velocity, time, and acceleration. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, or graphical representations. (S.P. 1.5, 2.1, 2.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express

the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) We might know that the greater the acceleration of, say, a car moving away from a stop sign, the greater the displacement in a given time. But we have not developed a specific equation that relates acceleration and displacement. In this section, we develop some convenient equations for kinematic relationships, starting from the definitions of displacement, velocity, and acceleration already covered. Notation: t, x, v, a First, let us make some simplifications in notation. Taking the initial time to be zero, as if time is measured with a stopwatch, is a great simplification. Since elapsed time is Δ = f − 0, taking 0 = 0 means that Δ = f, the final time on the stopwatch. When initial time is taken to be zero, we use the subscript 0 to denote initial values of position and velocity. That is, 0 is the initial position and 0 is the initial velocity. We put no subscripts on the final values. That is, is the final time, is the final position, and is the final velocity. This gives a simpler expression for elapsed time—now, Δ =. It also simplifies the expression for displacement, which is now Δ = − 0. Also, it simplifies the expression for change in velocity, which is now Δ = − 0. To summarize, using the simplified notation, with the initial time taken to be zero2.24) where the subscript 0 denotes an initial value and the absence of a subscript denotes a final value in whatever motion is under consideration. We now make the important assumption that acceleration is constant. This assumption allows us to avoid using calculus to find instantaneous acceleration. Since acceleration is constant, the average and instantaneous accelerations are equal. That is, - = = constant, (2.25) so we use the symbol for acceleration at all times. Assuming acceleration to be constant does not seriously limit the situations we can study nor degrade the accuracy of our treatment. For one thing, acceleration is constant in a great number of situations. Furthermore, in many other situations we can accurately describe motion by assuming a constant acceleration equal to the average acceleration for that motion. Finally, in motions where acceleration changes drastically, such as a car accelerating to top speed and then braking to a stop, the motion can be considered in separate parts, each of which has its own constant acceleration. Solving for Displacement ( Δ ) and Final Position

( ) from Average Velocity when Acceleration ( ) is Constant To get our first two new equations, we start with the definition of average velocity: - = Δ Δ. Substituting the simplified notation for Δ and Δ yields - = − 0. Solving for yields where the average velocity is = 0 +, - = 0 + 2 (constant ). This content is available for free at http://cnx.org/content/col11844/1.13 (2.26) (2.27) (2.28) (2.29) Chapter 2 | Kinematics The equation - = 0 + 2 reflects the fact that, when acceleration is constant, is just the simple average of the initial and 57 final velocities. For example, if you steadily increase your velocity (that is, with constant acceleration) from 30 to 60 km/h, then your average velocity during this steady increase is 45 km/h. Using the equation - = 0 + 2 to check this, we see that (2.30) - = 0 + 2 = 30 km/h + 60 km/h 2 = 45 km/h, which seems logical. Example 2.8 Calculating Displacement: How Far does the Jogger Run? A jogger runs down a straight stretch of road with an average velocity of 4.00 m/s for 2.00 min. What is his final position, taking his initial position to be zero? Strategy Draw a sketch. Figure 2.38 The final position is given by the equation To find, we identify the values of 0,, and from the statement of the problem and substitute them into the equation. Solution = 0 +. (2.31) 1. Identify the knowns. - = 4.00 m/s, Δ = 2.00 min, and 0 = 0 m. 2. Enter the known values into the equation. = 0 + = 0 + (4.00 m/s)(120 s) = 480 m (2.32) Discussion Velocity and final displacement are both positive, which means they are in the same direction. gives insight into the relationship between displacement, average velocity, and time. It shows, for The equation = 0 + example, that displacement is a linear function of average velocity. (By linear function, we mean that displacement depends on rather than on raised to some other power, such as. When graphed, linear functions look like straight lines with a constant slope.) On a car trip,

for example, we will get twice as far in a given time if we average 90 km/h than if we average 45 km/h. 58 Chapter 2 | Kinematics Figure 2.39 There is a linear relationship between displacement and average velocity. For a given time, an object moving twice as fast as another object will move twice as far as the other object. Solving for Final Velocity We can derive another useful equation by manipulating the definition of acceleration. Substituting the simplified notation for Δ and Δ gives us = Δ Δ Solving for yields = − 0 (constant ). = 0 + (constant ). (2.33) (2.34) (2.35) Example 2.9 Calculating Final Velocity: An Airplane Slowing Down after Landing An airplane lands with an initial velocity of 70.0 m/s and then decelerates at 1.50 m/s2 for 40.0 s. What is its final velocity? Strategy Draw a sketch. We draw the acceleration vector in the direction opposite the velocity vector because the plane is decelerating. Figure 2.40 Solution 1. Identify the knowns. 0 = 70.0 m/s, = −1.50 m/s2, = 40.0 s. 2. Identify the unknown. In this case, it is final velocity, f. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 3. Determine which equation to use. We can calculate the final velocity using the equation = 0 +. 4. Plug in the known values and solve. = 0 + = 70.0 m/s + −1.50 m/s2 (40.0 s) = 10.0 m/s 59 (2.36) Discussion The final velocity is much less than the initial velocity, as desired when slowing down, but still positive. With jet engines, reverse thrust could be maintained long enough to stop the plane and start moving it backward. That would be indicated by a negative final velocity, which is not the case here. Figure 2.41 The airplane lands with an initial velocity of 70.0 m/s and slows to a final velocity of 10.0 m/s before heading for the terminal. Note that the acceleration is negative because its direction is opposite to its velocity, which is positive. In addition to being useful in problem solving, the equation = 0 + gives us insight into

the relationships among velocity, acceleration, and time. From it we can see, for example, that • • • final velocity depends on how large the acceleration is and how long it lasts if the acceleration is zero, then the final velocity equals the initial velocity ( = 0), as expected (i.e., velocity is constant) if is negative, then the final velocity is less than the initial velocity (All of these observations fit our intuition, and it is always useful to examine basic equations in light of our intuition and experiences to check that they do indeed describe nature accurately.) Making Connections: Real-World Connection Figure 2.42 The Space Shuttle Endeavor blasts off from the Kennedy Space Center in February 2010. (credit: Matthew Simantov, Flickr) An intercontinental ballistic missile (ICBM) has a larger average acceleration than the Space Shuttle and achieves a greater velocity in the first minute or two of flight (actual ICBM burn times are classified—short-burn-time missiles are more difficult for an enemy to destroy). But the Space Shuttle obtains a greater final velocity, so that it can orbit the earth rather than come directly back down as an ICBM does. The Space Shuttle does this by accelerating for a longer time. Solving for Final Position When Velocity is Not Constant ( ≠ 0 ) We can combine the equations above to find a third equation that allows us to calculate the final position of an object experiencing constant acceleration. We start with Adding 0 to each side of this equation and dividing by 2 gives =. (2.37) (2.38) 60 Chapter 2 | Kinematics Since 0 + 2 - for constant acceleration, then = Now we substitute this expression for - = 0 + 1 2. - into the equation for displacementconstant )., yielding (2.39) (2.40) Example 2.10 Calculating Displacement of an Accelerating Object: Dragsters Dragsters can achieve average accelerations of 26.0 m/s2. Suppose such a dragster accelerates from rest at this rate for 5.56 s. How far does it travel in this time? Figure 2.43 U.S. Army Top Fuel pilot Tony “The Sarge” Schumacher begins a race with a controlled burnout. (credit: Lt. Col. William Thurmond. Photo Courtesy of U.S. Army.) Strategy Draw a sketch. Figure 2.44 We are asked to find displacement, which is if we take 0 to be zero.

(Think about it like the starting line of a race. It can be anywhere, but we call it 0 and measure all other positions relative to it.) We can use the equation = 0 + 0 + 1 2 2 once we identify 0,, and from the statement of the problem. Solution 1. Identify the knowns. Starting from rest means that 0 = 0, is given as 26.0 m/s2 and is given as 5.56 s. 2. Plug the known values into the equation to solve for the unknown : 2 2. = 0 + 0 + 1 Since the initial position and velocity are both zero, this simplifies to Substituting the identified values of and gives = 1 2 2. (2.41) (2.42) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics yielding Discussion = 1 2 26.0 m/s2 (5.56 s)2 = 402 m. 61 (2.43) (2.44) If we convert 402 m to miles, we find that the distance covered is very close to one quarter of a mile, the standard distance for drag racing. So the answer is reasonable. This is an impressive displacement in only 5.56 s, but top-notch dragsters can do a quarter mile in even less time than this. What else can we learn by examining the equation = 0 + 0 + 1 2 2? We see that: • displacement depends on the square of the elapsed time when acceleration is not zero. In Example 2.10, the dragster • covers only one fourth of the total distance in the first half of the elapsed time if acceleration is zero, then the initial velocity equals average velocity ( 0 = = 0 + 0 - ) and = 0 + 0 + 1 2 2 becomes Solving for Final Velocity when Velocity Is Not Constant ( ≠ 0 ) A fourth useful equation can be obtained from another algebraic manipulation of previous equations. If we solve = 0 + for, we get Substituting this and - = 0 + 2 into = 0 +, we get = − 0. 2 = 0 2 + 2( − 0) (constant). Example 2.11 Calculating Final Velocity: Dragsters Calculate the final velocity of the dragster in Example 2.10 without using information about time. Strategy Draw a sketch. Figure 2.45 The equation 2 = 0 displacement, and no time information is

required. 2 + 2( − 0) is ideally suited to this task because it relates velocities, acceleration, and Solution 1. Identify the known values. We know that 0 = 0, since the dragster starts from rest. Then we note that − 0 = 402 m (this was the answer in Example 2.10). Finally, the average acceleration was given to be = 26.0 m/s2. 2. Plug the knowns into the equation 2 = 0 2 + 2( − 0) and solve for. 2 = 0 + 2 26.0 m/s2 (402 m). (2.45) (2.46) (2.47) 62 Thus To get, we take the square root: Discussion 2 = 2.09×104 m2 /s2. = 2.09×104 m2 /s2 = 145 m/s. Chapter 2 | Kinematics (2.48) (2.49) 145 m/s is about 522 km/h or about 324 mi/h, but even this breakneck speed is short of the record for the quarter mile. Also, note that a square root has two values; we took the positive value to indicate a velocity in the same direction as the acceleration. An examination of the equation 2 = 0 physical quantities: 2 + 2( − 0) can produce further insights into the general relationships among • The final velocity depends on how large the acceleration is and the distance over which it acts • For a fixed deceleration, a car that is going twice as fast doesn't simply stop in twice the distance—it takes much further to stop. (This is why we have reduced speed zones near schools.) Putting Equations Together In the following examples, we further explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The box below provides easy reference to the equations needed. Summary of Kinematic Equations (constant ) = ( − 0) (2.50) (2.51) (2.52) (2.53) (2.54) Example 2.12 Calculating Displacement: How Far Does a Car Go When Coming to a Halt? On dry concrete, a car can decelerate at a rate of 7.00 m/s2, whereas on wet concrete it can decelerate at only 5.00 m/s2. Find the distances necessary to

stop a car moving at 30.0 m/s (about 110 km/h) (a) on dry concrete and (b) on wet concrete. (c) Repeat both calculations, finding the displacement from the point where the driver sees a traffic light turn red, taking into account his reaction time of 0.500 s to get his foot on the brake. Strategy Draw a sketch. Figure 2.46 This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 63 In order to determine which equations are best to use, we need to list all of the known values and identify exactly what we need to solve for. We shall do this explicitly in the next several examples, using tables to set them off. Solution for (a) 1. Identify the knowns and what we want to solve for. We know that 0 = 30.0 m/s ; = 0 ; = −7.00 m/s2 ( is negative because it is in a direction opposite to velocity). We take 0 to be 0. We are looking for displacement Δ, or − 0. 2. Identify the equation that will help up solve the problem. The best equation to use is 2 = 0 2 + 2( − 0). (2.55) This equation is best because it includes only one unknown,. We know the values of all the other variables in this equation. (There are other equations that would allow us to solve for, but they require us to know the stopping time,, which we do not know. We could use them but it would entail additional calculations.) 3. Rearrange the equation to solve for. 4. Enter known values. Thus = 02 − (30.0 m/s)2 −7.00 m/s2 2 = 64.3 m on dry concrete. (2.56) (2.57) (2.58) Solution for (b) This part can be solved in exactly the same manner as Part A. The only difference is that the deceleration is – 5.00 m/s2. The result is wet = 90.0 m on wet concrete. (2.59) Solution for (c) Once the driver reacts, the stopping distance is the same as it is in Parts A and B for dry and wet concrete. So to answer this question, we need to calculate how far the car travels during the reaction time, and then add that to the

stopping time. It is reasonable to assume that the velocity remains constant during the driver's reaction time. 1. Identify the knowns and what we want to solve for. We know that We take 0 − reaction to be 0. We are looking for reaction. - = 30.0 m/s ; reaction = 0.500 s ; reaction = 0. 2. Identify the best equation to use. = 0 + works well because the only unknown value is, which is what we want to solve for. 3. Plug in the knowns to solve the equation. = 0 + (30.0 m/s)(0.500 s) = 15.0 m. (2.60) This means the car travels 15.0 m while the driver reacts, making the total displacements in the two cases of dry and wet concrete 15.0 m greater than if he reacted instantly. 4. Add the displacement during the reaction time to the displacement when braking. braking + reaction = total (2.61) a. 64.3 m + 15.0 m = 79.3 m when dry b. 90.0 m + 15.0 m = 105 m when wet 64 Chapter 2 | Kinematics Figure 2.47 The distance necessary to stop a car varies greatly, depending on road conditions and driver reaction time. Shown here are the braking distances for dry and wet pavement, as calculated in this example, for a car initially traveling at 30.0 m/s. Also shown are the total distances traveled from the point where the driver first sees a light turn red, assuming a 0.500 s reaction time. Discussion The displacements found in this example seem reasonable for stopping a fast-moving car. It should take longer to stop a car on wet rather than dry pavement. It is interesting that reaction time adds significantly to the displacements. But more important is the general approach to solving problems. We identify the knowns and the quantities to be determined and then find an appropriate equation. There is often more than one way to solve a problem. The various parts of this example can in fact be solved by other methods, but the solutions presented above are the shortest. Example 2.13 Calculating Time: A Car Merges into Traffic Suppose a car merges into freeway traffic on a 200-m-long ramp. If its initial velocity is 10.0 m/s and it accelerates at 2.00 m/s2, how long does it take to travel the 200 m up the ramp

? (Such information might be useful to a traffic engineer.) Strategy Draw a sketch. Figure 2.48 We are asked to solve for the time. As before, we identify the known quantities in order to choose a convenient physical relationship (that is, an equation with one unknown, ). Solution 1. Identify the knowns and what we want to solve for. We know that 0 = 10 m/s ; = 2.00 m/s2 ; and = 200 m. 2. We need to solve for. Choose the best equation. = 0 + 0 + 1 equation is the variable for which we need to solve. 2 2 works best because the only unknown in the 3. We will need to rearrange the equation to solve for. In this case, it will be easier to plug in the knowns first. 200 m = 0 m + (10.0 m/s) + 1 2 This content is available for free at http://cnx.org/content/col11844/1.13 2.00 m/s2 2 (2.62) Chapter 2 | Kinematics 65 4. Simplify the equation. The units of meters (m) cancel because they are in each term. We can get the units of seconds (s) to cancel by taking = s, where is the magnitude of time and s is the unit. Doing so leaves 5. Use the quadratic formula to solve for. (a) Rearrange the equation to get 0 on one side of the equation. 200 = 10 + 2. This is a quadratic equation of the form 2 + 10 − 200 = 0 2 + + = 0, where the constants are = 1.00, = 10.0, and = −200. (b) Its solutions are given by the quadratic formula: This yields two solutions for, which are = − ± 2 − 4 2. = 10.0 and−20.0. In this case, then, the time is = in seconds, or = 10.0 s and − 20.0 s. (2.63) (2.64) (2.65) (2.66) (2.67) (2.68) A negative value for time is unreasonable, since it would mean that the event happened 20 s before the motion began. We can discard that solution. Thus, = 10.0 s. (2.69) Discussion Whenever an equation contains an unknown squared, there will be two solutions.

In some problems both solutions are meaningful, but in others, such as the above, only one solution is reasonable. The 10.0 s answer seems reasonable for a typical freeway on-ramp. With the basics of kinematics established, we can go on to many other interesting examples and applications. In the process of developing kinematics, we have also glimpsed a general approach to problem solving that produces both correct answers and insights into physical relationships. Problem-Solving Basics discusses problem-solving basics and outlines an approach that will help you succeed in this invaluable task. Making Connections: Take-Home Experiment—Breaking News We have been using SI units of meters per second squared to describe some examples of acceleration or deceleration of cars, runners, and trains. To achieve a better feel for these numbers, one can measure the braking deceleration of a car - = Δ / Δ. While traveling in a car, slowly apply the doing a slow (and safe) stop. Recall that, for average acceleration, brakes as you come up to a stop sign. Have a passenger note the initial speed in miles per hour and the time taken (in seconds) to stop. From this, calculate the deceleration in miles per hour per second. Convert this to meters per second squared and compare with other decelerations mentioned in this chapter. Calculate the distance traveled in braking. Check Your Understanding A manned rocket accelerates at a rate of 20 m/s2 during launch. How long does it take the rocket reach a velocity of 400 m/s? Solution To answer this, choose an equation that allows you to solve for time, given only, 0, and. Rearrange to solve for. = 0 + = − = 400 m/s − 0 m/s 20 m/s2 = 20 s (2.70) (2.71) 66 Chapter 2 | Kinematics 2.6 Problem-Solving Basics for One Dimensional Kinematics Figure 2.49 Problem-solving skills are essential to your success in Physics. (credit: scui3asteveo, Flickr) By the end of this section, you will be able to: Learning Objectives • Apply problem-solving steps and strategies to solve problems of one-dimensional kinematics. • Apply strategies to determine whether or not the result of a problem is reasonable, and if not, determine the cause. Problem-solving skills are obviously essential to success in a quantitative course in physics. More importantly,

the ability to apply broad physical principles, usually represented by equations, to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Such analytical skills are useful both for solving problems in this text and for applying physics in everyday and professional life. Problem-Solving Steps While there is no simple step-by-step method that works for every problem, the following general procedures facilitate problem solving and make it more meaningful. A certain amount of creativity and insight is required as well. Step 1 Examine the situation to determine which physical principles are involved. It often helps to draw a simple sketch at the outset. You will also need to decide which direction is positive and note that on your sketch. Once you have identified the physical principles, it is much easier to find and apply the equations representing those principles. Although finding the correct equation is essential, keep in mind that equations represent physical principles, laws of nature, and relationships among physical quantities. Without a conceptual understanding of a problem, a numerical solution is meaningless. Step 2 Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Many problems are stated very succinctly and require some inspection to determine what is known. A sketch can also be very useful at this point. Formally identifying the knowns is of particular importance in applying physics to real-world situations. Remember, “stopped” means velocity is zero, and we often can take initial time and position as zero. Step 3 Identify exactly what needs to be determined in the problem (identify the unknowns). In complex problems, especially, it is not always obvious what needs to be found or in what sequence. Making a list can help. Step 4 Find an equation or set of equations that can help you solve the problem. Your list of knowns and unknowns can help here. It is easiest if you can find equations that contain only one unknown—that is, all of the other variables are known, so you can easily solve for the unknown. If the equation contains more than one unknown, then an additional equation is needed to solve the problem. In some problems, several unknowns must be determined to get at the one needed most. In such problems it is especially important to keep physical principles in mind to avoid going astray in a sea

of equations. You may have to use two (or more) different equations to get the final answer. Step 5 Substitute the knowns along with their units into the appropriate equation, and obtain numerical solutions complete with units. This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made. However, be warned that correct units do not guarantee that the numerical part of the answer is also correct. This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics Step 6 67 Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important—the goal of physics is to accurately describe nature. To see if the answer is reasonable, check both its magnitude and its sign, in addition to its units. Your judgment will improve as you solve more and more physics problems, and it will become possible for you to make finer and finer judgments regarding whether nature is adequately described by the answer to a problem. This step brings the problem back to its conceptual meaning. If you can judge whether the answer is reasonable, you have a deeper understanding of physics than just being able to mechanically solve a problem. When solving problems, we often perform these steps in different order, and we also tend to do several steps simultaneously. There is no rigid procedure that will work every time. Creativity and insight grow with experience, and the basics of problem solving become almost automatic. One way to get practice is to work out the text's examples for yourself as you read. Another is to work as many end-of-section problems as possible, starting with the easiest to build confidence and progressing to the more difficult. Once you become involved in physics, you will see it all around you, and you can begin to apply it to situations you encounter outside the classroom, just as is done in many of the applications in this text. Unreasonable Results Physics must describe nature accurately. Some problems have results that are unreasonable because one premise is unreasonable or because certain premises are inconsistent with one another. The physical principle applied correctly then produces an unreasonable result. For example, if a person starting a foot race accelerates at 0.40 m/s2 for 100 s, his final speed will be 40 m/s (about 150 km/h)—clearly unreasonable because the time of 100 s is an unreasonable premise. The physics is correct in a

sense, but there is more to describing nature than just manipulating equations correctly. Checking the result of a problem to see if it is reasonable does more than help uncover errors in problem solving—it also builds intuition in judging whether nature is being accurately described. Use the following strategies to determine whether an answer is reasonable and, if it is not, to determine what is the cause. Step 1 Solve the problem using strategies as outlined and in the format followed in the worked examples in the text. In the example given in the preceding paragraph, you would identify the givens as the acceleration and time and use the equation below to find the unknown final velocity. That is, = 0 + = 0 + 0.40 m/s2 (100 s) = 40 m/s. (2.72) Step 2 Check to see if the answer is reasonable. Is it too large or too small, or does it have the wrong sign, improper units, …? In this case, you may need to convert meters per second into a more familiar unit, such as miles per hour. 40 m s 3.28 ft m 1 mi 5280 ft 60 s min 60 min 1 h = 89 mph (2.73) This velocity is about four times greater than a person can run—so it is too large. Step 3 If the answer is unreasonable, look for what specifically could cause the identified difficulty. In the example of the runner, there are only two assumptions that are suspect. The acceleration could be too great or the time too long. First look at the acceleration and think about what the number means. If someone accelerates at 0.40 m/s2, their velocity is increasing by 0.4 m/s each second. Does this seem reasonable? If so, the time must be too long. It is not possible for someone to accelerate at a constant rate of 0.40 m/s2 for 100 s (almost two minutes). 2.7 Falling Objects Learning Objectives By the end of this section, you will be able to: • Describe the effects of gravity on objects in motion. • Describe the motion of objects that are in free fall. • Calculate the position and velocity of objects in free fall. The information presented in this section supports the following AP® learning objectives and science practices: • 3.A.1.1 The student is able to express the motion of an object using narrative, mathematical, or graphical representations. (S.P. 1.5, 2.1,

2.2) • 3.A.1.2 The student is able to design an experimental investigation of the motion of an object. (S.P. 4.2) • 3.A.1.3 The student is able to analyze experimental data describing the motion of an object and is able to express the results of the analysis using narrative, mathematical, and graphical representations. (S.P. 5.1) 68 Chapter 2 | Kinematics Falling objects form an interesting class of motion problems. For example, we can estimate the depth of a vertical mine shaft by dropping a rock into it and listening for the rock to hit the bottom. By applying the kinematics developed so far to falling objects, we can examine some interesting situations and learn much about gravity in the process. Gravity The most remarkable and unexpected fact about falling objects is that, if air resistance and friction are negligible, then in a given location all objects fall toward the center of Earth with the same constant acceleration, independent of their mass. This experimentally determined fact is unexpected, because we are so accustomed to the effects of air resistance and friction that we expect light objects to fall slower than heavy ones. Figure 2.50 A hammer and a feather will fall with the same constant acceleration if air resistance is considered negligible. This is a general characteristic of gravity not unique to Earth, as astronaut David R. Scott demonstrated on the Moon in 1971, where the acceleration due to gravity is only 1.67 m/s2. In the real world, air resistance can cause a lighter object to fall slower than a heavier object of the same size. A tennis ball will reach the ground after a hard baseball dropped at the same time. (It might be difficult to observe the difference if the height is not large.) Air resistance opposes the motion of an object through the air, while friction between objects—such as between clothes and a laundry chute or between a stone and a pool into which it is dropped—also opposes motion between them. For the ideal situations of these first few chapters, an object falling without air resistance or friction is defined to be in free-fall. The force of gravity causes objects to fall toward the center of Earth. The acceleration of free-falling objects is therefore called the acceleration due to gravity. The acceleration due to gravity is constant, which means we can apply the kinematics equations to any falling object where air resistance and friction are negligible. This opens a broad class of interesting situations to us. The acceleration due to gravity is so

important that its magnitude is given its own symbol,. It is constant at any given location on Earth and has the average value = 9.80 m/s2. (2.74) Although varies from 9.78 m/s2 to 9.83 m/s2, depending on latitude, altitude, underlying geological formations, and local topography, the average value of 9.80 m/s2 will be used in this text unless otherwise specified. The direction of the acceleration due to gravity is downward (towards the center of Earth). In fact, its direction defines what we call vertical. Note that whether the acceleration in the kinematic equations has the value + or − depends on how we define our coordinate system. If we define the upward direction as positive, then = − = −9.80 m/s2, and if we define the downward direction as positive, then = = 9.80 m/s2. One-Dimensional Motion Involving Gravity The best way to see the basic features of motion involving gravity is to start with the simplest situations and then progress toward more complex ones. So we start by considering straight up and down motion with no air resistance or friction. These assumptions mean that the velocity (if there is any) is vertical. If the object is dropped, we know the initial velocity is zero. Once the object has left contact with whatever held or threw it, the object is in free-fall. Under these circumstances, the motion is onedimensional and has constant acceleration of magnitude. We will also represent vertical displacement with the symbol and use for horizontal displacement. Kinematic Equations for Objects in Free-Fall where Acceleration = -( − 0) (2.75) (2.76) (2.77) This content is available for free at http://cnx.org/content/col11844/1.13 Chapter 2 | Kinematics 69 Example 2.14 Calculating Position and Velocity of a Falling Object: A Rock Thrown Upward A person standing on the edge of a high cliff throws a rock straight up with an initial velocity of 13.0 m/s. The rock misses the edge of the cliff as it falls back to Earth. Calculate the position and velocity of the rock 1.00 s, 2.00 s, and 3.00 s after it is thrown, neglecting the effects of air resistance. Strategy Draw a sketch. Figure 2.51 We are asked to determine the position at various

times. It is reasonable to take the initial position 0 to be zero. This problem involves one-dimensional motion in the vertical direction. We use plus and minus signs to indicate direction, with up being positive and down negative. Since up is positive, and the rock is thrown upward, the initial velocity must be positive too. The acceleration due to gravity is downward, so is negative. It is crucial that the initial velocity and the acceleration due to gravity have opposite signs. Opposite signs indicate that the acceleration due to gravity opposes the initial motion and will slow and eventually reverse it. Since we are asked for values of position and velocity at three times, we will refer to these as 1 and 1 ; and 2 ; and 3 and 3. Solution for Position 1 1. Identify the knowns. We know that 0 = 0 ; 0 = 13.0 m/s ; = − = −9.80 m/s2 ; and = 1.00 s. 2. Identify the best equation to use. We will use = 0 + 0 + 1 here), which is the value we want to find. 3. Plug in the known values and solve for 1. 2 2 because it includes only one unknown, (or 1, = 0 + (13.0 m/s)(1.00 s) + 1 2 −9.80 m/s2 (1.00 s)2 = 8.10 m (2.78) Discussion The rock is 8.10 m above its starting point at = 1.00 s, since 1 > 0. It could be moving up or down; the only way to tell is to calculate 1 and find out if it is positive or negative. Solution for Velocity 1 1. Identify the knowns. We know that 0 = 0 ; 0 = 13.0 m/s ; = − = −9.80 m/s2 ; and = 1.00 s. We also know from the solution above that 1 = 8.10 m. 2. Identify the best equation to use. The most straightforward is = 0 − (from = 0 +, where = gravitational acceleration = − ). 3. Plug in the knowns and solve. 1 = 0 − = 13.0 m/s − 9.80 m/s2 (1.00 s) = 3.20 m/s (2.79) Discussion The positive value for 1 means that the rock is still heading upward at = 1.00 s. However, it has slowed from