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    Regents Physics--Physical Setting Power Pack Revised Edition
    Regents Physics--Physical Setting Power Pack Revised Edition
    Regents Physics--Physical Setting Power Pack Revised Edition
    Ebook1,984 pages16 hours

    Regents Physics--Physical Setting Power Pack Revised Edition

    By Barron's Educational Series, Miriam A. Lazar and Albert Tarendash

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    About this ebook

    Barron’s Regents Physics Power Pack provides comprehensive review, actual administered exams, and practice questions to help students prepare for the Physics Regents exam.

    This edition includes:
    • Two actual Regents exams online

    Regents Exams and Answers: Physics--Physical Setting
    • Four actual, administered Regents exams so students have the practice they need to prepare for the test
    • Review questions grouped by topic, to help refresh skills learned in class
    • Thorough explanations for all answers
    • Score analysis charts to help identify strengths and weaknesses
    • Study tips and test-taking strategies

    Let’s Review Regents: Physics--Physical Setting
    • Comprehensive review of all topics on the test
    • Extra practice questions with answers
    • One actual, administered Regents Physics exam with answer key
    LanguageEnglish
    PublisherBarrons Educational Services
    Release dateJan 5, 2021
    ISBN9781506277691
    Regents Physics--Physical Setting Power Pack Revised Edition

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      Regents Physics--Physical Setting Power Pack Revised Edition - Barron's Educational Series

      Regents Power Pack

      Physics—Physical Setting

      Revised Edition

      Miriam A. Lazar

      Principal

      Archimedes Academy

      for

      Math, Science & Technology Applications

      Bronx, New York

      formerly

      Chemistry/Physics Teacher

      Stuyvesant High School

      New York, New York

      © Copyright 2021, 2020, 2019, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2008, 2007, 2006, 2005, 2004, 2003, 2002, 2001, 2000, 1999, 1998, 1997, 1996, 1995, 1994, 1993, 1992 by Kaplan, Inc., d/b/a Barron’s Educational Series

      All rights reserved.

      No part of this book may be reproduced or distributed in any form

      or by any means without the written permission of the copyright owner.

      Published by Kaplan, Inc., d/b/a Barron’s Educational Series

      750 Third Avenue

      New York, NY 10017

      www.barronseduc.com

      ISBN: 978-1-5062-7769-1

      10 9 8 7 6 5 4 3 2 1

      Table of Contents

      Barron’s Regents Power Pack: Physics—Physical Setting

      Title Page

      Copyright Information

      Let’s Review: Physics—Physical Setting

      Cover

      Title Page

      Copyright Information

      Preface

      Chapter One: Introduction to Physics

      1.1 Physics Is …?

      1.2 Measurement and the Metric System

      1.3 Scientific Notation

      1.4 Accuracy, Precision, and Significant Digits

      1.5 Order of Magnitude

      1.6 Graphing Data

      1.7 Direct and Inverse Proportions

      1.8 Mathematical Relationships within Right Triangles

      Questions

      Chapter Two: Motion in One Dimension

      2.1 Motion Defined

      2.2 Graphing an Object’s Motion

      2.3 Displacement

      2.4 Velocity

      2.5 Acceleration

      2.6 The Equations of Uniformly Accelerated Motion

      2.7 Freely Falling Objects

      2.8 Motion Graphs Revisited

      Questions

      Chapter Three: Forces and Newton’s Laws

      3.1 Introduction

      3.2 What Is a Force?

      3.3 Measuring Forces Using Hooke’s Law

      3.4 Newton’s First Law of Motion

      3.5 Newton’s Second Law of Motion

      3.6 Weight

      3.7 Normal Forces

      3.8 Frictional Forces

      3.9 The Coefficient of Friction

      3.10 Newton’s Third Law of Motion—Action and Reaction

      Questions

      Chapter Four: Vector Quantities and Their Applications

      4.1 Introduction

      4.2 Displacement, and Representation of Vector Quantities

      4.3 Vector Addition

      4.4 Vector Subtraction

      4.5 Resolution of Vectors

      4.6 Using Resolution to Add Vectors

      4.7 Static Equilibrium

      4.8 Inclined Planes

      4.9 Motion in a Plane (Two-Dimensional Motion)

      Questions

      Chapter Five: Circular Motion and Gravitation

      5.1 Introduction

      5.2 Circular Motion

      5.3 Newton’s Law of Universal Gravitation

      5.4 Satellite Motion

      5.5 Weight and Gravitational Force

      5.6 The Gravitational Field

      Questions

      Chapter Six: Momentum and Its Conservation

      6.1 Momentum

      6.2 Newton’s Second Law and Momentum

      6.3 Conservation of Momentum

      6.4 Conservation of Momentum and Newton’s Third Law

      Questions

      Chapter Seven: Work and Energy

      7.1 Work

      7.2 Power

      7.3 Energy

      7.4 Internal Energy and Work

      Questions

      Chapter Eight: Static Electricity

      8.1 What Is Electricity?

      8.2 Electric Charges

      8.3 Conductors and Insulators

      8.4 Charging Objects

      8.5 The Electroscope

      8.6 Coulomb’s Law

      8.7 The Electric Field

      8.8 Electric Field Lines (Lines of Force)

      8.9 Electric Field Strength

      8.10 Potential Difference

      8.11 Electric Potential

      8.12 The Millikan Oil Drop Experiment

      Questions

      Chapter Nine: Electric Current and Circuits

      9.1 Introduction

      9.2 Electric Current

      9.3 Current and Potential Difference

      9.4 Resistance

      9.5 Electric Circuits and Ohm’s Law

      9.6 Power and Energy in Electric Circuits

      9.7 Series Circuits

      9.8 Parallel Circuits

      Questions

      Chapter Ten: Magnetism; Electromagnetism and Its Applications

      10.1 Introduction

      ✪ 10.2 General Properties of Magnets

      10.3 Magnetic Fields

      10.4 Electromagnetism

      ✪ 10.5 Electromagnetic Induction

      Questions

      Chapter Eleven: Waves and Sound

      11.1 Definition of Wave Motion

      11.2 Types of Waves

      11.3 Characteristics of Periodic Waves

      11.4 Speed of a Wave

      11.5 Reflection

      11.6 Refraction

      11.7 Interference

      11.8 Standing Waves

      11.9 Resonance

      11.10 Diffraction

      11.11 Doppler Effect

      Questions

      Chapter Twelve: Light and Geometric Optics

      12.1 Introduction

      ✪ 12.2 Polarization of Light

      12.3 Speed of Light

      12.4 Visible Light

      12.5 Reflection

      12.6 Refraction

      12.7 Diffraction and Interference of Light

      Questions

      Chapter Thirteen: Modern Physics

      13.1 Introduction

      13.2 Black-Body Radiation and Planck’s Hypothesis

      13.3 The Photoelectric Effect

      ✪ 13.4 The Compton Effect

      13.5 Models of the Atom

      13.6 Energy and Mass

      Questions

      Chapter Fourteen: Nuclear Energy

      14.1 Introduction

      14.2 Nucleons

      14.3 Nuclear Symbols

      ✪ 14.4 Isotopes

      14.5 Nuclear Masses

      ✪ 14.6 Average Binding Energy per Nucleon

      14.7 Nuclear Forces

      14.8 Nuclear Reactions

      ✪ 14.9 Induced Nuclear Reactions

      14.10 Fundamental Particles and Interactions

      Questions

      Glossary

      Appendix 1: New York State Reference Tables for Physical Setting: Physics

      Appendix 2: Summary of Equations for New York State Physical Setting: Physics Regents Examination

      Appendix 3: Answering Short-Constructed Response and Free-Response Questions

      Appendix 4: Physics: Physical Setting Regents Exam

      June 2019

      Barron’s Regents Exams and Answers: Physics: The Physical Setting

      Cover

      Title Page

      Copyright Information

      Preface

      How to Use This Book

      Test-Taking Techniques

      General Helpful Tips

      How to Answer Part C Questions

      What to Expect on the Regents Examination in Physics

      Format of the Physics Examination

      Topics Covered on the Regents Examination in Physics

      New York State Physical Setting/Physics Core

      Topic Outline

      Question Index

      Glossary of Important Terms

      Reference Tables for Physics

      Using the Equations to Solve Physics Problems

      Mechanics

      Electricity

      Waves

      Modern Physics

      Regents Examinations, Answers, and Self-Analysis Charts

      Examination June 2013

      Examination June 2014

      Examination June 2015

      Examination June 2016

      Examination June 2017

      Examination June 2018

      Examination June 2019

      Guide

      Table of Contents

      Let’s Review:

      Physics—Physical Setting

      Revised Edition

      Miriam A. Lazar

      Principal

      Archimedes Academy

      for

      Math, Science & Technology Applications

      Bronx, New York

      formerly

      Chemistry/Physics Teacher

      Stuyvesant High School

      New York, New York

      Illustration Acknowledgment

      Page 289 From Physics, by Halliday and Resnick. Copyright © 1962 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

      © Copyright 2021, 2020, 2018, 2017, 2016, 2015, 2014, 2013, 2012, 2011, 2010, 2009, 2007, 2006, 2005, 2004, 2002, 1996 by Kaplan, Inc., d/b/a Barron’s Educational Series

      All rights reserved.

      No part of this book may be reproduced or distributed in any form

      or by any means without the written permission of the copyright owner.

      Published by Kaplan, Inc., d/b/a Barron’s Educational Series

      750 Third Avenue

      New York, NY 10017

      www.barronseduc.com

      ISBN: 978-1-5062-7214-6

      10 9 8 7 6 5 4 3 2 1

      Preface

      To the Student

      This book has been written to help you understand and review high school physics. Physics is not a particularly easy subject, and no book—no matter how well written—can give you instant insight into the subject. Nevertheless, if you read this book carefully and do all of the problems and review questions, you will have a pretty good understanding of what physics is all about.

      I have designed this book to be your physics companion. It is somewhat more detailed than some review books but is probably much less detailed than your hardcover text. This book is divided into 14 chapters. Each chapter begins with Key Ideas and ends with multiple-choice questions (part A and B-1 Regents style questions) and some (where available) short-constructed response and free-response questions (part B-2 and C Regents style questions) drawn from past NYS Regents Physics examinations. The answers to these questions appear after the glossary. The appendices contain various items of information that are useful in the study of physics and in answering short-constructed response and free-response questions.

      This book covers all of the concepts and skills listed in the NYS Physical Setting: Physics core curriculum. In this edition, a significant amount of material that I had included in previous editions as review for a general physics course that went beyond the core (the core being only the minimum to be taught and what is testable by NYS) has at the request of reviewers and my publisher been deleted from this edition. There is only a small amount of material included that is not part of the core, but which I felt helped to clarify concepts that were a part of the core and so remain. A star icon (✪) appears next to chapter subheadings that are not part of the core curriculum. For those students taking the NYS Physical Setting: Physics Regents Examination at the end of your course, I have included general information about this examination and a summary of concepts to be mastered and skills to be demonstrated. These items can be found in Appendix 4. The correlation of chapter material with the NYS Physical Setting: Physics core curriculum Regents is clearly indicated in the Appendix.

      If you have any comments about this book, I would appreciate hearing from you. Please write to me in care of the publisher, whose name and address are given on the copyright page.

      To the Teacher

      Another review book in Physics? Every author feels that he or she has a unique contribution to make, and I’m certainly no exception to this rule. It is my belief that a physics review book should be more than an embellished outline of a particular syllabus with questions and problems added. Introductory physics is a difficult subject that requires diligent effort on the part of the student, and any book worth its salt must provide careful and detailed explanations of the material. I wrote the book with these thoughts in mind and I hope that it will be successful for both you and your students.

      This book covers all of the concepts and skills listed in the NYS Physical Setting: Physics core curriculum. I have divided the book into 14 chapters because I believe that the material is best presented in this fashion. In this edition, a significant amount of material, that I had included in previous editions as review for a general physics course that went beyond the core (the core being only the minimum to be taught and what is testable by NYS) has at the request of reviewers and my publisher been deleted from this edition. There is only a small amount of material included which is not part of the core but which I felt helped to clarify concepts that were a part of the core and so remain. A star icon (✪) appears next to chapter subheadings that are not part of the core curriculum. You may not feel that my division of the material is appropriate for your teaching style or course, but there is no one correct way to order the material in a physics text. If you are more comfortable with another approach, then use it by all means.

      I spent considerable time in working out short-constructed response and free-response problems throughout the text because I believe that is the area where students experience their greatest difficulties with physics. I chose not to pad each section with multiple-choice questions but reserved them for the end of each chapter. In this edition I have included a section of short-constructed response and free-response questions at the end of each chapter. These questions were drawn from past NYS Regents examinations spanning more than 30 years.

      I hope that this book will give you some new insights and enable you to plan and execute your physics lessons with a spirit of inquiry. If you have any comments or corrections, please write to me in care of the publisher, whose name and address are given on the copyright page or you can e-mail me at [email protected].

      I wish to thank my editors at Barron’s for their patience and understanding, Mr. Joseph Lazar (my father) for his excellent art work and all my former students at Stuyvesant, who contributed much to the first edition that continues to be included in this edition. I wish to thank Albert S. Tarendash, my teacher, my mentor, and my friend, for his invaluable hard work and effort that went into the creation of the first edition that lives on in this edition.

      I continue to dedicate this book to the memories of Dr. Murray Kahn and Dr. Matthew Litwin. Both men were friends and colleagues and outstanding chemistry teachers whose first and last thoughts were always of their students.

      Miriam Lazar

      Chapter

      one

      Introduction to Physics

      Key Ideas

      Science depends on our ability to measure quantities. The SI metric system is used internationally as the standard for scientific measurement. It consists of seven basic quantities (e.g., length and time) and many derived quantities (e.g., speed and density).

      Every measurement has a degree of uncertainty that is related to the limits of the measuring instrument. The concept of significant digits helps us to evaluate the degree of uncertainty in a particular measurement.

      A graph of data points can indicate whether there is regularity within a set of data and can be used to predict how variables will behave under given conditions.

      KEY OBJECTIVES

      At the conclusion of this chapter you will be able to:

      State the fundamental quantities of measurement in the Système International (SI) and the metric units associated with them.

      Perform calculations using scientific notation.

      Determine the number of significant digits in a measurement.

      Incorporate significant digits within calculations.

      Determine the order of magnitude of a measurement.

      Plot a graph from a series of data points.

      Determine proportional relationships within data.

      Calculate the slope of a straight-line graph.

      State the common mathematical relationships in a right triangle.

      1.1 Physics Is …?

      It’s not easy to come up with a precise definition of physics because this subject is so broad. The best one we’ve heard so far is this: Physics is what physicists do. But what do they do? Physicists study the universe, from the smallest parts of matter (the particles that make up atoms) to the largest (the galaxies and beyond). They study the interactions of matter and energy, using both experimental (laboratory) and theoretical (mathematical) techniques. In this chapter, we will introduce some of these techniques in order to prepare us for the material that is covered in later chapters.

      1.2 Measurement and the Metric System

      The basis of all science lies in the ability to measure quantities. For example, we can easily measure the length of a table. In principle, we can even design an experiment to measure the distance from Earth to the nearest star. Therefore, these lengths have scientific meaning to us. However, modern physics theorizes that an electron has no measurable diameter, and consequently no experiment or apparatus can ever measure this length. As a result, we say that the diameter of an electron has no scientific meaning.

      To be able to measure quantities we must have a system of measurement. We use the Système International (SI) because scientists all over the world express measurements in these metric units. The SI recognizes seven fundamental quantities upon which all measurement is based: (1) length, (2) mass, (3) time, (4) temperature, (5) electric current, (6) luminous intensity, and (7) number of particles. Each of these quantities has a unit of measure based on a standard that can be duplicated easily and does not vary appreciably.

      Length

      The unit of length is the meter (m), which is approximately 39 inches. The standard is based on the speed of light, which is absolutely constant (in a

      vacuum) and has an assigned value of 2.99792458 × 10⁸ meters per second.

      Mass

      The unit of mass is the kilogram (kg), which has an approximate weight (on Earth) of 2.2 pounds. The standard is a platinum-iridium cylinder that is kept at constant temperature and humidity in a dustless vault in Sèvres, France. (Note that mass and weight are not the same quantities. Mass is the measure of the matter an object contains, while weight is the force with which gravity attracts matter.)

      Time

      The unit of time is the second (s). The standard is based on the frequency of vibrations of cesium-133 atoms under certain defined conditions.

      ✪ Temperature

      The unit of temperature is the kelvin (K). The standard is based on the point at which solid, liquid, and gaseous water coexist simultaneously (the triple point, which has an assigned value of 273.16 K).

      Electric Current

      The unit of electric current is the ampere (A). The standard is based on the mutual forces experienced by parallel current-carrying wires.

      ✪ Luminous Intensity

      The unit of luminous intensity is the candela (cd). The standard is based on the amount of radiation emitted by a certain object, known as a black-body radiator, at the freezing temperature of platinum (2046 K).

      ✪ Number of Particles

      The unit of number of particles is the mole (mol). The standard is based on the number of atoms contained in 0.012 kilogram of carbon-12 (6.02 × 10²³ atoms).

      1.3 Scientific Notation

      Once we have established a system of measurement, we need to be able to express small and large numbers easily. Scientific notation accomplishes this purpose. In scientific notation, a number is expressed as a power of 10 and takes the form

      M is the mantissa; it is greater than or equal to 1 and is less than 10 (1 ≤ M < 10). The exponent, n, is an integer. For example, the number 2300 is written in scientific notation as 2.3 × 10³ (not as 23 × 10² or 0.23 × 10⁴). The number 0.0000578 is written as 5.78 × 10–5.

      To write a number in scientific notation, we move the decimal place until the mantissa is a number between 1 and 10. If we move the decimal place to the left, the exponent is a positive number; if we move it to the right, the exponent is a negative number.

      If we wish to multiply two numbers expressed in scientific notation, we multiply the mantissas and add the exponents. The final result must always be expressed in proper scientific notation. Here are two examples:

      To add two numbers expressed in scientific notation, both numbers must have the same exponent. The mantissas are then added together. The following example illustrates the application of this rule:

      To divide two numbers expressed in scientific notation, we divide the mantissas and subtract the exponents. The following example illustrates the application of this rule:

      1.4 Accuracy, Precision, and Significant Digits

      As stated in Section 1.2, every scientific discipline, including physics, is concerned with making measurements. Since no instrument is perfect, however, every measurement has a degree of uncertainty associated with it. A well-designed experiment will reduce the uncertainty of each measurement to the smallest possible value.

      Accuracy refers to how well a measurement agrees with an accepted value. For example, if the accepted density of a material is 1220 kilograms per meter³ and a student’s measurement is 1235 kilograms per meter³, the difference (15 kg/m³) is an indication of the accuracy of the measurement. The smaller the difference, the more accurate is the measurement.

      Precision describes how well a measuring device can produce a measurement. The limit of precision depends on the design and construction of the measuring device. No matter how carefully we measure, we can never obtain a result more precise than the limit of our measuring device. A good general rule is that the limit of precision of a measuring device is equal to plus or minus one-half of its smallest division. For example, in the diagram below:

      the smallest division of the meter stick is 0.1 meter, and the limit of its precision is ±0.05 meter. When we read any measurement using this meter stick, we must attach this limit to the measurement, for example, 0.27 ± 0.05 meter.

      Significant digits are the digits that are part of any valid measurement. The number of significant digits is a direct result of the number of divisions the measuring device contains. The following diagram represents a meter stick with no divisions on it. How should the measurement, indicated by the arrow, be reported?

      Since there are no divisions, all we know is that the measurement is somewhere between 0 and 1 meter. The best we can do is to make an educated guess based on the position of the arrow, and we report our measurement as 0.3 meter. This meter stick allows us to measure length to one significant digit.

      Suppose we now use a meter stick that has been divided into tenths and repeat the measurement:

      Now we can report the measurement with less uncertainty because we know that the indicated length lies between 0.3 and 0.4 meter. If we allow ourselves one guess, we could report the length as 0.33 meter. This measurement has two significant digits. The more significant digits a measurement has, the more confidence we have in our ability to reproduce the measurement because only the last digit is in doubt.

      Measurements that contain zeros can be particularly troublesome. For example, we say that the average distance between Earth and the Moon is 238,000 miles. Do we really know this number to six significant digits? If so, we would have measured the distance to the nearest mile. Actually, this measurement contains only three significant figures. The distance is being reported to the nearest thousand miles. The zeros simply tell us how large the measurement is.

      To avoid confusion, a number of rules have been established for determining how many significant digits a measurement has.

      Rules for Determining the Number of Significant Digits in a Measurement

      All nonzero numbers are significant. The measurement 2.735 meters has four significant digits.

      Zeros located between nonzero numbers are also significant. The measurements 1.0285 kilograms and 202.03 seconds each have five signifi­cant digits.

      For numbers greater than or equal to 1, zeros located at the end of the measurement are significant only if a decimal point is present. The measurement 60 amperes has one significant digit. In this case, the zero indicates the size of the number, not its significance. The measurements 60. amperes and 60.000 amperes, however, have two and five significant digits, respectively.

      For numbers less than 1, the leading zeros are not significant; they indicate the size of the number. Thus, the measurements 0.002 kilogram, 0.020 kilogram, and 0.000200 kilogram have one, two, and three sig­nificant digits, respectively. (The significant digits are indicated in bold type.)

      If we use scientific notation, we need not become involved with the preceding rules because the mantissas always contains the proper number of significant digits; the size of the number is absorbed into the exponent. For example, the measurement 3.10 × 10–4 meter has three significant digits.

      Using Significant Digits in Calculations

      Significant digits are particularly important in calculations involving measured quantities and it is crucial that the result of a calculation does not imply a greater precision than any of the individual measurements. Calculators routinely give us answers with ten digits. It is incorrect to believe that the results of most of our calculations have this many significant digits.

      When two measurements are multiplied (or divided), the answer should contain as many significant digits as the least precise measurement. For example, if the measurement 2.3 meters (two significant digits) is multiplied by 7.45 meters (three significant digits), the answer will contain two significant digits:

      (Note that the units are also multiplied together.)

      When two measurements are added (or subtracted), the answer should contain as many decimal places as the measurement with the smallest number of decimal places. For example, when 8.11 kilograms and 2.476 kilograms are added, the answer will be taken to the second decimal place:

      (Note that the answer has been rounded to two decimal places.)

      If counted numbers (such as 6 atoms) or defined numbers (such as 273.16 K) are used in calculations, they are treated as though they had an infinite number of significant digits or decimal places.

      1.5 Order of Magnitude

      There are times when we are interested in the size of a measurement rather than its actual value. The order of magnitude of a measurement is the power of 10 closest to its value. For example, the order of magnitude of 1284 kilograms (1.284 × 10³) is 10³, while the order of magnitude of 8756 kilograms (8.756 × 10³) is 10⁴. Orders of magnitude are very useful for comparing quantities, such as mass or distance, and for estimating the answers to problems involving complex calculations.

      For a real world example, the size of an oak leaf is on an order of mag­nitude of 10–1 m while the size of a single proton is on an order of magnitude of 10–15 m. By subtracting the exponents we can determine how many orders of magnitude difference there are between two quantities. In the above example there is a difference in order of magnitude on the scale of 10¹⁴. That is, an oak leaf is on order of magnitude 10¹⁴ times larger than a single proton.

      1.6 Graphing Data

      We have all heard the expression A picture is worth a thousand words. This adage is particularly true when we wish to present experimental data. The following table of data gives the stretched lengths of a spring (in meters) when different weights (in a unit called newtons) are placed on it.

      Now plot these points and draw a graph.

      Note how the graph is constructed. First, the axes are drawn so that the data can be displayed over the entire graph. Next, the axes are labeled with the names of the quantities and their units of measure. It is traditional to place the quantity that is varied by the experimenter (the independent variable) along the x-axis, and the result of the experiment (the dependent variable) along the y-axis. Each item of data is then entered in the correct place on the graph.

      Note that we do not play connect the dots. Rather, we look for some regular relationship between the data points. We then draw a smooth line (or curve) that will fit this relationship as closely as possible. The scatter of the data in this example implies that we are dealing with a straight-line relationship. Although in most cases the graphed line will pass through a maximum number of data points, it is entirely possible that the graph may not pass through any of the data points. We require only that the data points above and below the graph be evenly distributed, as shown in the example. This graph is known as a best-fit straight line.

      If a graph yields a straight line, we can conclude that the variables change uniformly. In the example given above, the length of the spring increases regularly as heavier weights are applied to the spring, and the steadily rising line of the graph reflects this direct relationship. If a graph is a curve, however, the variables (here, weight and length) change at a rate that is not uniform.

      1.7 Direct and Inverse Proportions

      Two quantities are directly proportional to one another if a change in one quantity is accompanied by an identical change in the other. For example, the table given below represents the ideal relationship between the mass of a substance and its corresponding volume. If the mass changes by a factor of 2 or 3, the volume changes by the same factor. This means that the ratio of the two quantities is a constant.

      When we plot this relationship, we obtain a straight-line graph that passes through the origin.

      Frequently, the slope of a straight line provides us with additional information. Recall that we calculate the slope of a line by selecting two data points (x1, y1 and x2, y2) and finding the ratio Δy/Δx:

      Physics Concepts

      *Note that this slope equation does not appear on the Reference Tables. However, it is essential when solving problems. Memorize this equation.

      The slope of the line representing a direct proportion is known as the constant of proportionality, and it provides the ratio between the variables. (In the mass-volume graph, the constant is known as the density of the substance.)

      Two quantities are inversely proportional to one another if a change in one quantity is accompanied by a reciprocal change in the other. For example, the table given below represents the ideal relationship between the pressure of a gas and its corresponding volume at constant temperature. If the pressure changes by a factor of 2 or 3, the volume changes by a factor of or . This means that the product of the two quantities is a constant.

      If we plot this relationship we will obtain a curve known as a hyperbola. As shown in the diagram, this curve will approach both the x- and the y-axis but will not intersect the axes.

      Other types of relationships include direct-squared proportions and inverse-squared proportions. Two quantities exhibit a direct-squared proportion if an increase in one causes a squared increase in the other.

      Two quantities exhibit an inverse-squared proportion if an increase in one causes a squared decrease in the other.

      1.8 Mathematical Relationships within Right Triangles

      Right triangles play an important part in the solution of physics problems. In this section, we summarize some of the more important relationships common to these triangles.

      Consider the right triangle drawn below, where h indicates the hypotenuse and x and y are the two shorter sides.

      Three well-known trigonometric relationships relate the value of acute angle θ to the lengths of the sides of the triangle:

      Physics Concepts

      In addition, the Pythagorean theorem relates the lengths of the sides of the triangle:

      Physics Concepts

      Review Questions

      Part A and B–1 Questions

      The unit of mass in the SI metric system is the

      gram

      kilogram

      newton

      meter

      A meter stick has millimeter divisions marked on it. The limit of precision of this instrument is

      0.005 m

      0.01 m

      0.0005 m

      0.001 m

      What is the order of magnitude for the measurement 72 meters per second?

      10–2

      10¹

      10²

      10⁴

      How many significant digits are contained in the measurement 500,000 kilometers?

      1

      2

      3

      6

      How many significant digits are contained in the measurement 406.200 seconds?

      6

      5

      3

      4

      How many significant digits are contained in the measurement 0.000300 volt?

      1

      2

      3

      6

      When the measurements 33.972 kilograms and 0.21 kilogram are added, the answer, to the correct number of significant digits, is

      34 kg

      34.2 kg

      34.18 kg

      34.182 kg

      When the measurements 8.14 meters and 2.1 meters are multiplied, the answer, to the correct number of significant digits, is

      17 m²

      17.0 m²

      17.09 m²

      17.094 m²

      How can the measurement 0.00567 liter be expressed to one significant digit?

      0.005 L

      0.0050 L

      0.006 L

      0.0060 L

      The approximate height of a high school physics student is

      10¹ m

      10² m

      10⁰ m

      10–2 m

      The mass of a paper clip is approximately

      1 × 10⁶ kg

      1 × 10³ kg

      1 × 10–3 kg

      1 × 10–6 kg

      What is the approximate length of a baseball bat?

      10–1 m

      10⁰ m

      10¹ m

      10² m

      The length of a dollar bill is approximately

      1.5 × 10–2 m

      1.5 × 10–1 m

      1.5 × 10¹ m

      1.5 × 10² m

      What is the approximate mass of an automobile?

      10¹ kg

      10² kg

      10³ kg

      10⁶ kg

      The reading of the ammeter in the diagram below should be recorded as

      1 A

      0.76 A

      0.55 A

      0.5 A

      Which is the most likely mass of a high school student?

      10 kg

      50 kg

      600 kg

      2500 kg

      What is the approximate thickness of this piece of paper?

      10¹ m

      10⁰ m

      10–2 m

      10–4 m

      What is the approximate mass of a pencil?

      5.0 × 10–3 kg

      5.0 × 10–1 kg

      5.0 × 10⁰ kg

      5.0 × 10¹ kg

      Answer Key

      Review Questions

      2

      3

      3

      1

      1

      3

      3

      1

      3

      3

      3

      2

      2

      3

      2

      2

      4

      1

      Chapter

      Two

      Motion in One Dimension

      Key Ideas

      Motion is the change of position in time. Displacement is the directed change of an object’s position. Velocity is the time rate of change of displacement, and acceleration is the time rate of change of velocity.

      Under conditions of constant acceleration (also known as uniform acceleration), the motion of an object is governed by a set of interrelated equations. Objects that fall freely near the surface of the Earth are uniformly accelerated by gravity.

      The motion of an object can be described by a series of motion graphs. The object’s position, velocity, and acceleration can be plotted as functions of time. These graphs can then be used to illustrate various aspects of the object’s motion at every point in time.

      KEY OBJECTIVES

      At the conclusion of this chapter you will be able to:

      Define the terms motion, distance, displacement, average velocity, speed, instantaneous velocity, and acceleration, and state their SI units.

      Solve problems involving average velocity and constant velocity.

      Distinguish between average velocity and instantaneous velocity, and relate these terms to a position-time graph.

      Solve problems involving the equations of uniformly accelerated motion.

      Solve problems involving freely falling objects.

      Interpret the data provided by motion graphs and solve problems related to them.

      2.1 Motion Defined

      How do we know when an object is in motion? If we look at the hour hand of a watch, it does not appear to be moving, yet over a period of time we see a change in its position. Therefore, a reasonable definition of motion is the change of an object’s position in time.

      2.2 Graphing an Object’S Motion

      Graphs are especially useful for analyzing an object’s motion. The position of the object is plotted along the y-axis, and the elapsed time along the x-axis. Here is a graph of very general motion in one dimension:

      The origin of the graph (0, 0) marks the reference point for both position and time. When we say zero time we mean the time when we begin the event—the time when we start the clock, so to speak. Similarly, zero position means the specific place where we begin measuring. It may be the ground or a table top or a spot on the wall. Generally, we use the letters d to represent position and t to represent time. If we know that we are measuring horizontal position, we may use the letter x, or dx in place of d. If we are measuring a vertical position we can substitute the letter y or dy in place of d.

      The dotted lines on the graph tell us where the object is at a given time. At time t1, the object is at position d1; at time t2, the object is at position d2.

      2.3 Displacement

      The displacement of an object is the change in its position and is measured in units of length (such as meters or inches). In the graph in Section 2.2, the displacement of the object between times t1 and t2 is given by the relationship d = d2 – d1. Since we are subtracting two coordinates (d1 from d2), we are not primarily concerned about the exact path taken by the object between these points. We assume that the magnitude of the displacement is given by the length of a straight line between d1 and d2 along the axis representing position.

      Displacement is known as a vector quantity because, in addition to magnitude, it has direction. (Vector quantities, which are discussed in detail in

      Chapter 4

      , are set in boldface type.) The magnitude of displacement is known as distance.

      In the first graph above, the displacement (d) from t1 to t2 is positive because its magnitude (indicated by the arrow) is measured in the positive direction; in the second graph, the displacement is negative. Whenever we are working with one-dimensional motion, we can use positive and negative signs to represent opposite directions.

      In each of the graphs, the elapsed time is given by the relationship t = t2 – t1 and is measured in units of time, such as seconds or hours. We always read the time axis from left to right since normally we do not travel backward in time.

      2.4 Velocity

      In the graph below, the slope of the straight line connecting the two points on the graph is given by the relationship d/t, and it indicates how rapidly the position of the object (d) has changed over the time interval (t). This quantity is known as the average velocity ( ) of the object and is measured in units such as meters per second (m/s) or miles per hour (mph). Mathematically, the average velocity of the object is defined by the equation

      Physics Concepts

      Velocity, like displacement, is a vector quantity because it has both magnitude and direction. As with displacement, we can use positive and negative signs to represent motion in opposite directions. The magnitude of the velocity is known as speed.

      Problem

      The position of an object is +35 meters at 2.0 seconds and is +87 meters at 15 seconds. Calculate the average velocity of the object.

      Solution

      This object is traveling in the positive direction with an average speed of 4.0 m/s.

      We use the term average velocity because we do not know exactly what is happening between the two points in question. For example, suppose we traveled by automobile due west 1000 miles and the trip took 20 hours. When we calculate our average velocity for the trip, we obtain

      Does this mean that we traveled the entire distance at a constant speed of 50 miles per hour? Not necessarily! We would probably have had to add fuel, eat, pay tolls, or engage in other activities on the trip. There might have been construction delays or reduced speed zones. All we can say with certainty is that our average speed was 50 miles per hour and our direction of travel was west.

      How then could we measure our velocity at any point on our trip—our instantaneous velocity? (This is the value that we read on the speedometer of our car.) One way would be to measure our average velocity over smaller and smaller time intervals. To accomplish this, however, we would need to use mathematical techniques that are beyond the scope of this book. The other way is to use a position versus time graph. The instantaneous velocity at any point on the graph is the slope of a line drawn tangent to the graph at that point, as shown in the diagram.

      2.5 Acceleration

      Consider the two graphs shown above. The first graph represents the position of an automobile as a function of time. Note that the graph becomes steeper (curves upward) as time passes. This occurs because the automobile’s instantaneous velocity is increasing.

      The second graph represents the instantaneous velocity of the same automobile as a function of time. Note that the graph is a straight line directed upward. This graph also shows that the instantaneous velocity of the automobile is increasing with time.

      Actually, the graphs represent the motion of the automobile from two different viewpoints: that of position (as measured by the automobile’s odometer) and that of velocity (as measured by the automobile’s speedometer).

      The slope of the velocity–time graph is given by the relationship △v/t and indicates the rate at which the velocity of the object (△v) has changed over the time interval (t). This quantity is known as the acceleration (a) of the object and is measured in units such as meters per second² (m/s²). Since the graph is a straight line, the acceleration in this case is constant or uniform. Mathematically, the uniform acceleration of the object is defined by the equation

      Physics Concepts

      Acceleration is also a vector quantity because it has both magnitude and direction. A positive acceleration means that the velocity of an object is becoming more positive with time; a negative acceleration, that the velocity of the object is becoming more negative with time.

      Problem

      The velocity of an object is +47 meters per second at 3.0 seconds and is +65 meters per second at 12.0 seconds. Calculate the acceleration of the object.

      Solution

      This object’s velocity is becoming more positive by 2.0 m/s each second. Since the magnitude of the velocity is increasing, the object is speeding up. The table below shows how this occurs.

      2.6 The Equations of Uniformly Accelerated Motion

      In the problem we solved in Section 2.5, an object accelerates uniformly, at 2.0 meters per second², from 47 meters per second to 65 meters per second in 9.0 seconds. There is a great deal of additional information about the object we might wish to learn. For example:

      What is the average velocity of the object over 9.0 seconds?

      What is the displacement of the object at the end of 9.0 seconds?

      What is the instantaneous velocity of the object at any given time (at 6.0 s, for example)?

      To solve these problems, we use a set of five equations that describe the motion of an object undergoing uniform acceleration. In each of these equations, we use the subscripts i (for initial value), f (for final value), and  to indicate average velocity. Your textbook or teacher may use different subscripts or notation, but they all yield the same results. The equations for uniformly accelerated motion are as follows:

      Physics Concepts

      *Note that equation 2 does not appear on the Reference Tables.

      Equation 1 is the definition of average velocity. Equation 2 tells us that, under uniform acceleration, the average velocity lies midway between the initial and final velocities. It should be noted that although this equation is not included on the Reference Table, it is essential to solving problems on the exam. Equation 3 is just the definition of acceleration (a = △v/t) rearranged in a more convenient form for solving problems. Equations 4 and 5 are relationships that have been derived from the first three equations.

      Which equation should you use to solve a particular problem? The answer depends on the data you are given. In the problem we have been considering, an object with an initial velocity of 47.0 meters per second accelerates uniformly at 2.0 meters per second² for 9.0 seconds. Suppose we wish to calculate the displacement of this object at the end of 9.0 seconds. We list the variables that are part of the problem, along with their values:

      If we examine the list of equations given above, we see that equation 4 contains the four variables that form the basis of our problem. We solve the problem by substituting the values and calculating the answer:

      2.7 Freely Falling Objects

      The following table represents the motion of an object falling from rest near the surface of the Earth when air resistance is ignored.

      If we analyze this motion, we see that the speed of the object increases uniformly by 9.8 meters per second for each second of travel. This suggests that the object is subject to a constant acceleration of 9.8 meters per second². The distance traveled by the object over time verifies that the object’s acceleration is constant.

      Problem

      How does the distance traveled by the object described above over time verify that the object’s acceleration is constant?

      Solution

      The distance traveled by an object under uniform acceleration is given by the equation:

      Since the object starts from rest, this equation reduces to:

      If we substitute each of the corresponding values of d and t given in the table, we find that the acceleration in each case is 9.8 m/s².

      If we were to investigate further, we would find that all objects falling near the surface of the Earth experience a constant acceleration of 9.8 meters per second² if air resistance is ignored. This phenomenon is due to the presence of gravity, which affects each and every object. If we were to travel to the Moon, we would find that all objects also fall to its surface with a uniform acceleration. However, this acceleration is only 1.6 meters per second² because of the Moon’s weaker gravitational forces.

      Since free fall involves uniform acceleration, the five equations we developed in Section 2.6 can be used to solve all free-fall problems. We need only remember that objects can move up as well as down in the presence of gravity. Therefore, we assign the up direction as positive and the down direction as negative. Since gravity always points downward (i.e., toward the Earth), its value is taken to be –9.8 meters per second². Gravitational acceleration is denoted by the lowercase letter g.

      Problem

      An object is dropped from rest from a height of 49 meters.

      How long does the object take to hit the ground?

      What is its speed as it hits the ground?

      Solution

      Since the initial velocity is zero, we can use the equation

      The displacement is –49 m (since we measure the distance in a downward direction), and the acceleration due to gravity is –9.8 m/s² (since gravity points downward). Therefore:

      We can use the relationship vf = vi + a · t, which reduces to vf = a · t since the initial velocity is zero. Then

      2.8 Motion Graphs Revisited

      Throughout this chapter we have used motion graphs as aids to understanding the concept of motion. In this section, we take a more detailed look at these graphs and the information they can provide. We shall examine three types of graphs: position–time, velocity–time, and acceleration–time.

      Position–Time Graphs

      The following graph illustrates the position of an object as a function of time.

      We recall from Section 2.2 that zero time represents the start of an event and that zero position represents some arbitrary reference point. We have divided the position–time graph into five sections: A, B, C, D, and E. Since each section is a straight-line segment, the velocity within each section is constant and the acceleration over each section is zero. We will learn how to interpret this graph by considering the following problem.

      Problem

      What is the displacement over each section of the graph?

      What is the velocity over each section of the graph?

      What is the displacement over the entire trip (0–10 seconds)?

      What is the average velocity over the entire trip (0–10 seconds)?

      Solution

      To calculate the displacement (d) we subtract the initial position from the final position.

      The displacement over section A is 0 m because the object has not changed its position.

      The displacement over section B is +2 m because the object has changed its position from +2 m to +4 m.

      The displacement over section C is –1 m because the object has changed its position from +4 m to +3 m.

      The displacement over section D is 0 m because the object has not changed its position.

      The displacement over section E is –4 m because the object has changed its position from +3 m to –1 m.

      The velocity over each section is found by dividing the displacement by the elapsed time.

      The velocity over section A is 0 m/s (0 m/3 s).

      The velocity over section B is +2 m/s (+2 m/1 s).

      The velocity over section C is –1 m/s (–1 m/1 s).

      The velocity over section D is 0 m/s (0 m/2 s).

      The velocity over section E is –1.3 m/s (–4 m/3 s).

      The displacement over the entire trip is –3 m because the object changed position from +2 m at t = 0 s to –1 m at t = 10 s.

      The average velocity over the entire trip is –0.3 m/s (–3 m/10 s).

      It is important to note that the slope of a position–time graph or a d vs. t graph is equal to the velocity of the object. Assuming as in the example above that there are straight-line segments, the velocity is constant for each segment. A curved graph or a graph with a changing slope would indicate changing velocity or the presence of acceleration.

      Velocity–Time and Acceleration–Time Graphs

      The graph below illustrates the velocity of an object as a function of time.

      The values on the y-axis represent the instantaneous velocities of the object at the times marked on the x-axis. It is as though we were looking at a car’s speedometer at various times. We have divided the graph into six sections: A, B, C, D, E, and F. Since each section is a straight-line segment, the object’s acceleration within each section is constant. We will learn how to interpret this graph by considering the following problem.

      Problem

      What is the average velocity within each section of the graph?

      What is the acceleration within each section of the graph?

      When does the object come to rest?

      When does the object reverse the direction of its motion?

      What is the displacement within each section of the graph?

      What is the displacement over the entire trip (0–14 seconds)?

      What is the average velocity over the entire trip (0–14 seconds)?

      What is the shape of the corresponding acceleration versus time graph?

      Solution

      The average velocity for each section is calculated by finding the midpoint of each line, that is, by adding the initial and final velocities within each section and dividing this sum by 2:

      We must remember to take both positive and negative signs into account when we add the velocities. For example, in section E, the initial velocity is +3.0 m/s and the final velocity is –2.0 m/s. Therefore, the average velocity is

      The table summarizes the results of the calculations for the six sections:

      The acceleration within each section is found by calculating the slope of each of the lines:

      For example, in section A, vi = 1.0 m/s, vf = 2.0 m/s, and △t = 2.0 s. The acceleration is calculated to be

      The table summarizes the results of the calculations for the six sections:

      The object comes to rest when its velocity is zero. Referring to the graph, we estimate that zero velocity corresponds to an approximate time of 9.5 s. (Actually, the time is 9.4 s; we could have calculated this value by using the graph and the equation vf = vi + a · t.)

      Before 9.4 s, the velocity of the object is always positive; after 9.4 s, its velocity is negative. Therefore the object reverses the direction of its motion at 9.4 s.

      There are two ways to calculate the displacement of the object.

      First, we could multiply the average velocity of each section by the time elapsed in that section. For example, in section E the average velocity is +0.5 m/s and the elapsed time is 4.0 s. Therefore, the displacement of the object is +2.0 m (+0.5 m/s · 2.0 s). A positive displacement means that the object traveled the distance in the positive direction.

      Second, we could calculate the displacement by measuring the area between the section line and the x-axis. (In mathematics, this is known as calculating the area under the curve.). We will use this method to calculate the displacement for section F of the graph, as follows.

      The shaded area is defined by a rectangle whose length is 3.0 s and whose height is –2.0 m/s (this value is negative because section F lies under the x-axis). The area is the product of these two values (3.0 s · –2.0 m/s), that is, –6.0 m.

      The table summarizes the results of the calculations for the six sections:

      The displacement over the entire trip is found by adding the displacements for all the sections:

      (Refer to the table above.)

      The average velocity for the entire trip is found by dividingdtotal by the total time (14 s):

      The acceleration versus time graph for this object is constructed by referring to the accelerations over all of the sections. (See the table on page 25 for part 2 of this problem.)

      Note that the accelerations are drawn as straight-line segments within each section, but they are not connected between sections. The reason is that the change in velocity between sections is so abrupt that the acceleration cannot be calculated accurately. (Contrast the velocity versus time graph shown in part 5 of this problem.)

      The first point of importance to note is that the slope of a velocity–time graph or a v vs. t graph is equal to the acceleration of the object. Assuming as in the graph on page 26 that there are straight-line segments, the acceleration is constant for each segment. A curved graph or a graph with a changing slope would indicate changing acceleration.

      The second point of importance to note is that the area under the curve of a velocity–time graph is equal to the displacement of the object.

      Lastly, in an acceleration–time graph or an a vs. t graph, it is important to note that the area under the curve is equal to the object’s change in velocity.

      Review Questions

      indicates material that is not part of the core curriculum.

      Part A and B–1 Questions

      star

      A flashing light of constant 0.20-second period is situated on a lab cart. The diagram below represents a photograph of the light as the cart moves across a tabletop.

      How much time elapsed as the cart moved from position A to position B?

      1.0 s

      5.0 s

      0.80 s

      4.0 s

      Approximately how much time does it take light to travel from the Sun to Earth?

      2.00 × 10–³ s

      1.28 × 10⁰ s

      5.00 × 10² s

      4.50 × 10¹⁹ s

      An object travels for 8.00 seconds with an average speed of 160. meters per second. The distance traveled by the object is

      20.0 m

      200. m

      1280 m

      2560 m

      A blinking light of constant period is situated on a lab cart. Which diagram best represents a photograph of the light as the cart moves with constant velocity?

      A car is accelerated at 4.0 meters per second² from rest. The car will reach a speed of 28 meters per second at the end of

      3.5 s

      7.0 s

      14 s

      24 s

      Base your answers to questions 6 through 8 on the information and diagram below. The diagram represents a block sliding along a frictionless surface between points A and G.

      As the block moves from point A to point B, the speed of the block will be

      decreasing

      increasing

      constant, but not zero

      zero

      Which expression represents the magnitude of the block’s acceleration as it moves from point C to point D?

      Which formula represents the velocity of the block as it moves along the horizontal surface from point E to point F?

      An object that is originally moving at a speed of 20. meters per second accelerates uniformly for 5.0 seconds to a final speed of 50. meters per second. What is the acceleration of the object?

      14 m/s²

      10. m/s²

      6.0 m/s²

      4.0 m/s²

      A block starting from rest slides down the length of an 18-meter plank with a uniform acceleration of 4.0 meters per second². How long does the block take to reach the bottom?

      4.5 s

      2.0 s

      3.0 s

      9.0 s

      Base your answers to questions 11 through 15 on the information below.

      A toy projectile is fired from the ground vertically upward with an initial velocity of +29 meters per second. The projectile arrives at its maximum altitude in 3.0 seconds. [Neglect air resistance.]

      The greatest height the projectile reaches is approximately

      23 m

      44 m

      87 m

      260 m

      What is the velocity of the projectile when it hits the ground?

      0. m/s

      –9.8 m/s

      –29 m/s

      +29 m/s

      What is the displacement of the projectile from the time it left the ground until it returned to the ground?

      0. m

      9.8 m

      44 m

      88 m

      Which graph best represents the relationship between velocity (v) and time (t) for the projectile?

      As the projectile rises and then falls back to the ground, its acceleration

      decreases, then increases

      increases, then decreases

      increases, only

      remains the same

      A freely falling object near the Earth’s surface has a constant

      velocity of –1.00 m/s

      velocity of –9.81 m/s

      acceleration of –1.00 m/s²

      acceleration of –9.81 m/s²

      The speed of an object undergoing constant acceleration increases from 8.0 meters per second to 16.0 meters per second in 10. seconds. How far does the object travel during the 10. seconds?

      3.6 × 10² m

      1.6 × 10² m

      1.2 × 10² m

      8.0 × 10¹ m

      A 2.0-kilogram stone that is dropped from the roof of a building takes 4.0 seconds to reach the ground. Neglecting air resistance, the maximum speed of the stone will be approximately

      8.0 m/s

      9.8 m/s

      29 m/s

      39 m/s

      Base your answers to questions 19 and 20 on the diagram below, which shows a 1-kilogram aluminum sphere and a 3-kilogram brass sphere, both having the same diameter and both at the same height above the ground. Both spheres are allowed to fall freely. [Neglect air resistance.]

      Both spheres are released at the same instant. They will reach the ground at

      the same time but with different speeds

      the same time with the same speeds

      different times but with the same speeds

      different times and with different speeds

      If the spheres are 19.6

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