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physics For Scientists and Engineers An Interactive Approach Second Edition physics For Scientists and Engineers An Interactive Approach Second Edition Robert Hawkes Mount Allison University Javed Iqbal University of British Columbia Firas Mansour University of Waterloo Marina Milner-Bolotin University of British Columbia Peter Williams Acadia University This is an electronic version of the print textbook Due to electronic rights restrictions, some third party content may be suppressed The publisher reserves the right to remove content from this title at any time if subsequent rights restrictions require it For valuable information on pricing, previous editions, changes to current editions, and alternate formats, please visit www.nelson.com to search by ISBN#, author, title, or keyword for materials in your areas of interest Important Notice: Media content referenced within the product description or the product text may not be available in the eBook version Physics for Scientists and Engineers: An Interactive Approach, Second Edition by Robert Hawkes, Javed Iqbal, Firas Mansour, Marina Milner-Bolotin, and Peter Williams VP, Product Solutions, K–20: Claudine O’Donnell Production Project Manager: Wendy Yano Interior Design: Brian Malloy Senior Publisher: Paul Fam Production Service: Cenveo Publisher Services Cover Design: Courtney Hellam Marketing Manager: Kimberley Carruthers Copy Editor: Julia Cochrane Technical Reviewer: Simon Friesen Karim Jaffer Anna Kiefte Kamal Mroue Proofreader: Subash.J Cover Image: Yuichi Takasaka/ Blue Moon Promotions Content Manager: Suzanne Simpson Millar Design Director: Ken Phipps Photo and Permissions Researcher: Kristiina Paul Higher Education Design PM: Pamela Johnston Copyright © 2019, 2014 by Nelson Education Ltd All rights reserved No part of this work covered by the copyright herein may be reproduced, transcribed, or used in any form or by any means—graphic, electronic, or mechanical, including photocopying, recording, taping, Web distribution, or information storage and retrieval systems— without the written permission of the publisher Printed and bound in Canada 20 19 18 17 For more information contact Nelson Education Ltd., 1120 Birchmount Road, Toronto, Ontario, M1K 5G4 Or you can visit our Internet site at nelson.com Cognero and Full-Circle Assessment are registered trademarks of Madeira Station LLC Möbius is a trademark of Waterloo Maple Inc Indexer: Robert A Saigh For permission to use material from this text or product, submit all requests online at cengage.com/permissions Further questions about permissions can be emailed to permissionrequest@cengage.com Every effort has been made to trace ownership of all copyrighted material and to secure permission from copyright holders In the event of any question arising as to the use of any material, we will be pleased to make the necessary corrections in future printings Art Coordinator: Suzanne Peden Illustrator(s): Crowle Art Group, Cenveo Publisher Services Compositor: Cenveo Publisher Services Library and Archives Canada Cataloguing in Publication Data Hawkes, Robert Lewis, 1951–, author Physics for scientists and engineers : an interactive approach / Robert Hawkes, Mount Allison University, Javed Iqbal, University of British Columbia, Firas Mansour, University of Waterloo, Marina Milner-Bolotin, University of British Columbia, Peter Williams, Acadia University — Second edition Includes index Issued in print and electronic formats ISBN 978-0-17-658719-2 (hardcover).—ISBN 978-0-17680985-0 (PDF) Physics—Textbooks. Textbooks. I Iqbal, Javed, 1953–, author II Mansour, Firas, author III Milner-Bolotin, Marina, author IV Williams, Peter (Peter J.), 1959–, author V Title QC23.2.H38 2018 530 C2017-906981-0 C2017-906982-9 ISBN-13: 978-0-17-658719-2 ISBN-10: 0-17-658719-5 Brief Table of Contents Preface xvi About the Authors xxv Text Walkthrough xxvii Acknowledgments xxx Section 1 Mechanics Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter Chapter 10 Chapter 11 Chapter 12 Introduction to Physics Scalars and Vectors 31 Motion in One Dimension 55 Motion in Two and Three Dimensions 111 Forces and Motion 141 Work and Energy 191 Linear Momentum, Collisions, and Systems of Particles 223 Rotational Kinematics and Dynamics 265 Rolling Motion 311 Equilibrium and Elasticity 345 Gravitation 383 Fluids 421 Section Waves and Oscillations 465 Chapter 13 Oscillations 465 Chapter 14 Waves 507 Chapter 15 Sound and Interference 561 Section Thermodynamics 591 Chapter 16 Temperature and the Zeroth Law of Thermodynamics 591 Chapter 17 Heat, Work, and the First Law of Thermodynamics 613 Chapter 18 Heat Engines and the Second Law of Thermodynamics 635 Section Electricity, Magnetism, and Optics Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 657 Electric Fields and Forces 657 Gauss’s Law 693 Electrical Potential Energy and Electric Potential 735 Capacitance 773 Electric Current and Fundamentals of DC Circuits 801 Magnetic Fields and Magnetic Forces 839 Electromagnetic Induction 893 Alternating Current Circuits 937 Electromagnetic Waves and Maxwell’s Equations 957 Geometric Optics 987 Physical Optics 1027 v NEL Section Modern Physics Chapter 30 Chapter 31 Chapter 32 Chapter 33 Chapter 34 Chapter 35 1057 Relativity 1057 Fundamental Discoveries of Modern Physics 1099 Introduction to Quantum Mechanics 1123 Introduction to Solid-State Physics 1163 Introduction to Nuclear Physics 1187 Introduction to Particle Physics 1227 A-1 Appendix A Answers to Selected Problems Appendix B SI Units and Prefixes B-1 Appendix C Geometry and Trigonometry C-1 Appendix D Key Calculus Ideas D-1 Appendix E Useful Mathematic Formulas and Mathematical Symbols E-1 Appendix F Periodic Table F-1 Index I-1 vi Brief Table of Contents NEL Table of Contents Preface xvi About the Authors xxv Text Walkthrough xxvii Acknowledgments xxx Section Mechanics 1 Chapter 1 Introduction to Physics What Is Physics? Experiments, Measurement, and Uncertainties Mean, Standard Deviation, and SDOM Significant Digits Scientific Notation SI Units 12 14 15 Base SI Units 15 Other Units 15 SI Prefixes 16 Writing SI 16 Dimensional Analysis 17 Unit Conversion 19 Approximations in Physics 20 Fermi Problems 21 What Is a Fermi Problem? 21 Probability 22 Advice for Learning Physics 23 Key Concepts and Relationships 24 Applications 24 Key Terms 24 Questions 24 Problems by Section 25 Comprehensive Problems 28 29 Data-Rich Problem Open Problems 30 Chapter 2 Scalars and Vectors 31 Definitions of Scalars and Vectors Vector Addition: Geometric and Algebraic Approaches 32 The Geometric Addition of Vectors Algebraic Addition of Vectors Cartesian Vector Notation The Dot Product of Two Vectors The Dot Product and Unit Vectors The Cross Product of Vectors The Cross Product and Unit Vectors 34 34 35 39 42 43 45 46 Key Concepts and Relationships 48 Applications 49 Key Terms 49 Questions 49 Problems by Section Comprehensive Problems Chapter 3 Motion in One Dimension 50 52 55 Distance and Displacement Speed and Velocity 56 59 Motion Diagrams 59 Average Speed and Average Velocity 59 Instantaneous Velocity 62 Acceleration 66 Instantaneous Acceleration 67 Acceleration Due to Gravity 69 Mathematical Description of One-Dimensional Motion with Constant Acceleration 72 Velocity as a Function of Time for Objects Moving with Constant Acceleration Position as a Function of Time for Objects Moving with Constant Acceleration Analyzing the Relationships between x(t), y(t), and a(t) Plots 72 73 76 Applicability of the Principle of Graphical Integration 80 Free Fall Relative Motion in One Dimension Reference Frames Relative Velocity Derivation of the General Kinematics Equations for Relative Motion 82 87 87 87 89 90 General Framework for Kinematics Equations 90 Key Concepts and Relationships 93 Applications 94 Key Terms 94 Questions 94 Problems by Section 98 Comprehensive Problems 105 Data-Rich Problems 108 Open Problems 108 Calculus of Kinematics Chapter 4 Motion in Two and Three Dimensions Position, Velocity, and Acceleration Projectile Motion A Graphical Vector Perspective Projectile Motion in Component Form Circular Motion Uniform Circular Motion Non-uniform Circular Motion Relative Motion in Two and Three Dimensions 111 112 115 115 118 124 124 127 128 Formal Development of the Relative Motion Equations in Two Dimensions Relative Acceleration 128 131 Key Concepts and Relationships 132 Applications 133 Key Terms 133 vii NEL Questions 133 135 Problems by Section Comprehensive Problems 137 Data-Rich Problem 140 Chapter 5 Forces and Motion 141 Dynamics and Forces Mass and the Force of Gravity Newton’s Law of Motion 142 143 146 Newton’s First Law 146 Newton’s Second Law 146 Net Force and Direction of Motion 148 Newton’s Third Law 148 151 Applying Newton’s Laws Multiple Connected Objects 158 161 Component-Free Solutions Friction 162 Spring Forces and Hooke’s Law 168 171 Fundamental and Non-fundamental Forces Uniform Circular Motion 172 Reference Frames and Fictitious Forces 176 Momentum and Newton’s Second Law 178 Key Concepts and Relationships 180 Applications 181 Key Terms 181 Questions 181 Problems by Section 184 Comprehensive Problems 188 Data-Rich Problem 190 Open Problem 190 Chapter 6 Work and Energy 191 What Is Energy? Work Done by a Constant Force in One Dimension 192 Units for Work Work Done by a Constant Force in Two and Three Dimensions Work Done by Variable Forces Graphical Representation of Work Work Done by a Spring Work Done by the External Agent Kinetic Energy—The Work–Energy Theorem Total or Net Work The Work–Energy Theorem for Variable Forces Conservative Forces and Potential Energy Potential Energy Gravitational Potential Energy near Earth’s Surface Elastic Potential Energy 193 193 194 198 198 200 201 202 203 208 209 209 210 210 Conservation of Mechanical Energy 212 Force from Potential Energy 217 Energy Diagrams 219 Power 220 Key Concepts and Relationships 221 Applications 222 Key Terms 222 Questions 222 Problems by Section 224 Comprehensive Problems 228 Data-Rich Problem 232 Open Problem 232 viii Table of Contents Chapter 7 Linear Momentum, Collisions, and Systems of Particles Linear Momentum Momentum and Kinetic Energy 233 234 235 Rate of Change of Linear Momentum and Newton’s Laws 236 Impulse 237 The Force of Impact 239 Linear Approximation for the Force of Impact 239 241 Systems of Particles and Centre of Mass Systems of Particles and Conservation of Momentum 244 Internal Forces and Systems of Particles 244 Defining the System 245 Collisions 247 Inelastic Collisions 247 Elastic Collisions 251 Conservation of Momentum 251 253 Variable Mass and Rocket Propulsion Key Concepts and Relationships 256 Applications 257 Key Terms 257 Questions 257 Problems by Section 259 261 Comprehensive Problems Data-Rich Problem 263 Open Problem 263 Chapter 8 Rotational Kinematics and Dynamics 265 Angular Variables 266 266 Kinematic Equations for Rotation 268 Constant Acceleration 268 Torque 270 What Is Torque? 270 What Does Torque Depend Upon? 270 Pivot and Axis of Rotation 271 The Force 271 The Distance 271 The Angle 272 The Perpendicular Component of the Force 272 From Translation to Rotation The Perpendicular Component of the Distance: The Moment Arm Torque Has Direction Torque Is a Vector Quantity “Curl” Right-Hand Rule for Torque Direction “Three-Finger” Right-Hand Rule for Torque Direction Torque: Vector Components as Vectors Connection to the Right-Hand Rule Moment of Inertia of a Point Mass Moment of Inertia of a Point Mass Systems of Particles and Rigid Bodies A System of Point Masses Moment of Inertia for Continuous Objects A Thin Ring A Solid Disk Moment of Inertia for Composite Objects The Parallel-Axis Theorem The Perpendicular-Axis Theorem 272 273 274 274 275 276 277 277 278 279 279 280 281 282 284 285 287 NEL 36 ★★ Solve the following vector equations: (a) F i^ j^ "2 i^ F j^ F k^ 5k^ x y z (b) i^ j^ Fx i^ 2Fy j^ Fzk^ 3k^ Section 2-4 The Dot Product of Two Vectors> Mechanics 37 ★ Calculate the dot product of vectors A 3, 4, 252 and B^ 12, 22, 42 What does this product represent? 38 ★ Solve the following vector equation to find Fz: 13 i^ j^ Fzk^ ~ 14 i^ j^ Fzk^ > > 39 ★ Two vectors, A and B , have magnitudes 10 units and units, respectively, and are located in the xy-plane The angle between them is 20° Find the dot product of these two vectors What does this dot product represent? 40 ★ (a) Without doing any calculations, rank the values of> the > following > > in > > > dot > >products > Figure > 2-25: > A ~ B , B ~> C ,> A ~ C , >B ~ A , C ~ B , and C ~ A for vectors A , B , and C Justify your ranking > > > > (b) Now A ~B, B ~C, > > > the> dot products > > calculate > > A ~ C , B ~ A , C ~ B , and C ~ A , and check if the values you obtained support your ranking in (a) z A(]2, 0, 3) x C (]5, 0, 0) y B (0, 4, 0) Figure 2-25 Problem 40 41 ★★ Find a unit vector located in the xy-plane and per> pendicular to the vector A i^ 2 j^ 5k^ How many answers you have? Explain why 42 ★★ Find the projection of the 2-D force vector > F 3.00, 4.002 N onto the line y 22x (Hint: Consider how the dot product can be used to calculate the projection of a vector onto a given direction.) 43 ★★★ Suggest at least two different > methods> to find the A and B , where > angle between two vectors, > A 22 i^ j^ 5k^ and B 22 i^ j^ 5k^ Explain each method and compare the angles you found using each one Section 2-5 The Cross Product of Vectors 44 ★★ Use two different methods to >calculate the cross (vector) product of the vectors A 3, ,252 and > B 2, 22, 42 Note that to find the cross product one must find both the magnitude and the direction of the cross product vector Based on your calculations, which method you prefer and> why? > 45 ★ Two vectors, A and B , have magnitudes of 10 units and units, respectively, and are located in the xz-plane The angle between them is 20° Find the magnitude and direction of the vector product of these two vectors 52 SECTION 1 | MECHANICS > > 46 ★★ Three vectors, A 14, 3, 12 , B 122, 4, 222 , and > C 121, 23, 52 , form the sides of a parallelogram prism (a) Find the volume of the prism (b) Find the surface area > > of the prism > 47 ★★ For the vectors A , B ,> and> C> in Figure > > > 2-24, > > calculate A 3B , B 3C , A 3C , B 3A , > > the cross > products > C B , and C A , using two different methods: (a) Use Cartesian vector notation to calculate the cross product using its determinant form (b) Find the angles between the corresponding vectors and calculate the cross product using the original definition of the cross product—Equation 2-26 (c) Compare the results in (a) and (b) What does your comparison tell you? 48 ★★ Solve the following vector equations: (a) 2k^ a Fx i^ j^ b Fy i^ "2 j^ (b) j^ 1Fx i^ 5k^ Fzk^ "3Fx i^ 49 ★★ Use the determinant form for calculating > cross> A and B product to find the cross product of vectors > > (A B ) in each of the following: > (a) A i^ j^ 5k^ > B 24 i^ j^ 4k^ > (b) A j^ 5k^ > B i^ 4k^ > (c) A i^ j^ 5k^ > B 24 i^ j^ 10k^ Could you have predicted some of the results without calculations? Explain 50 ★★ Determine the angle between the diagonal of a cube and each one of the edges that shares a vertex with the diagonal COMPREHENSIVE PROBLEMS 51 ★★ An orthogonal coordinate system, xy, undergoes the following transformation: it moves parallel to itself such that the new location of its origin is O9(4, 5) It is then rotated 60° counterclockwise The new coordinate system is x9y9 (a) Derive an expression for the coordinates of an arbitrary point under this transformation Express how the new coordinates depend on the old coordinates (b) Derive an expression that describes how vector components change under this transformation Express how the new vector components depend on the old vector components > 52 ★★ > For the three vectors > defined as A (Ax, Ay, Az), B (Bx, By, Bz), and C (Cx, Cy, Cz), prove that Ax > > > A ~ 1B C Bx Cx Ay By Cy Az Bz Cz NEL NEL where u is the angle between sides A and B of the triangle Prove the law of cosines using the triangle construction method of vector addition and the fact that the square of the magnitude of a vector is equal to the dot product of the vector with itself 63 ★★★ Prove the formula of the > > for >the >differentiation > > scalar product: d 1A ~ B dA ~ B A ~ 1dB Use your > > > proof to find an expression for d 1y ~ y , where y is the velocity of a particle 64 ★★★ The speed of a particle moving in the xy-plane can > be described as follows: y y0x i^ 1y0y gt j^ , where g is a positive constant called the acceleration due to gravity, t is time, and y0x and y0y are the components of the velocity on the x- and y-axes when t For Earth, g ≈ 9.81 m/s2 (a) Find the expression for the speed of the particle as a function of time (b) Draw a diagram that illustrates how the speed of the particle changes with time (c) Determine the time when the speed of the particle is the smallest (d) This mathematical model is appropriate to describe a common, everyday phenomenon Suggest what that phenomenon might be 65 ★★★ Newton’s second law of motion states that the acceleration of a moving object is proportional to the net force acting on it> and> inversely proportional to the object’s a F Fnet S mass: a 5 There are three forces acting on a m m particle: > F1 13.00 N2 i^ 12.00 N2 j^ 15.00 N2 k^ ; > F2 121.00 N2 i^ 14.00 N2 j^ 12.00 N2 k^ ; > F 122.00 N2 i^ 15.00 N2 j^ 17.00 N2 k^ The particle’s mass is 0.100 kg (a) Find the magnitude of the particle’s acceleration (b) Find the coordinate direction angles of its acceleration vector (c) Write the expression for the particle’s acceleration in Cartesian notation (d) Determine the projections of the particle’s acceleration onto each of the coordinate planes xy, xz, and yz > 66 ★★★ (a) Three vectors are described as A 2, ,252 , > > B 121, 4, 222 , and C 11, 2, 12 Find the following products, if possible: > > (i) A B > > > (ii) A B C > > > (iii) 2A B C > > > (iv) 1A B ~ C > > > (v) 1A B C > > > (vi) 1A ~ B C (b) Explain why some products are vectors, while others are scalars If you think it is impossible to calculate a product, explain why Chapter 2 | Scalars and Vectors 53 Mechanics > > 53 ★★ Prove that> for any two non-zero vectors A and B , > both A > > and B are perpendicular to their cross product A 3B 54 ★★ Prove the following relationships: > > > > > > (a) 1A B C B A C > > > > > > > > > (b) 1A B C 2C A B C B A > > > > > > (c) 1A B C A B C > > > > > > (d) 1A B C A C B 55 ★★ Prove that the algebraic expression for the cross product of two vectors using the determinant form (Equation 2-34) is equivalent to the cross product calculation using the cross product definition (Equation 2-26) (Hint: Find the magnitude of each vector, and then find the angle between them using the dot product.) 56 ★★ Prove that the algebraic expression for the cross product of two vectors using the determinant form (Equation 2-34) is equivalent to the cross product calculation using Cartesian notation and the expressions for the cross product for the unit vectors (Equations 2-30 and 2-32) 57 ★ (a) Find the cross product > > of the two vectors A i^ j^ 5k^ and B 26 i^ j^ 10k^ using the determinant form, Equation 2-34 (b) Explain how you could have predicted this result without doing the calculation 58 ★★ Two straight lines, a and b, are oriented in the xyplane and described as a: y m1x b1 and b: y m2x b2, where neither m1 nor m2 is Prove that, to be perpendicular to each other, the following must be true: m1m2 21 (Hint: Prove that any two arbitrary vectors directed along these lines are perpendicular to each other.) > 59 ★★ The work (WF) by a constant force F on an object can be described as a scalar> product of the force and the object’s disS placement, WF F ~ Dr , and is measured in joules when the force is measured in> newtons and the displacement in metres ^ A constant force F 3.00 N i^ 1 4.00 N j^ 5.00 > N2 k > > acts on an object moving along the line > r a tb , with S a 2.00 i^ 3.00 j^ 4.00k^ and b 3.00 i^ 2.00 j^ 4.00k^ (t is a parameter represented by a real number: t H R) (a) Prove that points A(5.00 m, 25.00 m, 0.00 m) and B(8.00 m, 27.00 m, 4.00 m) are located along this line > (b) Find the work done on the object by force F while the object moved from point A(5.00 m, 25.00 m, 0.00 m) to point A(5.00 m, 25.00 m, 0.00 m) along > > > the line r a tb 60 ★★★ A triangle is built on the diameter of a circle such that all three of its vertices lie on the circle’s circumference Prove that the triangle is a right triangle (Hint: Express the sides of the triangle as vectors, and use a scalar product to prove that these vectors are orthogonal.) > > > > > > 61 ★★★ Show that A B A B 2 1A ~ B 2 62 ★★★ The law of cosines states that for any triangle, the following holds true: C 2 A2 B2 2AB cos u, Mechanics 67 ★★★ A trigonometric identity involving the cosine function can be proven using the dot product operation Imagine a unit circle, which is a circle whose radius is unit Unit vectors u^ and y^ form angles a and b, respectively, with the horizontal axis Use the unit circle to prove that cos (b a) cos a cos b sin a >sin b 68 ★★★ Three vectors are described as A 2, 23, 262 , > > B 1, 24, 22 , and C 1, 22, 32 > Find all> the vectors that are perpendicular to vectors > A and B and have the same magnitude as vector C 69 ★★ The carbon tetrachloride molecule, CCl4 (Figure 2-26(a)), has the shape of a tetrahedron Chlorine atoms are located at the vertices of the tetrahedron, and the carbon atom is located in the centre The bond angle (u) is an angle formed by three connected atoms: Cl2C2Cl 54 SECTION 1 | MECHANICS A tetrahedron is formed by connecting alternating vertices of a cube, as shown in Figure 2-26(b) Find the bond angle (u) of the carbon tetrachloride molecule (a) (b) Cl u Cl Cl C Cl Cl C Cl Cl Cl Figure 2-26 Problem 69 (a) A carbon tetrachloride molecule (b) Atom locations in the carbon tetrachloride molecule NEL Chapter Learning Objectives Motion in One Dimension When you have completed this chapter, you should be able to LO5 Construct and analyze displacement, velocity, and acceleration time plots LO1 Define, calculate, and distinguish between distance and displacement LO6 Use kinematics equations to analyze the motion of free-falling objects LO2 Define, calculate, and distinguish between average and instantaneous speed and velocity LO7 Describe relative motion in one dimension qualitatively and quantitatively using the kinematics equations LO3 Define and calculate acceleration, and distinguish between average and instantaneous acceleration LO8 Use calculus to analyze the motion of objects with constant and variable acceleration LO4 Develop and apply the kinematics equations for motion with constant acceleration he 100 m dash is a sprint race in track and field competitions It is one of the most popular and prestigious events in the world of athletics It has been contested since the first Summer Olympic Games in 1896 for men and the ninth Summer Olympic Games in 1928 for women The first winner of a modern Olympic 100 m race was American Francis Lane, whose time in 1896 was 12.2 s The current male world champion in the 100 m dash is the legendary Jamaican sprinter Usain Bolt (Figure 3-1), who has improved the world record three times from 9.74 s to 9.58 s Since 1969, the men’s 100 m dash record has been revised 13 times, from 9.95 s to 9.58 s (an increase in performance of 3.72%) Bolt’s 2009 record-breaking margin from 9.69 s (his own previous world record) to 9.58 s is the highest since the start of fully automatic time measurements in 1977 The fastest ever woman sprinter was American Florence Griffith-Joyner (1959–1998), whose 1988 record of 10.49 s hasn’t been broken to this day Since automatic time measurements allow for increased accuracy, athletes fight for every split second The continuous improvement in their performance would not have been possible without the detailed analysis of every parameter of an athlete’s motion, such as sprinting speed, acceleration, and the length and frequency of their stride Coaches and athletes combine their knowledge of kinematics with their knowledge of human kinetics to optimize PCN Photography/Alamy Stock Photo T Figure 3-1 World 100 m dash champion Usain Bolt sprints during the 2012 Summer Olympics in London performance and improve the world record (Krzysztof, M., & Mero, A 2013 A kinematics analysis of the three best 100 m performances ever Journal of Human Kinetics, volume 36, 149–160) 55 NEL 3-1 Distance and Displacement Mechanics Kinematics is the study of motion, which allows us to predict how an object will move, where it will be at a certain time, when it will arrive at a certain location, or how long it will take to cover a certain distance In other words, in kinematics, we analyze how an object’s position, velocity, and acceleration relate to one another, and how they change with time However, in kinematics, we purposely ignore the why questions that will require an understanding of forces In Chapter 5, when we consider the related topic of dynamics, we will focus on why objects move in a certain way We restrict the discussion in this chapter to motion in one dimension (along a straight line), extending it to two and three dimensions in Chapter Here, we derive the general relationships for kinematics and the equations that describe the motion of objects moving with constant (uniform) acceleration To describe motion, we have to define the concepts of position, displacement, and distance, originally introduced in Chapter (Section 2-3) Let us expand on these concepts and describe the relationships between them Consider the girl in Figure 3-2, who is racing a dogsled across the finish line The figure shows her at locations corresponding to two different times, one before and one after she has crossed the finish line We call these the initial and final positions: they describe where the object was located at the initial and final times of its motion The SI unit for describing an object’s position is the metre (m) Displacement is a vector physical quantity that refers to “how far out of place” (thus displacement) an object is The displacement can be represented by a vector connecting the object’s initial and final positions As a vector, displacement has both magnitude and direction The SI unit for displacement is also the metre Since we are considering motion in one dimension, the direction of all the vectors describing an object’s motion in this chapter will be aligned with the chosen coordinate axis We will describe motion as directed in the positive or in the negative x-direction by using respectively positive (which can be omitted for simplicity) and negative signs before the vec> tors For example, a velocity of y 13 mys (in this chapter we will represent it as yx 13 m/s ; m/s) is directed in the positive x-direction, while a velocity > of y 24 mys (we will represent it as yx 24 m/s) is directed in the negative x-direction Index x means that we are talking about velocity, not speed This means we won’t have to use formal vector notation in this chapter, but it is still important to remember that displacements and velocities are always vectors 56 SECTION 1 | MECHANICS Dx xf – x0 x0 xf x Figure 3-2 The displacement of an object (a dogsled racer in this case) represents the change in its position in terms of both its magnitude and its direction The x-axis is chosen along the direction of the girl’s motion In this chapter, for convenience, we label the onedimensional axis as the x-axis Returning to the dogsled racer example shown in Figure 3-2, we define the x-axis as being oriented horizontally, with right representing positive, left representing negative, and the origin chosen to be the finish line Her initial and final positions, x0 and xf, are measured from the origin, and her displacement is shown by the vector quantity Dx connecting x0 and xf (Figure 3-2) The displacement can be expressed as the difference between the final and initial positions, Dx xf x0 and its SI unit is the metre Since the final position of an object is often denoted as x, the expression for displacement becomes KEY EQUATION Dx x x0 3Dx4 m (3-1) For this example and our choice of reference frame (the x-axis, the origin, and the unit of measurement), the initial position is negative and the final position is positive; hence, according to Equation 3-1, the displacement is positive In the one-dimensional case, positive displacement means that the object moved in the positive x-direction, negative displacement means that it moved in the negative x-direction, and zero displacement means that the object’s final position is the same as its initial position Let’s consider an example to help differentiate the terms distance and displacement You are sitting at your desk at home, your coffee machine indicates a fresh brew is ready, you walk 10 steps to fill your coffee mug, and then you bring it back to your desk Your displacement in this case is zero, since your final position is the same as your initial position However, the distance you have covered, there and back, is a total of 20 steps As we discussed in the dogsled racer example above, displacement is a vector that represents the straight-line path between the end points of the t rajectory (the path in space followed by a moving object) For one-dimensional motion, that displacement can take on positive or negative values The distance, on the other hand, is a positive NEL scalar representing the actual distance travelled The SI unit for measuring distance is also the metre One way to define distance is to consider the sum of the magnitudes of the small displacements that make up the journey For example, if each step you take along a certain path is dx, we can say that the distance you cover, d, is given by I always remember that the odometer of a car indicates the distance travelled by the vehicle For example, when my friends and I go for a weekend-long camping trip, the odometer of our car indicates how long our driving path (our trajectory) is and eventually how much gas we have to buy for the trip While at the end of the long journey the car will hopefully end up at the same location as it started (our home), the odometer will show a different value than what we started with, but the displacement of the vehicle will be zero The word odometer derives from the Greek words hodós (“path or gateway”) and métron (measure) d a dxi 0 (3-2) i Example 3-1 Get That Fly Therefore, the initial position of the tarantula as measured from the nail is negative (x0 221 cm), and its final position is positive (x2 32 cm) The displacement shown in Figure 3-3 is given by A tarantula is on a wall 21 cm below a nail It moves down to a point 64 cm below the nail to catch a fly, and then takes the fly straight up to a position 32 cm above the nail, as shown in Figure 3-3 (a) Find the displacement of the tarantula (b) Find the distance covered by the tarantula x x2 Dx x2 x0 32 cm 1221 cm2 53 cm (b) On the way down, the tarantula moves from its initial position, 21 cm below the nail (x0 5 221 cm), to the location of the fly, 64 cm below the nail (x1 264 cm) The distance covered by the tarantula on the way down is d1 x1 x0 264 cm 1221 cm2 32 cm 21 cm x0 d1 x1 On the way up, the tarantula moves from position x1, 64 cm below the nail, to position x2, 32 cm above the nail, for a total upward distance of d2 x2 x1 32 cm 1264 cm2 d2 64 cm 0264 cm 21 cm 43 cm 32 cm 64 cm 96 cm The total distance covered by the tarantula is the sum of the two distances: d d1 d2 43 cm 96 cm 139 cm < 140 cm 1.4 m Making sense of the result Figure 3-3 Example 3-1 Solution (a) We define a vertical x-axis with the origin located at the position of the nail, and up as the positive direction NEL For displacement, only the initial and final positions matter Positive displacement in this case means that the tarantula ended up above where it started The distance travelled, however, is a positive scalar quantity that takes into account the details of the path taken, such as turns of the tarantula along its path Lastly, notice that the final answer has only two significant figures, as the quantities given in the problem had only two significant figures Chapter 3 | Motion in One Dimension 57 Mechanics KEY EQUATION PEER TO PEER INTERACTIVE ACTIVITY 3-1 Checkpoint 3-1 Mechanics Race Car: Distance or Displacement? The Scale of the Universe When a Formula One race car drives a full lap around a circular track that has a radius of 210 m, (a) the displacement of the car is the circumference of the track (b) the distance travelled by the car is the circumference of the track (c) the distance travelled by the car is zero (d) the displacement and distance travelled are equal In this activity, you will use the computer simulation called “The Scale of the Universe,” designed by Cary Huang (htwins.net/scale2/) to help you gain some useful intuition about the vast range of sizes in our universe The simulation will allow you to change the scale (zoom in and out) and experience very small and very big objects on a grand scale Work through the simulation and accompanying questions to deepen your understanding of the scale of the universe you live in Answer: (b) Displacement is the difference between the final and initial positions and is, therefore, zero The distance covered is the circumference of the track MAKING CONNECTIONS The Scale of the Universe Table 3-1 lists some common ranges of distances in nature and everyday life Table 3-1 Common Ranges of Distances 58 Description Distance Equivalent Distance in Metres Planck length, the shortest distance that can theoretically be measured 1.6162 10235 m 1.6162 10235 m Radius of a nucleon 0.8768 fm 8.768 10216 m Upper limit for the wavelength of gamma rays 12.4 pm 12.4 10212 m Radius of a hydrogen atom 52.9 pm 52.9 10212 m Covalent bond length ,0.1 nm ,1 10210 m Hydrogen bond length ,0.2 nm ,2 10210 m Wavelength of X-rays 0.1–10 nm 10210 m to 1028 m Wavelength of yellow light 580–590 nm 5.8 1027 m to 5.9 1027 m Diameter of strand of human hair 100 mm 1024 m Thickness of human skin on palm 1.5 mm 1.5 1023 m ft 30.48 cm 3.048 1021 m Distance travelled by light in 1/299 752 458 s 1m 1m Wingspan of Airbus 380-800 79.8 m 7.98 102 m mile 1.609 km 1.609 103 m nautical mile 1.852 km 1.852 103 m Mean radius of the Moon 1737 km 1.737 103 m Mean radius of Earth 6371 km 6.371 103 m Mean distance between Earth and the Moon 384 400 km 3.844 105 m 108 Radius of the Sun 6.955 m 6.955 108 m Radius of R136a1, the most massive known star 35.4 solar radii 2.46 1010 m astronomical unit, the mean Earth–Sun distance 149 597 871 km 1.495 978 71 1011 m Radius of VY Canis Majoris, possibly the largest known star 1800–2100 solar radii 1.25 1012 m to 1.46 × 1012 m 1015 9.460 528 1015 m light year (ly), the distance that light travels in one year 9.460 528 Approximate diameter of the Milky Way galaxy 100 000 light years (ly) 9.460 528 1020 m Approximate diameter of the observable universe 93 billion light years (ly) 8.8 1026 m SECTION 1 | MECHANICS m NEL 3-2 Speed and Velocity Motion Diagrams As we discussed earlier, one of the main goals of kinematics is to describe how the object’s position changes with time We can this by plotting an x(t) graph To this we have to collect detailed information about the object’s position at different times One way of doing this is to use a video camera A video analysis of motion allows us to collect position versus time data of the object’s motion and then plot the corresponding graphs, such as x(t), y(t), etc The video recording is nothing else but time-lapse photography of the object’s motion We will discuss the advantages of computerized video analysis later in the chapter For now let us imagine a person walking along a straight line oriented in the positive x-direction Let us record his position by making a series of snapshots of his locations along the coordinate axis at equal time intervals (for example, every s) and recording the clock reading when these snapshots were taken Then we can visualize his motion using a motion diagram, as shown in Figure 3-4 If he walks at a constant pace (Figure 3-4(a)), the distances between his adjacent positions on the diagram will be equal However, when he is speeding up, the distances between his adjacent positions will continuously increase (Figure 3-4(b)) Figure 3-4(c) shows the case when the person is slowing down; in this case the distances will decrease The motion diagram illustrates the pace of the person’s motion In physics, to describe the pace, or the rate of change of the object’s position, we use the concepts of velocity and speed, discussed below KEY EQUATION (a) 3m 4s 4m 6s 5m 8s 6m x 10 s 1m 2m 3m 4s 0s 2s 4m 6s 5m 6m x 8s 1m 0s 4m 1m 0s 2m 2s (b) (c) 2m 3m 2s 4s 5m 6m x 6s 8s Figure 3-4 Motion diagrams representing one-dimensional motion of a person walking in the positive x-direction (a) Walking at a constant speed (constant pace) (b) Speeding up (c) Slowing down The motion diagram represents both the person’s position (top scale) and the corresponding clock reading (bottom scale) NEL Two cars that start at the same location and travel for an hour at the same speed but heading in different directions will end up at different final destinations Consequently, their displacements will also be different To be able to predict where an object will end up, it is important to know not only where it started (initial position, x0) and how fast it is moving, but also where it is headed Unfortunately, in everyday life, the terms speed and velocity are often used interchangeably to describe the pace of motion However, as discussed in Chapter 2, the meanings of these terms in physics are distinctly different (see the Chapter introduction) Speed is a scalar quantity describing how fast an object is moving, while velocity is a vector indicating not only how fast an object is moving, but also where it is headed (the direction of its motion) Average speed (a scalar) is the distance covered by the object divided by the time it took the object to cover it (elapsed time)—since distance is a positive scalar (if the object moved), so is average speed Notice the difference between the concepts of elapsed time, Dt, and clock reading, t If an object starts moving when the clock reading (initial time) was t0 5 s and finishes moving when tf s, the elapsed time for the object’s motion can be calculated as Dt tf t0 s s s As in the case of displacement (Equation 3-1), the index f in the final clock reading is often omitted, so the expression for elapsed time can be written as Dt t t0 The SI unit for time measurement and for elapsed time is the second (s) Average velocity (a vector) is given by the displacement divided by the elapsed time For motion along the x-axis, with initial position x0 and final position x, the average speed and the average velocity are given by the following expressions, where t0 and t are the initial and final clock readings, respectively: d d Dt t t0 m 3yavg s x x0 Dx Average velocity yx,avg 5 Dt t t0 m 3yx,avg s Average speed yavg (3-3) (3-4) Here d is the distance, a scalar quantity that is always positive; Dx is the displacement, a vector quantity that can take on positive or negative values in onedimensional motion; and Dt is the elapsed time When the two cars are moving in opposite directions with the same speed, their velocities will be equal in magnitude but opposite in sign The SI units for both speed and Chapter 3 | Motion in One Dimension 59 Mechanics Average Speed and Average Velocity Mechanics velocity are metres per second (m/s), but in everyday life it is often convenient to use kilometres per hour (km/h) or miles per hour (mph or mi/h) For example, the speed limit in school zones in Canada is 30 km/h to 50 km/h, which translates to approximately 20 mi/h to 30 mi/h (1.0 mi < 1.6 km) For historical reasons, the speed of aircraft and boats is measured in knots (kn): knot nautical mile/h 6072 ft/h 1852 m/h < 0.514 m/s Pay attention to the differences in the notations for average speed (yavg) and for average velocity (yx,avg) We use the index x for average velocity to emphasize that velocity is a vector quantity and we are considering the one-dimensional case along the x-axis Sometimes the notation y is used to indicate average speed and velocity, but we will generally use yavg and yx,avg, respectively In Chapter 4, we consider velocity in more complex two- and three-dimensional cases There, we will express velocity in > its full two- and three-dimensional vector form, y Let us examine the difference between speed and velocity in the following scenario You instruct the courier truck for your company to leave the garage of your office building, pick up a package from the post office 4.80 km down the road, and bring it to a lab facility that is 1.20 km away from your office (Figure 3-5) From your office, the lab is in the opposite direction down the road as the post office from which the package was picked up It takes the driver 15.0 to finish the task Let us choose the positive direction of the x-axis to the right (in the direction of the post office) and the origin to be where your office is located As illustrated in Figure 3-5, the total displacement of the courier, the change in her position, in this case is 21.20 km Dividing this by the total elapsed time gives us the average velocity for the trip (Equation 3-4) Since the total trip time is 15.0 (900 s), we calculate the average velocity to be yx,avg 21.20 103 m 21.33 mys 900 s Since the courier’s displacement was directed to the left, in the negative x-direction, the average velocity is also negative The average speed, however, is given by the total distance (not the displacement!) divided by the elapsed time (Equation 3-3) The total distance covered by the truck is 4.80 km 4.80 km 1.20 km 10.80 km 1.080 103 m, so the average speed is given by yavg 1.080 103 m 12.0 mys 900 s Notice that since the truck turned around, the values of the average speed and the magnitude of the average velocity differ PEER TO PEER I know that the speedometer in a car shows how fast I am moving but not where I am heading This helps me remember that speed is a scalar quantity, while velocity is a vector I also like to use the fact that a speed of 10 m/s is equivalent to 36 km/h It allows me to fast and easy mental calculations of speed conversions from m/s into km/h and vice versa For example, a speed limit of 70 km/h is equivalent to about 20 m/s The average velocity of an object during a specific time interval, yx,avg, can be represented on a motion diagram as an arrow pointing in the direction of motion (we use red to denote velocity vectors) The length of the arrow should be proportional to the distance between the adjacent locations of an object (Figure 3-6) In addition to using motion diagrams, it is convenient to describe the motion of an object using a position versus time plot, x(t), as shown in Figure 3-7 In an x(t) plot, position is displayed along the vertical POST OFFICE LABORATORY (a) 21.20 km Lab Office Dx 21.20 km 4.80 km Post office 4.80 km (b) 21.20 km x (km) 4.80 km x (km) Figure 3-5 (a) Schematic representation of the problem including the coordinate system (b) The trajectory of a truck moving from the office building, to the post office, and back to the lab The distance covered by the truck is different from the magnitude of its displacement 60 SECTION 1 | MECHANICS NEL x 1m 0s 2m 2s 3m 4s 4m 6s 6m x 10 s 5m 8s 1m 2m 3m 4s 0s 2s 4m 6s 6m x 8s 5m (c) 1m 0s 2m 3m 2s 4m 4s 5m Position (m) (b) 6m x 6s 8s Slope of a chord represents the average velocity between two given points in time: y x,avg Dx rise Dt run x1 Mechanics (a) x4 x2 x3 Figure 3-6 Motion diagrams including average velocity vectors representing the one-dimensional motion of a person walking in the positive x-direction: (a) Walking at a constant pace (b) Speeding up (c) Slowing down axis and time is displayed along the horizontal axis Let us use the x(t) plot to visualize the average velocity of the moving object between two given points in time Dx According to Equation 3-4, yx,avg Since Dx Dt represents the rise and Dt represents the run of the x(t) function, the average velocity between any two points in time is represented by the slope (rise over run) of the chord (a segment between two points on a curve) t1 t2 t3 t4 t Time (s) Figure 3-7 The average velocity of an object during a certain time interval is represented by the slope (rise over run) of the chord connecting the two points on the x(t) plot that correspond to the endpoints of the time interval The three chords represent the average velocities of an object during the time intervals (t1, t2), (t1, t3), and (t1, t4), respectively connecting corresponding points on the x(t) plot For example, in Figure 3-7 the slopes of the three chords represent the average velocities of an object between the times t1 and t2, t1 and t3, and t1 and t4 Example 3-2 Distinguishing Speed and Velocity A meteorological observation plane flies 1240 km straight southeast in 1.70 h from Stockholm, Sweden, to Moscow, Russia The plane then flies to Oslo, Norway, 425 km to the northwest of Stockholm The trip to Oslo takes 2.10 h The three cities lie along a straight line, as shown in Figure 3-8 (a) Find the average velocity of the plane in km/h and in knots (b) Find the average speed of the plane for the entire trip in km/h and in knots (1 knot 1.852 km/h) on the net displacement of 425 km and the total elapsed time: yx,avg We used the following relationship to convert km/h into knots: knot 1.852 km/h (b) The average speed is the total distance, d, travelled divided by the total time: yavg x Oslo Stockholm Dx Dx1 Dx2 Moscow Figure 3-8 Flight path of the plane in Example 3-2 Solution (a) Let us pick the direction from Moscow to Oslo to be along the positive x-axis The average velocity depends NEL Dx 425 km 5 112 kmyh < 60.4 knots Dt 1.70 h 2.10 h d 1240 km 1240 km 425 km Dt 1.70 h 2.10 h 2905 km 5 764 kmyh < 413 knots 3.80 h where we have chosen the positive x-axis to point from Moscow to Oslo Making sense of the result Since the overall displacement of the plane is relatively small, the average velocity is expected to be small The average speed, however, represents how fast the plane was going in the air, which is a typical speed for a plane Next time you are flying, pay attention to the interactive inflight map available on many modern commercial aircraft It will indicate a lot of interesting information about the flight, including the distance covered by the aircraft and the time it took to cover it (flight time) This will allow you to estimate your average speed during the flight Chapter 3 | Motion in One Dimension 61 between points t0 and t and then shrink the time interval Dt to almost zero, which is equivalent to considering a limiting case, t S t0 or Dt S 0: CHECKPOINT 3-2 Average Speed Answer: (f ) The distance covered is 270 m 230 m 500 m Covering 500 m in 14.0 s gives an average speed of 35.7 m/s, which is about 130 km/h Notice that this is a scalar quantity, so it has no direction The average velocity is obtained by dividing the displacement by the time, in this case 40.0 m (downward) divided by 14.0 s, which gives the magnitude of the average velocity as 2.86 m/s and its direction as downward (you end up below the point where you started) Since the question asks for average speed, only (f ) is correct Mechanics You make a bungee jump off the famous 321 m high Royal Gorge Bridge in Cañon City, Colorado It takes you 14.0 s to fall 270 m and bounce 230 m back up What is your average speed during the jump? (a) 2.86 m/s (b) 2.86 m/s down (c) 17.9 m/s (d) 19.3 m/s up (e) 35.7 m/s (f ) 35.7 m/s down Instantaneous Velocity If we want to know the details of how an object’s position is changing with time at a given point along its trajectory, we consider the concept of instantaneous velocity, the velocity, yx(t), of an object at a given moment in time For example, to calculate the instantaneous velocity of an object at point t in the case described in Figure 3-9, we can find the average velocity yx t2 lim t S t0 x x0 t t0 (3-5) The instantaneous velocity is given by the limit of the average velocity when the time interval between the two points goes to zero Mathematically, this means that the instantaneous velocity is given by the derivative of the position with respect to time: KEY EQUATION yx t2 lim yx,avg lim Dt S Dt S dx t Dx 5 xr t2 (3-6) Dt dt Figure 3-9 shows the position versus time plot for a moving object, x(t) The instantaneous velocity at a given point in time is represented by the slope of the tangent to the position versus time graph at that point Conceptually, you can see that when the time interval between any two given points becomes very small, the average velocity becomes almost identical to the instantaneous velocity in the middle of this time interval In Figure 3-9, when both t1 and t4 approach t such that Dt t4 t1 S 0, the average velocity between t1 and t4 approaches the value of the instantaneous velocity at time t: yx,avg1t ,t S y t as Dt S Slope of a chord represents the average velocity between two given points in time: x y x,avg Dx rise Dt run Position (m) x1 Slope of a tangent to an x(t) graph at point t represents the instantaneous velocity at this time: x4 x2 x y x (t) lim y x,avg lim Dx DtS0 DtS0 Dt x3 y x (t) t1 t2 t t3 t4 dx(t) 5x(׳t) dt t Time (s) Figure 3-9 The average velocity of an object is represented by the slope of the chord connecting the two ends of the time interval on the x(t) plot, while the instantaneous velocity at time t is represented by the slope of the tangent line to the x(t) plot at time t The instantaneous velocity is the time derivative of the x(t) plot 62 SECTION 1 | MECHANICS NEL of one-dimensional motion, this means it can be negative, positive, or zero), while instantaneous speed is always positive or zero For example, in Figure 3-9, the instantaneous velocity at time t3 is zero (the slope of the tangent to the curve at this point in time is zero), which means that at t3 the object will momentarily stop MAKING CONNECTIONS Range of Speeds in Nature and Everyday Life Table 3-2 lists some common ranges of speeds in nature and everyday life Table 3-2 Range of Speeds in Nature and Everyday Life Description NEL Speed Speed in m/s 1010 Theoretical rate of flow of glass at room temperature ,1 m in Speed of continental drift ,1 cm/y ,3 10210 m/s Speed of a small earthworm ,0.2 cm/s ,2 1023 m/s Speed of a molecule in the Bose–Einstein condensate (occurs at very low temperatures) cm/s 1022 m/s knot, the customary unit for the speed of ships and aircraft 1.85 km/h 0.514 m/s Average walking speed ,5 km/h ,1.4 m/s Sprint speed in a 100 m race ,10 m/s ,10 m/s Top running speed of a bear ,52 km/h ,14 m/s Top speed of cheetahs and sailfish ,112 km/h ,31.1 m/s Top swooping speed of a peregrine falcon ,325 km/h ,90.3 m/s Top speed of a Bugatti Veyron race car 431 km/h 120 m/s Speed record for the MLX01 magnetic levitation train (Japan, 2003) 581 km/h 161 m/s Maximum operating speed of an Airbus 380 at cruising altitude 945 km/h 262 m/s Speed of sound at sea level at 15 °C 340 m/s 340 m/s Equatorial rotation speed of Earth 1674 km/h 465 m/s Top speed of the Concorde supersonic jet at cruising altitude 2172 km/h 603 m/s Orbital speed of the Moon around Earth 3600 km/h 1000 m/s Root mean square speed of a room-temperature helium atom 1352 m/s 1352 m/s Equatorial rotation speed of the Sun ,2.0 km/s ,2000 m/s Top speed of the X-15, the fastest piloted rocket plane 7275 km/h 2,020 m/s Speed of the space shuttle on re-entry ,28 000 km/h ,7,800 m/s Mean orbital speed of Earth around the Sun 29.8 km/s 2.98 × 104 m/s Range of meteor speeds in Earth’s atmosphere 11–72 km/s 1.1 104 m/s to 7.2 104 m/s Speed of the Sun around the centre of the Milky Way galaxy ,230 km/s ,2.3 105 m/s Speed of the Milky Way galaxy (relative to the cosmic microwave background) ,600 km/s ,6 105 m/s Equatorial rotation speed of the Crab Pulsar neutron star ,2100 km/s 2.1 106 m/s Speed of light 3.00 108 m/s 3.00 108 m/s y ,3 10218 m/s Chapter 3 | Motion in One Dimension 63 Mechanics The instantaneous speed is the magnitude of the instantaneous velocity It corresponds to the absolute value of the slope of the tangent to the x(t) plot at a given point in time The instantaneous velocity can be found as the derivative of the position versus time graph Instantaneous velocity is a vector quantity (in the case CHECKPOINT 3-3 Relating Motion Diagrams to Position versus Time Plots Which position versus time graph represents the motion diagram shown in Figure 3-10? x Mechanics x x x x x t t t (a) (b) t (c) (d) t (e) Figure 3-10 Checkpoint 3-3 Answer: (d) The object’s initial position is to the left of the origin (it is negative) It has a very small (if not zero) initial velocity, and then it speeds up in the positive direction, moves at a constant velocity, and then slows down The only graph that starts at a negative initial position and has a nearly zero slope at the beginning, then an increasing positive slope, and then a decreasing positive slope is shown in (d) Example 3-3 Figure 3-11 shows a position versus time plot for a car Represent this graph algebraically and then use it to derive the velocity versus time and speed versus time graphs for the interval shown Assume all the information is known to two significant figures Position (m) 300 250 200 y 100 Dx Dt g5 s m/ Dx 250 m v x,a 50 25 Dt 10 s Time (s) 10 Figure 3-11 Position versus time plot for a car moving at a x 1t2 x0 yx,avgt 50 m 25t m 150 25t2 m The units of this expression are metres, which is the proper unit for position The graphs of both the instantaneous velocity and the instantaneous speed versus time will be represented by a horizontal line with a y-intercept of 25 m/s, as shown in Figure 3-12 Both the instantaneous and the average velocities for the car during this time interval can be expressed as yx yx,avg 25 m/s 40 30 20 10 Time (s) Figure 3-12 Velocity versus time graph for a car moving at a con- constant velocity stant velocity of 25 m/s Solution Making sense of the result The x(t) plot is represented by a straight line Therefore, it can be expressed algebraically using the expression for a straight line, y mx b The y-intercept in this case represents the initial position of the car, which we will represent as x0 According to the plot, the initial position of the car is x0 50 m Since the slope of the line doesn’t change, the velocity of the car and consequently its speed are constant Thus, the average velocity of the car is also equal to the instantaneous velocity of the car during this 10 s time interval: yx,avg yx Therefore, the motion of this car can be represented algebraically as x(t) x0 yxt During this time interval, the displacement of the car is 250 m in the positive x-direction Therefore, we can calculate the average velocity of the car for this time interval: Notice that constant velocity does not mean that the values of speed and velocity are the same An object can move with a constant but negative velocity Its speed is constant but not equal to the value of its velocity (as the speed is always positive) We also discussed earlier that the instantaneous velocity is the time derivative of x(t): yx,avg 64 This gives us Velocity (m/s) Deriving Velocity from a Position versus Time Plot Dx 250 m 5 25 mys Dt 10 s SECTION 1 | MECHANICS yx 1t2 xr 1t2 d m 150 25t2 m 25 dt s Since the x(t) plot is linear, its derivative is a constant, which means the speed and the velocity are also constant Notice that the initial position of an object (50 m in this case) has no impact on its velocity NEL Instantaneous and Average Velocities In this example, we highlight how the average velocity approaches the instantaneous velocity as the time interval separating two events gets smaller A toy remote-controlled car moves in a straight line along the x-axis such that its position (in metres) is given by x(t) (0.250t3) m, where t is measured in seconds and the coefficient 0.250 has units of m/s3 (a) Plot the car’s position versus time graph for the interval t1 1.00 s to t2 2.00 s (b) Find the average velocity of the car during the interval t1 1.00 s to t2 2.00 s, and include the corresponding line on your graph (c) Find the car’s average velocity during the interval t3 1.25 s to t4 1.75 s, and include the corresponding line on your graph (d) Find the car’s instantaneous velocity at t5 1.50 s, and include the corresponding line on your graph Comment on the difference between average and instantaneous velocities as Dt gets smaller corresponding to t3 1.25 s and t4 1.75 s This chord is also shown in orange (d) To get the instantaneous velocity at t5 1.50 s, we use Equation 3-6 Let us first find the expression for instantaneous velocity: yx 1t2 Now we can find the value for the instantaneous velocity at time t5: yx 1t5 a 0.250 As expected, the instantaneous velocity of the car at t5 1.50 s corresponds to the slope of the tangent to the x(t) graph at t5 1.50 s, shown in green x Dx x2 x1 yx,avg1t ,t Dt t2 t1 0.250 12.002 0.250 11.002 m 5a b 1.75 mys 2.00 1.00 s As we discussed earlier, the average velocity between t1 1.00 s and t2 2.00 s can be represented by the slope of the chord shown in orange connecting the points on the x(t) plot corresponding to t1 1.00 s and t2 2.00 s Notice, since the car is moving in the positive x-direction, the average velocity is positive The slope of the chord is also positive (c) We can use the same logic to find the average velocity of the car during the time interval between t3 1.25 s and t4 1.75 s: yx,avg1t ,t NEL x4 x3 Dx Dt t4 t3 5a 0.250 11.752 0.25 11.252 m b 1.70 mys 1.75 1.25 s Once again, the average velocity corresponds to the slope of the chord connecting the points on the x(t) plot x2 Position (m) (a) We will take the positive x-axis to point along the direction of the car's motion To create the plot, we can pick convenient values along the horizontal t-axis, calculate the corresponding x-coordinates, and indicate them on the vertical axis and sketch a smooth curve from the resulting points, as shown in Figure 3-13 You can also use graphing software or a spreadsheet to visualize this plot (b) We can apply Equation 3-4 to find the average velocity of the car between t1 1.00 s and t2 2.00 s: m m b 1t 2 a 0.750 b 11.50 s2 s3 s 1.69 mys Solution dx 1t2 d 10.250t m 5 a 0.250 b t dt dt s y x,avg (t1,t2) x4 x5 y x,avg (t3,t4) x3 x1 y x (t5) t1 t3 t5 Time (s) t4 t2 t Figure 3-13 The closer the two points t1 and t2 are on a p osition versus time plot, the closer the average velocity approaches the instantaneous velocity As the chord gets shorter, the slope of the chord approaches the slope of the tangent Making sense of the result We see that as the time, Dt, separating the two points gets smaller, the chord representing the average velocity during this time interval becomes parallel to the tangent at the point in the middle of this time interval Therefore, the average velocity during a very short time interval is for practical p urposes equivalent to the instantaneous velocity in the middle of this interval Mathematically, this means that the instantaneous velocity is represented by the derivative of the position dx versus time function: yx xr 1t2 ; This means that the limit dt of the average velocity when the time interval approaches zero approaches the instantaneous velocity at that point in time Chapter 3 | Motion in One Dimension 65 Mechanics Example 3-4 CHECKPOINT 3-4 Average Velocity Mechanics The average velocity of an object (a) must be greater than its instantaneous velocity (b) must be less than its instantaneous velocity (c) cannot be equal to its instantaneous velocity (d) none of the above acceleration using either positive or negative signs, as we did with velocity The letter subscript in the notation of acceleration, such as ax or ay, indicates that we are talking about the vector of acceleration in one dimension For example, for one-dimensional motion along the x-axis, ax denotes the acceleration vector along the x-axis However, if we want to talk about the absolute value of acceleration, we will denote it as a We will use full three-dimensional vector acceleration notation in the following chapters The average acceleration of an object is the change in its velocity divided by the elapsed time: Answer: (d) The average velocity can be greater, less than, or equal to the instantaneous velocity For example, Figure 3-13 illustrates the case when the average velocity is greater than an instantaneous velocity KEY EQUATION Measuring the Speed of Neutron Stars Chandra: NASA/CXC/Middlebury College/ F.Winkler; ROSAT: NASA/GSFC/S.Snowden et al.; Optical: NOAO/CTIO/Middlebury College/F.Winkler et al The Chandra X-ray Observatory detected a neutron star, RX J0822-4300, which is moving away from the centre of Pupis A, a supernova remnant about 7000 ly away (Figure 3-14) Believed to be propelled by the strength of the lopsided supernova explosion that created it, this neutron star is moving at a speed of about 4.8 million km/h (0.44% of the speed of light, 0.44c), putting it among the fastestmoving stars ever observed At this speed, its trajectory will take it out of the Milky Way galaxy in a few million years Astronomers were able to estimate its speed by measuring its position over a period of years Figure 3-14 Supernova remnant RX J0822-4300. 3-3 Acceleration Just as velocity represents the rate of change of position in time (Equation 3-6), acceleration represents the rate of change of velocity in time Since velocity is a vector quantity (it has both magnitude and direction), the rate of change of velocity is also a vector In the case of one-dimensional motion, we indicate the direction of 66 SECTION 1 | MECHANICS Dyx Dt yx yx,0 t t0 3a4 m (3-7) s2 The average acceleration of an object between two points in time depends only on its final and initial velocities and the elapsed time Since acceleration indicates by how much the velocity is changing every second, its units are metres per second per second, or m/s2 Sometimes the notation a is used to denote average acceleration, but we will use ax,avg, analogous to average velocity Figure 3-15 shows the velocity versus time plot for an object The average acceleration, given by Equation 3-7, is the change in velocity (rise) divided by the elapsed time (run) between the two points in time The average acceleration is represented by the slope of the chord connecting these two points on the yx (t) plot yx Velocity (m/s) MAKING CONNECTIONS ax,avg Slope of a chord represents the average acceleration between two given points in time: Dy ax,avg x rise run Dt yx1 yx4 yx2 yx3 t1 t2 t3 Time (s) t4 t Figure 3-15 The average acceleration between any two points in time is given by the slope (rise over run) of the chord connecting the two points on the yx(t) graph NEL ... in the eBook version Physics for Scientists and Engineers: An Interactive Approach, Second Edition by Robert Hawkes, Javed Iqbal, Firas Mansour, Marina Milner-Bolotin, and Peter Williams VP, Product.. .physics For Scientists and Engineers An Interactive Approach Second Edition physics For Scientists and Engineers An Interactive Approach Second Edition Robert Hawkes Mount... Physics for scientists and engineers : an interactive approach / Robert Hawkes, Mount Allison University, Javed Iqbal, University of British Columbia, Firas Mansour, University of Waterloo, Marina