Physics concepts and connections book two

4.6 Linear Momentum and Centre of Mass 211STSE — Recreational Vehicle Safety and Collisions 214 Lab 4.1 — Linear Momentum in One Dimension: Dynamic Laboratory Carts 222 Lab 4.2 — Linear

Henri M van Bemmel

Don Bosomworth, Physics Advisor

Copyright © 2002 by Irwin Publishing Ltd.

National Library of Canada Cataloguing in Publication Data

Heimbecker, Brian

Physics: concepts and connections two

For use in grade 12

ISBN 0-7725-2938-8

1 Physics I Nowikow, Igor II Title

QC23.N683 2002 530 C2002-900508-6

All rights reserved It is illegal to reproduce any portion of this book in any form or

by any means, electronic or mechanical, including photocopy, recording or anyinformation storage and retrieval system now known or to be invented, without theprior written permission of the publisher, except by a reviewer who wishes to quotebrief passages in connection with a review written for inclusion in a magazine,newspaper, or broadcast

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ON M5E 1E5

Cover and text design: Dave Murphy/ArtPlus Ltd

Page layout: Leanne O’Brien, Beth Johnston/ArtPlus Ltd

Illustration: Donna Guilfoyle, Sandy Sled, Joelle Cottle, Nancy Charbonneau/ArtPlus Ltd., Dave McKay, Sacha Warunkiw, Jane Whitney

ArtPlus Ltd production co-ordinator: Dana Lloyd

Publisher: Tim Johnston

Project developer: Doug Panasis

Editor: Lina Mockus-O’Brien

Photo research: Imagineering, Martin Tooke

Indexer: May Look

The authors and the publisher would like to thank the following reviewers for their

insights and suggestions

Bob Wevers, Teacher, Toronto, Toronto District School Board

Vince Weeks, Teacher, Burlington, Halton District School Board

Peter Mascher, Department of Engineering Physics, McMaster University

Andy Auch, Teacher, Windsor-Essex District School Board

Peter Stone, Teacher, Simcoe County District School Board

George Munro, Teacher, District School Board of Niagara

Brendan Roberts, Teacher, Windsor-Essex Catholic District School Board

To my wife Laurie and my children Alyssa and Emma for making it possible for me

to do this one more time

I would like to thank David Badregon and Vanessa Mann for their contributions

to the problems and their solutions

Brian Heimbecker

I would like to dedicate this book to my family: my wife Jane, my children Melissa

and Cameron, my mom Alla, and my brother Alex, as well as all my students

Special thanks to the students who worked on various aspects of solutions and

research: Ashley Pitcher, Roman and Eugene Zassoko, Teddy Lazongas, and

Katherine Wetmore

Igor Nowikow

Dedicated to my wife Marcy and daughter Alison, for their never-ending love and

support In memory of the late Violet Howes and her passion for teaching

I would like to thank Devin Smith (Queen’s University), Kristen Koopmans

(McMaster University), Jon Ho (University of Waterloo), and Paul Finlay

(University of Guelph) for their solutions to the problems

Christopher T Howes

To my wife Lynda for her support and encouragement, and to all my students who

make physics fun I would like to thank Tyler Samson, a student at Confederation

Secondary School in Val Caron, for his contribution as a problem solver

Jacques Mantha

I would like to thank my wife Judy and daughter Erin for their valuable

sugges-tions, and my son Brad for his careful solutions to the problems

Brian P Smith

I would like to dedicate my portion of this effort to my wife Nadine for her love and

Table of Contents

1 Kinematics and Dynamics

1.6 An Algebraic Description of Uniformly

The Guinea and Feather Demonstration 19

1.8 A Graphical Analysis of Linear Motion 24

1.11 Newton’s First Law of Motion:

Inertial and Non-inertial Frames

1.12 Newton’s Second Law of Motion: F

net ma 361.13 Newton’s Third Law: Action–Reaction 39

1.14 Friction and the Normal Force 44

1.15 Newton’s Law of Universal Gravitation 48

Calculating Gravitational Forces 50

STSE — New Respect for the Humble Tire 52

Lab 1.1 — Uniform Acceleration: The Relationship

Lab 1.2 — Uniform Acceleration: The Relationship

between Angle of Inclination and Acceleration 62

2 Kinematics and Dynamics

Lab 2.2 — Centripetal Force and Centripetal

Lab 2.3 — Amusement Park Physics 126

3 Extension: Statics — Objects and Structures in Equilibrium 127

3.1 Keeping Things Still: An Introduction

3.2 The Centre of Mass — The Gravity Spot 128

3.5 Static Equilibrium: Balancing Forces

3.6 Static Equilibrium and the Human Body 148

3.9 Stress and Strain — Cause and Effect 161

Strain: The Effect of Stress 1633.10 Stress and Strain in Construction 170STSE — The Ultimate Effect of Stress on

4.1 Introduction to Linear Momentum 189

4.6 Linear Momentum and Centre of Mass 211

STSE — Recreational Vehicle Safety and Collisions 214

Lab 4.1 — Linear Momentum in

One Dimension: Dynamic Laboratory Carts 222

Lab 4.2 — Linear Momentum in

Two Dimensions: Air Pucks (Spark Timers) 224

Lab 4.3 — Linear Momentum in

5.4 Gravitational Potential Energy 243

5.5 Elastic Potential Energy and Hooke’s Law 249

5.7 Elastic and Inelastic Collisions 260

Equations for One-dimensional

Graphical Representations of Elastic

STSE — The Physics Equation — The Basis

Lab 5.3 — Inelastic Collisions (Dry Lab) 282

Lab 5.4 — Conservation of Kinetic Energy 283

Kinetic Energy Considerations 290

Escape Energy and Escape Speed 292

Implications of Escape Speed 293

Kepler’s Laws of Planetary Motion 298

Kepler’s Third Law for Large Masses 300

Extension: Orbital Parametres 301

6.3 Simple Harmonic Motion —

6.4 Damped Simple Harmonic Motion 308

7.3 Angular Velocity and Acceleration 322

Relating Angular Variables to Linear Ones 323More About Centripetal Acceleration 3257.4 The Five Angular Equations of Motion 327

Extension: The Parallel-axis Theorem 337

8 Electrostatics and Electric Fields 370

8.1 Electrostatic Forces and Force Fields 3718.2 The Basis of Electric Charge — The Atom 371

Electricity, Gravity, and Magnetism:

Forces at a Distance and Field Theory 398

8.7 Electric Potential and Electric

8.8 Movement of Charged Particles in

a Field — The Conservation of Energy 404

The Electric Potential around a

Lab 8.1 — The Millikan Experiment 430

Lab 8.2 — Mapping Electric Fields 433

9 Magnetic Fields and Field Theory 435

9.1 Magnetic Force — Another Force

9.2 Magnetic Character — Domain Theory 437

9.4 Artificial Magnetic Fields —

Magnetic Character Revisited 442

A Magnetic Field around a Coiled

9.5 Magnetic Forces on Conductors

and Charges — The Motor Principle 447

The Field Strength around a

Current-carrying Conductor 451

The Unit for Electric Current

Magnetic Force on Moving Charges 456

9.6 Applying the Motor Principle 460

The Mass of an Electron and a Proton 462

9.7 Electromagnetic Induction —

From Electricity to Magnetism

STSE — Magnetic Resonance Imaging (MRI) 472

Lab 9.1 — The Mass of an Electron 479

The Production of Electromagnetic

10.4 Electromagnetic Wave Phenomena:

The Refractive Index, n — A Quick Review 500

Snell’s Law: A More In-depth Look 502Refraction in an Optical Medium 504

Liquid Crystal Displays (LCDs) 516

Polarization in the Insect World 518

Measuring Concentrations of Materials

Two-dimensional Cases 536

11.4 Young’s Double-slit Equation 538

More Single-slit Equations (but they

The Diffraction-grating Equation 564

Lab 11.1 — Analyzing Wave Characteristics

Lab 11.2 — Qualitative Observations of the

Lab 11.3 — Comparison of Light, Sound, and

Lab 11.4 — Finding the Wavelength of Light

using Single Slits, Double Slits, and

12.5 De Broglie and Matter Waves 606

The Conservation of Angular Momentum 610

12.8 Heisenberg’s Uncertainty Principle 617

A Hypothetical Mechanical Example

Heisenberg’s Uncertainty Principle

12.9 Extension: Quantum Tunnelling 622STSE — The Scanning Tunnelling Microscope 624

Lab 12.2 — The Photoelectric Effect I 631Lab 12.3 — The Photoelectric Effect II 632

13 The World of Special Relativity 633

13.1 Inertial Frames of Reference and Einstein’s

First Postulate of Special Relativity 63413.2 Einstein’s Second Postulate of Special

13.3 Time Dilation and Length Contraction 640

Moving Objects Appear Shorter 64313.4 Simultaneity and Spacetime Paradoxes 646

Electrons Moving in Magnetic Fields 656

13.6 Velocity Addition at Speeds Close to c 659

Lab 13.1 — A Relativity Thought Experiment 683

14 Nuclear and Elementary Particles 685

14.1 Nuclear Structure and Properties 686

Mass Defect and Mass Difference 688

Nuclear Binding Energy and Average

Binding Energy per Nucleon 688

Decay (Electron Emission) 693

Decay (Positron Emission) 695

Electron Capture and Gamma Decay 695

14.3 Half-life and Radioactive Dating 697

Comparing Energy Sources — A Debate 717

Hadrons (Baryons and Mesons) 723

14.8 Fundamental Forces and Interactions —

What holds these particles together? 727

Quantum Chromodynamics (QCD): Colour

Charge and the Strong Nuclear Force 730

The Weak Nuclear Force — Decay and

Statistical Deviation of the Mean 753

Making Measurements with Stated

Manipulation of Data with Uncertainties 756Addition and Subtraction of Data 756Multiplication and Division of Data 757

Creating an Equation from a Proportionality 758Finding the Correct Proportionality

Finding the Constant of Proportionality

in a Proportionality Statement 761Other Methods of Finding Equations

Substitution Method of Solving Equations 766

Appendix J: Some Elementary Particles

Numerical Answers to Applying the Concepts 776 Numerical Answers to End-of-chapter

Flowcharts

The flowcharts in this book are visual summaries that graphically show youthe interconnections among the concepts presented at the end of each sectionand chapter They help you organize the methods and ideas put forward inthe course The flowcharts come in three flavors: Connecting the Concepts,Method of Process, and Putting It All Together They are introduced as youneed them to help you review and remember what you have learned

Examples

The examples in this book are loaded with both textual and visual cues, soyou can use them to teach yourself to do various problems They are thenext-best thing to having the teacher there with you

Applying the Concepts

At the end of most subsections, we have included a few simple practice tions that give you a chance to use and manipulate new equations and try outnewly introduced concepts Many of these sections also include extensions ofnew concepts into the areas of society, technology, and the environment toshow you the connection of what you are studying to the real world

ques-e x a m p l ques-e 1

app

ly lyingthe

o c e ss

of

co

nnectct in g the

C o

End-of-chapter STSE

Every chapter ends with a feature that deals exclusively with how our

stud-ies impact on society and the environment These articles attempt to

intro-duce many practical applications of the chapter’s physics content by

challenging you to be conscious of your responsibility to society and the

envi-ronment Each feature presents three challenges The first and most

impor-tant is to answer and ask more questions about the often-dismissed societal

implications of what we do These sections also illustrate how the knowledge

and application of physics are involved in various career opportunities in

Canada Second, you are challenged to evaluate various technologies by

per-forming correlation studies on related topics Finally, you are challenged to

design or build something that has a direct correlation to the topic at hand

Exercises

Like a good musician who needs to practise his or her instrument regularly,

you need to practise using the skills and tools of physics in order to become

good at them Every chapter ends with an extensive number of questions to

give you a chance to practise Conceptual questions challenge you to think

about the concepts you have learned and apply them to new situations The

problems involve numeric calculations that give you a chance to apply the

equations and methods you have learned in the chapter In many cases, the

problems in this textbook require you to connect concepts or ideas from

other sections of the chapter or from other parts of the book

Labs

“Physics is for everyone” is re-enforced by moving learning into the

practi-cal and tactile world of the laboratory You will learn by doing labs that

stress verification and review of concepts By learning the concepts first and

applying them in the lab setting, you will internalize the physics you are

studying During the labs, you will use common materials as well as more

high-tech devices

Appendices

The appendices provide brief, concise summaries of mathematical methods

that have been developed throughout the book They also provide you with

detailed explanations on how to organize a lab report, evaluate data, and

make comparisons and conclusions using results obtained experimentally

They explain uncertainty analysis techniques, including some discussion

E X E R C I S E S

1 Kinematics and Dynamics

in One Dimension

2 Kinematics and Dynamics

in Two Dimensions

3 Extension: Statics —Objects and Structures

in Equilibrium

Forces and

UN IT

of nature and the universe that would last for 2000 years.

Greeks suggested that all matter is composed of tiny atoms bumping and clumping in empty space.

Euclid put together

300 years of Greek mathematics in 13

books of The Elements,

still in use in the early 20th century.

Ptolemy—mathematician, astronomer, and geographer

Books by this epitome of Greek science informed students for the next

1400 years.

Copernicus improved Ptolemy’s astronomy by proposing that Earth revolves around the Sun.

Copernicus—published results of 30 years’ analysis

of the planetary system with Sun at the centre of planets’ orbits; Earth has daily rotation on axis.

Kepler began 30-year study of the orbits of the planets.

Italian engineers published studies of mechanical devices following principles

of Archimedes.

Archimedes made substantial analysis of the physics of floating bodies and of levers

Also conducted great engineering projects.

The classical physics we study today was mostly developed from themid-16th to the late 19th centuries The scientific method was formallydeveloped and applied during the Enlightenment (17th and 18th centuries)

As a result, many important advances were made in many scientific fields

Nicolas Copernicus (1473–1543), a Polish mathematician, explained thedaily motion of the Sun and stars by suggesting that Earth rotates on anaxis Galileo Galilei (1564–1642), an Italian mathematician, experimentedextensively to test ancient theories of motion His famous experiment ofdropping two stones, a large one and a small one, from the Tower of Pisadisproved the ancient idea that mass determined the properties of motion

The understanding of celestial mechanics grew quickly with JohannesKepler (1571–1630), who explained celestial results using Tycho Brahe’sdata (1546–1601) Sir Isaac Newton (1642–1727) developed the concepts ofgravity and laid the foundations of our current concepts of motion in his

published book, Principia Mathematica With his three laws and the

devel-opment of the mathematical methods now called calculus, Newton isresponsible for our understanding of dynamics and kinematics Newtonand Galileo created a new approach for scientific analysis — testing andexperimentation — which we still use today

square of the time.

After being condemned for Earth’s motion in

1633, Galileo published result of a lifetime of motion studies in his

Two New Sciences.

Huygens in Holland published mechanical study

of his new pendulum clock, accurate to 10 s per day —

a gigantic improvement.

The 14 th General Conference on Weights and Measure picked seven quantities as base quantities, forming the basis of the International System of Units (SI), also called the metric

Republic of France established a new system

of weights and measures, defining the metre for the first time It also tried a 10-h day.

In this unit, we will learn various methods for

studying a variety of forces ranging from simple motion,

to motion with friction, to orbital motion We will also

explain the motion of human beings, the development

of a variety of vehicles, and the reasons behind the

designs of different types of equipment, such as skis and

car tires, in terms of the classical laws of physics This

unit lays the foundation for later units on momentum,

energy, fields, and modern physics

Motion Dynamics







1 Kinematics and Dynamics

in One Dimension

By the end of this chapter, you will be able to

• analyze the linear motion of objects using graphical and algebraic methods

• solve problems involving forces by applying Newton’s laws of motion

• add and subtract vector quantities in one dimension

• solve problems involving Newton’s law of universal gravitation

Chapter Outline

1.1 Introduction

1.2 Distance and Displacement

1.3 Unit Conversion and Analysis

1.4 Speed and Velocity

1.5 Acceleration

1.6 An Algebraic Description of Uniformly

Accelerated Linear Motion

1.7 Bodies in Free Fall

1.8 A Graphical Analysis of Linear Motion

1.9 Dynamics

1.10 Free-body Diagrams

1.11 Newton’s First Law of Motion:

The Law of Inertia

1.12 Newton’s Second Law of Motion: Fnet ma

1.13 Newton’s Third Law: Action–Reaction

1.14 Friction and the Normal Force

1.15 Newton’s Law of Universal Gravitation

New Respect for the Humble Tire

1.1 Uniform Acceleration: The Relationship

between Displacement and Time

1.2 Uniform Acceleration: The Relationship

between Angle of Inclination and Acceleration

S

S E

1.1 Introduction

Every day, we observe hundreds of moving objects Cars drive down the

street, you walk your dog through the park, leaves fall to the ground These

events are all part of our everyday experience It’s not surprising, then, that

one of the first topics physicists sought to understand was motion

The study of motion is called mechanics It is broken down into two

parts, kinematics and dynamics Kinematics is the “how” of motion, that

is, the study of how objects move, without concerning itself with why they

move the way they do Dynamics is the “why” of motion In dynamics, we

are concerned with the causes of motion, which is the study of forces In the

next two chapters, we will consider the aspects of kinematics and

dynam-ics in relation to motion around us

In any field of study, using precise language is important so that people can

understand one another’s work Every field has certain concepts that are

considered the fundamental building blocks of that discipline When we

begin the study of physics, our first task is to define some fundamental

con-cepts that we’ll use throughout this text

Fig.1.2 Moving objects are part of our daily livesFig.1.1 Uniform or non-uniform motion?

Your answer is a scalar A scalar is a quantity that has a magnitude only, in

this case, 400 km An answer such as “North Bay is 400 km east of here”would answer the question much more clearly This answer is a vector

answer A vector is a quantity that has both a magnitude and a direction.

“400 km east” is an example of a displacement vector, where the magnitude

of the displacement is 400 km and the direction is east Displacement is

the change in position of an object The standard SI (Système International

d’Unités) or metric unit is the metre (m), and the variable representing

dis-placement is d Examples of scalars are: 10 minutes, 30°C, 4.0 L, 10 m.Examples of vectors are: 100 km [E], 2.0 m [up], 3.5 m [down]

Displacement is commonly confused with distance Distance is the length

of the path travelled and has no direction, so it is a scalar

A cyclist travels around a 500-m circular track 10 times (Figure 1.3) What

is the distance travelled, and what is the cyclist’s final displacement?

Fig.1.3

S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

The cyclist travels a distance of 500 m each time she completes one lap

Since she completes 10 laps, her total distance is 5000 m To find her

dis-placement, we draw a line segment from the starting point to the end

point of her motion Because she starts and ends at the same point, herdisplacement has a magnitude of zero

In this example, we obtain very different answers for distance anddisplacement It is a good reminder of how important it is to clearly dif-ferentiate between vector and scalar quantities

Position is a vector quantity that

gives an object’s location relative

to an observer.

Total displacement

is zero after

1 complete loop Position

Defining Directions

In two-dimensional vector problems, directions are often given in terms

of the four cardinal directions: north, south, east, and west For

one-dimensional or linear problems, we use the directions of the standard

Cartesian coordinate system: vectors to the right and up are positive, and

vectors to the left and down are negative

In the past, when the Imperial system of measurement was in common use,

it was often necessary to convert from one set of units to another Today, by

using the SI or metric system, conversions between units need only be done

occasionally To convert the speed of a car travelling at 100 km/h to m/s, we

multiply the original value by a series of ratios, each of which is equal to

one We set up these ratios such that the units we don’t want cancel out,

leaving the units of the correct answer For example,

100 km/h 100

h

km

60

1m

hin

1

6

m0

is

n

11

00k

0m

d

21

4d

h

601

mh

in

1

6m

0i

sn

 5.7  108

sThere are 5.7  108

s in 18 years

1 How many seconds are there in a month that has 30 days?

2 A horse race is 7 furlongs long How many kilometres do the horses

run? (Hint: 8 furlongs  1 mile, 1 km  0.63 miles.)

3 Milk used to be sold by the quart An Imperial quart contains

Table 1.1

Prefixes of the Metric System

Factor Prefix Symbol

0.001 km



 36 1 00

 h

If you were to walk east along Main Street for a distance of 1.0 km in a time

of 1 h, you could say that your average velocity is 1.0 km/h [E] However,

en route, you may have stopped to look into a shop window, or even satdown for 10 minutes and had a cold drink So, while it’s true that your aver-age velocity was 1.0 km/h [E], at any given instant, your instantaneousvelocity was probably a different value It is important to differentiatebetween instantaneous velocity, average velocity, and speed

Average speed is the total change in distance divided by the total elapsed

time Average speed is a scalar quantity and is represented algebraically bythe equation



d t

 (eq 1)

Average velocity is change in displacement over time Average velocity is a

vector quantity and is represented algebraically by the equation

Instantaneous velocity is the velocity of an object at a specific time Note

that speed is a scalar and velocity is a vector, but both use the same variable,

v, and have the same units, m/s To distinguish velocity from speed, we

place an arrow over the velocity variable to show that it’s a vector Similarly,

an arrow is placed over the displacement variable, d, to distinguish it fromdistance, d Later, they will be distinguished in the final statement only.Average and instantaneous velocities can be calculated algebraically Wewill revisit these two terms in Section 1.8 using graphical methods

1 What is the velocity of the train if it travels a displacement of 25 km

[N] in 30 minutes?

2 A ship sails 3.0 km [W] in 2.0 h, followed by 5.0 km [E] in 3.0 h a) What is the ship’s average speed?

b) What is the ship’s average velocity?

3 The table below shows position–time data for a toy car

d

(m) [E] 0 2.0 4.0 6.0 8.0 8.0 8.0 9.0 9.0

t(s) 0 1.0 2.0 3.0 4.0 5.0 6.0 7 0 8.0

a) What is the average velocity of the toy car’s motion?

b) What is the instantaneous velocity of the car at time t 5.0 s?

Fig.1.4 Why doesn’t the sign say

“Velocity Limit”?

Fig.1.5 Motion is everywhere

1.5 Acceleration

The simplest possible type of motion that an object can undergo (short of being

at rest) is uniform motion Uniform motion is motion at a constant speed in

a straight line Another name for uniform motion is uniform velocity

When an object’s motion isn’t uniform, the object’s velocity changes

Because velocity is a vector, its magnitude as well as its direction can

change An example of a change of magnitude only occurs when a car speeds

up as it pulls away from a stoplight A change in the direction only of an

object’s velocity occurs when a car turns a corner at a constant speed

Acceleration is the change in velocity per unit time Velocity can change

in magnitude or direction or both A negative acceleration in horizontal

motion is an acceleration to the left If an object’s initial velocity is to the left,

the negative acceleration will cause it to speed up If an object’s initial

veloc-ity is to the right, the negative acceleration will cause it to slow down.

Algebraically, we can express acceleration as

The SI unit for acceleration is a derived unit; that is, it is a unit created by

dividing a velocity unit (such as m/s) by a time unit (such as s), giving

units  m

s 1

s or m

s2Writing acceleration units as m/s2 doesn’t mean that we have measured a

second squared It is simply a short form for the unit (m/s)/s, which means

that the velocity is changing so many m/s each second

When struck by a hockey stick, a hockey puck’s velocity changes from

15 m/s [W] to 10 m/s [E] in 0.30 s Determine the puck’s acceleration

Recall that in our standard coordinate system, we can represent west as

negative and east as positive

S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

ms

s

Fig.1.6 Slapshot!

S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

If we use our standard coordinate system and assume that the initialmotion of the car was in the positive direction, its acceleration is in thedirection opposite to its initial motion Therefore, the car’s acceleration isnegative If in our example the acceleration of the car is 4.0 m/s2

, the car

is losing 4.0 m/s of speed every second The negative value for tion doesn’t mean that the car is going backwards It means that the car ischanging its speed by 4.0 m/s2in the negative direction Since the car wastravelling in a positive direction, it is slowing down

accelera-For motion in one dimension, we will designate the direction by using  and

 signs Thus, 12 km [N] becomes 12 km (written as 12 km) and 12 km [S]

is written as 12 km

We will also omit vector arrows in the equations for displacement, velocity,

and acceleration Instead, we will convey direction by using  and  signs

We will place vector arrows over variables only if the full vector quantity is

referred to (e.g., d 12 km [N])

Accelerated Linear Motion

Thus far, we have defined two algebraic equations that apply to objectsundergoing uniform acceleration These two equations are

If the initial velocity of the car in

Example 4 had been 15.0 m

s  , an acceleration of 1.0 m/s 2 would

mean that the car was speeding up

in the negative direction.

Fig.1.7

From equation 2, we can isolate d:

Even though the vector arrows have been left off of these equations, they

are still vector equations! For linear motion, we will leave the vector arrows

off, but still indicate direction as positive or negative In general (i.e., when

solving two-dimensional problems), we leave the vector arrows on,

other-wise we might forget to add and subtract these values vectorially

Equations 4 and 5 are both very useful for solving problems in which

objects are accelerating uniformly in a straight line If we look carefully at

these two equations, we will notice that many of the variables are common

The only variables not common to both equations are changes in

displace-ment, d, and acceleration, a We can combine equations 4 and 5 by

substi-tuting the common variables to form other new and useful equations First,

isolate v2in equation 4:

v2 at  v1 (eq 6)Now, substitute equation 6 into equation 5:

d  v2t 1

2 a t2and

v2  v1  2ad

The derivation of these equations is left as an exercise in the Applying the

Concepts section The five equations for uniform linear acceleration are

listed in Table 1.2

A physics teacher accelerates her bass boat from 8.0 m/s to 11 m/s at a rate

Figure 1.8 below summarizes how to choose the correct kinematics equation.

Fig.1.8 Choosing Kinematics Equations

Jane Bond runs down the sidewalk, accelerating uniformly at a rate of

0.20 m/s2from her initial velocity of 3.0 m/s How long will it take Jane

solve this problem either by factoring or by using the quadratic formula

01

2 a t2 v1t  d

Determine which variables you are given values for, and which variables you

are required to find

Check each of the five kinematics equations in order

Do you have a value for each variable in the equation except for the variable that you are required

to find?

YES

NO Choose another equation

Use this equation

Fig.1.9 Jane Bond

t 

t  3.0

0

2

3.7

Therefore, t  3.5 s or t  33.5 s

We use the positive value because time cannot be negative Therefore,

t  3.5 s It takes Jane Bond 3.5 s to run 12 m.

S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

The first step is to break the problem down into simpler parts or stages.This problem asks us to find the total displacement and average velocity Wecan solve the problem by first finding the displacement, time, and velocity

at each stage of Bounder’s trip, then adding the results of each stage together

to obtain the final answer The table below illustrates the different stages ofBounder’s trip and the information we are given at each stage

Fig.1.10 A sport utility vehicle (SUV)

Checking the Units for t

To calculate the displacement, we use equation 3:

The initial velocity during stage C is the same as the velocity during stage

B because the SUV hasn’t slowed down yet; therefore,

Before we can calculate the average velocity, we need to find the totaltime of the trip:

0s

m



vavg 29 m/sTherefore, Bounder’s total displacement is 780 m and his average veloc-ity is 29 m/s

Fred and his friend Barney are at opposite ends of a 1.0-km-long dragstrip in their matching racecars Fred accelerates from rest toward Barney

at a constant 2.0 m/s2

Barney travels toward Fred at a constant speed of

10 m/s How much time elapses before Fred and Barney collide?

S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

Given

d  1000 m aF 2.0 m/s2

v1F 0 vB 10 m/s

To solve this problem, we must note two things First, the distance travelled

by Barney plus the distance travelled by Fred must add up to 1000 m.Second, Fred is accelerating uniformly, while Barney is undergoing uni-form motion

We will assume that Fred is moving in the positive direction At any time

t, his distance from his starting point is

Barney’s displacement from the same point is 1000 m plus his

Jack, who is running at 6.0 m/s to catch a bus, sees it start to move when

he is 20 m away from it If the bus accelerates at 1.0 m/s2

, will Jack take it? If so, how long will it take him?

over-S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

Given

vJack 6.0 m/s v1bus 0 abus 1.0 m/s2

aJack 0 d  20 m

We will consider Jack’s initial position as our origin and assume that he

is running in the positive direction His displacement at any time t is

The displacement of the bus from the same origin at any time t is

dbus 20 m  v1t 1

2 a t2

dbus 20 m 1

2 (1.0 m/s2)t2When Jack overtakes the bus, the two displacements are equal:

(6.0 m/s)t  20 m  (0.5 m/s2

)t2(0.5 m/s2)t2  (6.0 m/s)t  20 m  0

Using the quadratic equation to solve for t,

t 

t 

There are no real roots for this equation; therefore, there is no real time

at which Jack and the bus have the same position Jack will have to walk

or wait for the next bus!

1 A CF-18 fighter jet flying at 350 m/s engages its afterburners and

accelerates at a rate of 12.6 m/s2

to a velocity of 600 m/s How fardoes the fighter jet travel during acceleration?

2 A butterfly accelerates over a distance of 10 cm in 3.0 s, increasing

its velocity to 5.0 cm/s What was its initial velocity?

3 During a football game, Igor is 8.0 m behind Brian and is running

at 7.0 m/s when Brian catches the ball and starts to accelerate away

at 2.8 m/s2

from rest

a) Will Igor catch Brian? If so, after how long?

b) How far down the field will Brian have run?

4 A bullet is fired into a tree trunk (Figure 1.12), striking it with an

ini-tial velocity of 350 m/s If the bullet penetrates the tree trunk to a depth

of 8.0 cm and comes to rest, what is the acceleration of the bullet?

5 A delivery truck accelerates uniformly from rest to a velocity of

8.0 m/s in 3.0 s It then travels at a constant speed for 6.0 s Finally,

it accelerates again at a rate of 2.5 m/s2

, increasing its speed for 10 s

Determine the truck’s average velocity

6 While undergoing pilot training, a candidate is put in a rocket sled

that is initially travelling at 100 km/h When the rocket is ignited,

the sled accelerates at 30 m/s2

At this rate, how long will it take therocket sled to travel 500 m down the track?

7 A parachutist, descending at a constant speed of 17 m/s,

acciden-tally drops his keys, which accelerate downward at 9.8 m/s2

a) Determine the time it takes for the keys to reach the ground if

they fall 80 m

b) What is the final velocity of the keys just before they hit the ground?

8 Derive the following equations from first principles:

a) v2  v1  2ad

b)d  v2t 1

2 a t2

Galileo Galilei (1564–1642), an Italian astronomer and physicist, is credited

with being the father of modern experimental science because he combined

experiment and calculation rather than accepting the statements of an

authority, namely Aristotle, regarding the study of nature His greatest

con-tributions were in the field of mechanics, especially dynamics His

experi-ments on falling bodies and inclined planes disproved the accepted

Aristotelean idea that a body’s rate of descent is proportional to its weight

Galileo’s conclusions greatly upset Aristotelean scholars of his day

The Guinea and Feather Demonstration

Galileo experimented in many different fields One of his experiments in

mechanics involved rolling spheres down a wooden ramp (Figure 1.13b)

He found that the square of the time a sphere took to reach the bottom of a

ramp was proportional to the length of the ramp He also observed that the

time a sphere took to reach the bottom of the ramp was independent of its

mass; that is, less massive objects and more massive objects both reach the

bottom of the ramp at the same time when released from the same height

Fig.1.13a Galileo Galilei

Fig.1.13b The inclined plane used

by Galileo Galilei

Today, we can easily confirm Galileo’s findings by performing the guineaand feather demonstration A guinea (or any coin) and a feather are placed

in a long glass tube with a hole at one end, which is connected to a vacuumpump If the guinea and feather are allowed to fall through the tube full ofair, they will not strike the bottom at the same time The guinea will landfirst and the feather will flutter slowly to the bottom due to air resistance

If the vacuum pump is used to remove the air from the tube, both objectswill strike the bottom at the same time

Acceleration due to Gravity

Today we know that when objects are dropped from a height close toEarth’s surface, they accelerate downward at a rate of 9.8 m/s2

This

num-ber is known as the acceleration due to gravity It doesn’t depend on the

object’s mass For this value to be valid, we must assume that air resistance

is negligible and that Earth is a sphere of constant density and radius InSection 1.15, we will study gravity in greater depth

A marble is dropped from the top of the CN Tower, 553 m above the ground

a) How long does it take the marble to reach the ground?

b) What is the marble’s final speed just before it hits the ground?

c) What is the marble’s speed at the halfway point of its journey?

S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

Given

d  553 m v1 0 a  g  9.8 m/s2

a) We choose down to be the positive direction To calculate the time, we

use the equation

Fig.1.14 A hammer and a feather

are dropped on the Moon Which

will land first?

Fig.1.15 The guinea and feather

demonstration

Fig.1.16 The CN Tower in

Toronto, Ontario

Therefore, the marble takes 11 s to reach the ground

b) To find the final speed, we use the equation

a) How high does the ball go?

b) How long will it take the ball to reach maximum height?

c) How long will it take before the ball returns to the thrower’s hand?

2) At its maximum height, the ball will come to rest After that, it will fall

back down into the thrower’s hand This problem is an example ofsymmetry because the amount of time it takes the ball to travelupward to maximum height equals the amount of time it takes the ball

to fall back down Also because of symmetry, the velocity with whichthe ball strikes the thrower’s hand equals its initial upward velocity

3) The acceleration is constant in both magnitude and direction for the

entire motion For this reason, the ball slows down as it goes up andspeeds up as it falls down

(89

.0.8

mm

//

ss

)2 2)

88

.0m

m/s

/2s



t  0.82 s

Therefore, the ball reaches maximum height in 0.82 s

c) Because of symmetry, we know that the time to go up equals the time

to come down The time for the ball to go up and come back down issimply twice the answer in b); that is, 1.6 s

In this problem, we are ignoring

the effects of air resistance.

A rock is thrown vertically upward from the edge of a cliff at an initial

velocity of 10.0 m/s It hits the beach below the cliff 4.0 s later How far

down from the top of the cliff is the beach? Consider up to be positive

Therefore, the beach is 38.4 m below the top of the cliff

1 An arrow is shot straight up in the air at 80.0 m/s Find

a) its maximum height.

b) how long it will take the arrow to reach maximum height.

c) the length of time the arrow is in the air.

2 Tom is standing on a bridge 30.0 m above the water.

a) If he throws a stone down at 4.0 m/s, how long will it take to

reach the water?

b) How long will the stone take to reach the water if Tom throws it

up at 4.0 m/s?

3 A ball thrown from the edge of a 35-m-high cliff takes 3.5 s to reach

the ground below What was the ball’s initial velocity?

1.8 A Graphical Analysis of Linear Motion

So far, the examples we have studied have been algebraic problems We havetherefore used algebraic solutions Often in physics, especially while per-forming experiments, data is presented in graphical form So, physicistsneed to be able to analyze graphical data

There are three main types of graphs used in kinematics: position–timegraphs, velocity–time graphs, and acceleration–time graphs The relationshipsamong these graphs provide us with some of our most powerful analytical tools

Velocity

Figure 1.18 shows the position–time graph for an air-hockey puck movingdown the table This simple example provides us with a considerableamount of information about the motion of the object Recall that

slope  

r

ru

isn

ms)

By calculating the slope of the linear graph, we can determine the velocity

of the air-hockey puck in metres per second From this result, we can clude that:

con-The slope of a position–time graph gives the velocity of the object

25

10 15 20

If the slope of a position–time graph gives velocity, and uniform motion is

constant velocity, then the graph must have a constant slope (i.e., be a

straight line) In other words,

If an object is undergoing uniform motion, its position–time graph must

be a straight line

Not all position–time graphs are straight lines Some are curves, and some are

a complex combination of curves and straight lines Regardless of the graph’s

shape, the slope of the position–time graph gives the velocity of the object

Figure 1.19 summarizes the information we can obtain from position–

Positive acceleration

of the line

 0?

Constant velocity (uniform motion)

Increasing positive velocity

Decreasing negative velocity

Negative

acceleration

Decreasing positive velocity

Increasing negative velocity





d t

Slowing down

Speeding up

co

nnectctingthe

C

o

n c e p ts

Figure 1.20 shows the slope of the tangent at points A and B on an

increasing position–time graph At point B, the velocity of the object (i.e.,the slope of the tangent) is greater than at point A The graph also shows a

line joining points A and B The slope of this secant gives us the average

velocity between points A and B

t



Average velocity is the slope of a line connecting two points on a

position–time graph For position–time graphs representing uniformacceleration, the instantaneous velocity of an object can be determined

by finding the slope of the tangent to the curve

The graph in Figure 1.21 represents the motion of a lime-green AMCPacer, which has started to roll downhill after its parking brake hasdisengaged Using this data, determine the slope of the tangent to theposition–time graph at four different points Then plot the correspondingvelocity–time graph, and find its slope Consider positive values to bedown the hill

S o l u t i o n a n d C o n n e c t i o n t o T h e o r y

When we calculate the slope (i.e., the velocity) at four different pointsalong the curve in Figure 1.22a, we find that these values are increasing

An increasing slope indicates acceleration Since the velocity–time graph

is a straight line (Figure 1.22c), we know that the acceleration is uniform

Time t (s)

50 40 30 20 10

Fig.1.20 The slope of the secant

joining A to B is the average velocity

of that portion of the motion That

slope lies between the values of the

slopes of the tangents at A and B.

(c)

Now we can find the slope of the

velocity–time graph (Figure 1.22c):

/s

slope  4.0 m/s2 acceleration

From this example, we have determined that:

The slope of a straight-line velocity–time graph is the constant

accelera-tion of the object

By analogy,

If the velocity–time graph is a curve (Fig.1.22d), the slope of its tangent

at any given point is the instantaneous acceleration of the object

What can we learn by finding the area under a velocity–time graph? Let’s

look at the following example:

Fig.1.22d The slope of a tangent

drawn to a point on a v–t graph gives

the instantaneous acceleration at that time

e x a m p l e 1 4 The area under a velocity–time graph

What is the area under the graph in Figure 1.23 for the first 3.5 s? (Be sure

to include the correct units.)