The Free High School Science Texts: Textbooks for High School Students Physics

The Free High School Science Texts: Textbooks for High School Students Physics
FHSST Authors
The Free High School Science Texts:
Textbooks for High School Students
Studying the Sciences
Physics
Grades 10 - 12
Version 0
November 9, 2008
ii
Copyright 2007 “Free High School Science Texts”
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FHSST Editors
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iii
iv
Contents
I
Introduction
1
1 What is Physics?
3
II
5
Grade 10 - Physics
2 Units
9
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Unit Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.1
SI Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.2
The Other Systems of Units . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3
Writing Units as Words or Symbols . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4
Combinations of SI Base Units . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5
Rounding, Scientific Notation and Significant Figures . . . . . . . . . . . . . . . 12
2.5.1
Rounding Off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5.2
Error Margins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5.3
Scientific Notation
2.5.4
Significant Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6
Prefixes of Base Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.7
The Importance of Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8
How to Change Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8.1
2.9
Two other useful conversions . . . . . . . . . . . . . . . . . . . . . . . . 19
A sanity test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.11 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Motion in One Dimension - Grade 10
23
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2
Reference Point, Frame of Reference and Position . . . . . . . . . . . . . . . . . 23
3.3
3.4
3.2.1
Frames of Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.2
Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Displacement and Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1
Interpreting Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2
Differences between Distance and Displacement . . . . . . . . . . . . . . 29
Speed, Average Velocity and Instantaneous Velocity . . . . . . . . . . . . . . . . 31
v
CONTENTS
3.4.1
CONTENTS
Differences between Speed and Velocity . . . . . . . . . . . . . . . . . . 35
3.5
Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6
Description of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.6.1
Stationary Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6.2
Motion at Constant Velocity . . . . . . . . . . . . . . . . . . . . . . . . 41
3.6.3
Motion at Constant Acceleration . . . . . . . . . . . . . . . . . . . . . . 46
3.7
Summary of Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.8
Worked Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.9
Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.9.1
Finding the Equations of Motion . . . . . . . . . . . . . . . . . . . . . . 54
3.10 Applications in the Real-World . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.12 End of Chapter Exercises: Motion in One Dimension . . . . . . . . . . . . . . . 62
4 Gravity and Mechanical Energy - Grade 10
4.1
Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1
4.2
67
Differences between Mass and Weight . . . . . . . . . . . . . . . . . . . 68
Acceleration due to Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1
Gravitational Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.2
Free fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3
Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.4
Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.1
4.5
Checking units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Mechanical Energy
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.5.1
Conservation of Mechanical Energy . . . . . . . . . . . . . . . . . . . . . 78
4.5.2
Using the Law of Conservation of Energy . . . . . . . . . . . . . . . . . 79
4.6
Energy graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.8
End of Chapter Exercises: Gravity and Mechanical Energy . . . . . . . . . . . . 84
5 Transverse Pulses - Grade 10
87
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2
What is a medium? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3
What is a pulse? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.4
5.3.1
Pulse Length and Amplitude . . . . . . . . . . . . . . . . . . . . . . . . 88
5.3.2
Pulse Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Graphs of Position and Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.4.1
Motion of a Particle of the Medium . . . . . . . . . . . . . . . . . . . . 90
5.4.2
Motion of the Pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.5
Transmission and Reflection of a Pulse at a Boundary . . . . . . . . . . . . . . . 96
5.6
Reflection of a Pulse from Fixed and Free Ends . . . . . . . . . . . . . . . . . . 97
5.6.1
Reflection of a Pulse from a Fixed End . . . . . . . . . . . . . . . . . . . 97
vi
CONTENTS
5.6.2
CONTENTS
Reflection of a Pulse from a Free End . . . . . . . . . . . . . . . . . . . 98
5.7
Superposition of Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.8
Exercises - Transverse Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6 Transverse Waves - Grade 10
105
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2
What is a transverse wave? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.1
Peaks and Troughs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2.2
Amplitude and Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2.3
Points in Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.2.4
Period and Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2.5
Speed of a Transverse Wave . . . . . . . . . . . . . . . . . . . . . . . . 111
6.3
Graphs of Particle Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.4
Standing Waves and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 118
6.4.1
Reflection of a Transverse Wave from a Fixed End . . . . . . . . . . . . 118
6.4.2
Reflection of a Transverse Wave from a Free End . . . . . . . . . . . . . 118
6.4.3
Standing Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
6.4.4
Nodes and anti-nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.4.5
Wavelengths of Standing Waves with Fixed and Free Ends . . . . . . . . 122
6.4.6
Superposition and Interference . . . . . . . . . . . . . . . . . . . . . . . 125
6.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.6
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7 Geometrical Optics - Grade 10
129
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.2
Light Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3
7.4
7.5
7.2.1
Shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.2.2
Ray Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.3.1
Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3.2
Law of Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.3.3
Types of Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7.4.1
Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
7.4.2
Snell’s Law
7.4.3
Apparent Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.5.1
Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.5.2
Plane Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
7.5.3
Ray Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.5.4
Spherical Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7.5.5
Concave Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
vii
CONTENTS
7.6
CONTENTS
7.5.6
Convex Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.5.7
Summary of Properties of Mirrors . . . . . . . . . . . . . . . . . . . . . 154
7.5.8
Magnification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Total Internal Reflection and Fibre Optics . . . . . . . . . . . . . . . . . . . . . 156
7.6.1
Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.6.2
Fibre Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.8
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
8 Magnetism - Grade 10
167
8.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.2
Magnetic fields
8.3
Permanent magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
8.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.3.1
The poles of permanent magnets . . . . . . . . . . . . . . . . . . . . . . 169
8.3.2
Magnetic attraction and repulsion . . . . . . . . . . . . . . . . . . . . . 169
8.3.3
Representing magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . 170
The compass and the earth’s magnetic field . . . . . . . . . . . . . . . . . . . . 173
8.4.1
The earth’s magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.5
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.6
End of chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
9 Electrostatics - Grade 10
177
9.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.2
Two kinds of charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.3
Unit of charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.4
Conservation of charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
9.5
Force between Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
9.6
Conductors and insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
9.6.1
9.7
The electroscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
Attraction between charged and uncharged objects . . . . . . . . . . . . . . . . 183
9.7.1
Polarisation of Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.8
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
9.9
End of chapter exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
10 Electric Circuits - Grade 10
187
10.1 Electric Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10.1.1 Closed circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
10.1.2 Representing electric circuits . . . . . . . . . . . . . . . . . . . . . . . . 188
10.2 Potential Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.2.1 Potential Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
10.2.2 Potential Difference and Parallel Resistors . . . . . . . . . . . . . . . . . 193
10.2.3 Potential Difference and Series Resistors . . . . . . . . . . . . . . . . . . 194
10.2.4 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
viii
CONTENTS
CONTENTS
10.2.5 EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
10.3 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.3.1 Flow of Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.3.2 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
10.3.3 Series Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
10.3.4 Parallel Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
10.4 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.4.1 What causes resistance? . . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.4.2 Resistors in electric circuits . . . . . . . . . . . . . . . . . . . . . . . . . 202
10.5 Instruments to Measure voltage, current and resistance . . . . . . . . . . . . . . 204
10.5.1 Voltmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.5.2 Ammeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.5.3 Ohmmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
10.5.4 Meters Impact on Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 205
10.6 Exercises - Electric circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
III
Grade 11 - Physics
11 Vectors
209
211
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.2 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
11.3.1 Mathematical Representation . . . . . . . . . . . . . . . . . . . . . . . . 212
11.3.2 Graphical Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.4 Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.4.1 Relative Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
11.4.2 Compass Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11.4.3 Bearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
11.5 Drawing Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
11.6 Mathematical Properties of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.6.1 Adding Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
11.6.2 Subtracting Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
11.6.3 Scalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.7 Techniques of Vector Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.7.1 Graphical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
11.7.2 Algebraic Addition and Subtraction of Vectors . . . . . . . . . . . . . . . 223
11.8 Components of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
11.8.1 Vector addition using components . . . . . . . . . . . . . . . . . . . . . 231
11.8.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
11.8.3 End of chapter exercises: Vectors . . . . . . . . . . . . . . . . . . . . . . 236
11.8.4 End of chapter exercises: Vectors - Long questions . . . . . . . . . . . . 237
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12 Force, Momentum and Impulse - Grade 11
CONTENTS
239
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.2 Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.2.1 What is a force? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
12.2.2 Examples of Forces in Physics . . . . . . . . . . . . . . . . . . . . . . . 240
12.2.3 Systems and External Forces . . . . . . . . . . . . . . . . . . . . . . . . 241
12.2.4 Force Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
12.2.5 Free Body Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
12.2.6 Finding the Resultant Force . . . . . . . . . . . . . . . . . . . . . . . . . 244
12.2.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
12.3 Newton’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
12.3.1 Newton’s First Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
12.3.2 Newton’s Second Law of Motion . . . . . . . . . . . . . . . . . . . . . . 249
12.3.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
12.3.4 Newton’s Third Law of Motion . . . . . . . . . . . . . . . . . . . . . . . 263
12.3.5 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
12.3.6 Different types of forces . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
12.3.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
12.3.8 Forces in equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
12.3.9 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
12.4 Forces between Masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
12.4.1 Newton’s Law of Universal Gravitation . . . . . . . . . . . . . . . . . . . 282
12.4.2 Comparative Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
12.4.3 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
12.5 Momentum and Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
12.5.1 Vector Nature of Momentum . . . . . . . . . . . . . . . . . . . . . . . . 290
12.5.2 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
12.5.3 Change in Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
12.5.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
12.5.5 Newton’s Second Law revisited . . . . . . . . . . . . . . . . . . . . . . . 293
12.5.6 Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
12.5.7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
12.5.8 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 297
12.5.9 Physics in Action: Impulse . . . . . . . . . . . . . . . . . . . . . . . . . 300
12.5.10 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
12.6 Torque and Levers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
12.6.1 Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
12.6.2 Mechanical Advantage and Levers . . . . . . . . . . . . . . . . . . . . . 305
12.6.3 Classes of levers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
12.6.4 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
12.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
12.8 End of Chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
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13 Geometrical Optics - Grade 11
327
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
13.2 Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
13.2.1 Converging Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
13.2.2 Diverging Lenses
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
13.2.3 Summary of Image Properties
. . . . . . . . . . . . . . . . . . . . . . . 343
13.3 The Human Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
13.3.1 Structure of the Eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
13.3.2 Defects of Vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
13.4 Gravitational Lenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
13.5 Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
13.5.1 Refracting Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
13.5.2 Reflecting Telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
13.5.3 Southern African Large Telescope . . . . . . . . . . . . . . . . . . . . . 348
13.6 Microscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
13.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
13.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352
14 Longitudinal Waves - Grade 11
355
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
14.2 What is a longitudinal wave? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
14.3 Characteristics of Longitudinal Waves . . . . . . . . . . . . . . . . . . . . . . . 356
14.3.1 Compression and Rarefaction . . . . . . . . . . . . . . . . . . . . . . . . 356
14.3.2 Wavelength and Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 357
14.3.3 Period and Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
14.3.4 Speed of a Longitudinal Wave . . . . . . . . . . . . . . . . . . . . . . . 358
14.4 Graphs of Particle Position, Displacement, Velocity and Acceleration . . . . . . . 359
14.5 Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
14.6 Seismic Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
14.7 Summary - Longitudinal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
14.8 Exercises - Longitudinal Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
15 Sound - Grade 11
363
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
15.2 Characteristics of a Sound Wave . . . . . . . . . . . . . . . . . . . . . . . . . . 363
15.2.1 Pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
15.2.2 Loudness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
15.2.3 Tone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
15.3 Speed of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
15.4 Physics of the Ear and Hearing . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
15.4.1 Intensity of Sound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
15.5 Ultrasound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
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15.6 SONAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
15.6.1 Echolocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
15.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
15.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
16 The Physics of Music - Grade 11
373
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
16.2 Standing Waves in String Instruments . . . . . . . . . . . . . . . . . . . . . . . 373
16.3 Standing Waves in Wind Instruments . . . . . . . . . . . . . . . . . . . . . . . . 377
16.4 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
16.5 Music and Sound Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
16.6 Summary - The Physics of Music . . . . . . . . . . . . . . . . . . . . . . . . . . 385
16.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
17 Electrostatics - Grade 11
387
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
17.2 Forces between charges - Coulomb’s Law . . . . . . . . . . . . . . . . . . . . . . 387
17.3 Electric field around charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
17.3.1 Electric field lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
17.3.2 Positive charge acting on a test charge . . . . . . . . . . . . . . . . . . . 393
17.3.3 Combined charge distributions . . . . . . . . . . . . . . . . . . . . . . . 394
17.3.4 Parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
17.4 Electrical potential energy and potential . . . . . . . . . . . . . . . . . . . . . . 400
17.4.1 Electrical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
17.4.2 Real-world application: lightning . . . . . . . . . . . . . . . . . . . . . . 402
17.5 Capacitance and the parallel plate capacitor . . . . . . . . . . . . . . . . . . . . 403
17.5.1 Capacitors and capacitance . . . . . . . . . . . . . . . . . . . . . . . . . 403
17.5.2 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
17.5.3 Physical properties of the capacitor and capacitance . . . . . . . . . . . . 404
17.5.4 Electric field in a capacitor . . . . . . . . . . . . . . . . . . . . . . . . . 405
17.6 Capacitor as a circuit device . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
17.6.1 A capacitor in a circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
17.6.2 Real-world applications: capacitors . . . . . . . . . . . . . . . . . . . . . 407
17.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
17.8 Exercises - Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
18 Electromagnetism - Grade 11
413
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
18.2 Magnetic field associated with a current . . . . . . . . . . . . . . . . . . . . . . 413
18.2.1 Real-world applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
18.3 Current induced by a changing magnetic field . . . . . . . . . . . . . . . . . . . 420
18.3.1 Real-life applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422
18.4 Transformers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
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18.4.1 Real-world applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
18.5 Motion of a charged particle in a magnetic field . . . . . . . . . . . . . . . . . . 425
18.5.1 Real-world applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
18.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
18.7 End of chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
19 Electric Circuits - Grade 11
429
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
19.2 Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
19.2.1 Definition of Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 429
19.2.2 Ohmic and non-ohmic conductors . . . . . . . . . . . . . . . . . . . . . 431
19.2.3 Using Ohm’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
19.3 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
19.3.1 Equivalent resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
19.3.2 Use of Ohm’s Law in series and parallel Circuits . . . . . . . . . . . . . . 438
19.3.3 Batteries and internal resistance . . . . . . . . . . . . . . . . . . . . . . 440
19.4 Series and parallel networks of resistors . . . . . . . . . . . . . . . . . . . . . . . 442
19.5 Wheatstone bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
19.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
19.7 End of chapter exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
20 Electronic Properties of Matter - Grade 11
451
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
20.2 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
20.2.1 Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
20.2.2 Insulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
20.2.3 Semi-conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
20.3 Intrinsic Properties and Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
20.3.1 Surplus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
20.3.2 Deficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
20.4 The p-n junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
20.4.1 Differences between p- and n-type semi-conductors . . . . . . . . . . . . 457
20.4.2 The p-n Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
20.4.3 Unbiased . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
20.4.4 Forward biased
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
20.4.5 Reverse biased . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
20.4.6 Real-World Applications of Semiconductors . . . . . . . . . . . . . . . . 458
20.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459
IV
Grade 12 - Physics
21 Motion in Two Dimensions - Grade 12
461
463
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
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21.2 Vertical Projectile Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463
21.2.1 Motion in a Gravitational Field . . . . . . . . . . . . . . . . . . . . . . . 463
21.2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
21.2.3 Graphs of Vertical Projectile Motion . . . . . . . . . . . . . . . . . . . . 467
21.3 Conservation of Momentum in Two Dimensions . . . . . . . . . . . . . . . . . . 475
21.4 Types of Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
21.4.1 Elastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
21.4.2 Inelastic Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
21.5 Frames of Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
21.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
21.5.2 What is a frame of reference? . . . . . . . . . . . . . . . . . . . . . . . 491
21.5.3 Why are frames of reference important? . . . . . . . . . . . . . . . . . . 491
21.5.4 Relative Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
21.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494
21.7 End of chapter exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
22 Mechanical Properties of Matter - Grade 12
503
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
22.2 Deformation of materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
22.2.1 Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
22.2.2 Deviation from Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . 506
22.3 Elasticity, plasticity, fracture, creep . . . . . . . . . . . . . . . . . . . . . . . . . 508
22.3.1 Elasticity and plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
22.3.2 Fracture, creep and fatigue . . . . . . . . . . . . . . . . . . . . . . . . . 508
22.4 Failure and strength of materials . . . . . . . . . . . . . . . . . . . . . . . . . . 509
22.4.1 The properties of matter . . . . . . . . . . . . . . . . . . . . . . . . . . 509
22.4.2 Structure and failure of materials . . . . . . . . . . . . . . . . . . . . . . 509
22.4.3 Controlling the properties of materials . . . . . . . . . . . . . . . . . . . 509
22.4.4 Steps of Roman Swordsmithing . . . . . . . . . . . . . . . . . . . . . . . 510
22.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
22.6 End of chapter exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
23 Work, Energy and Power - Grade 12
513
23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
23.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
23.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
23.3.1 External and Internal Forces . . . . . . . . . . . . . . . . . . . . . . . . 519
23.3.2 Capacity to do Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520
23.4 Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
23.5 Important Equations and Quantities . . . . . . . . . . . . . . . . . . . . . . . . 529
23.6 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
xiv
CONTENTS
CONTENTS
24 Doppler Effect - Grade 12
533
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
24.2 The Doppler Effect with Sound and Ultrasound . . . . . . . . . . . . . . . . . . 533
24.2.1 Ultrasound and the Doppler Effect . . . . . . . . . . . . . . . . . . . . . 537
24.3 The Doppler Effect with Light . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
24.3.1 The Expanding Universe . . . . . . . . . . . . . . . . . . . . . . . . . . 538
24.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
24.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
25 Colour - Grade 12
541
25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
25.2 Colour and Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
25.2.1 Dispersion of white light . . . . . . . . . . . . . . . . . . . . . . . . . . 544
25.3 Addition and Subtraction of Light . . . . . . . . . . . . . . . . . . . . . . . . . 544
25.3.1 Additive Primary Colours . . . . . . . . . . . . . . . . . . . . . . . . . . 544
25.3.2 Subtractive Primary Colours . . . . . . . . . . . . . . . . . . . . . . . . 545
25.3.3 Complementary Colours . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
25.3.4 Perception of Colour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546
25.3.5 Colours on a Television Screen . . . . . . . . . . . . . . . . . . . . . . . 547
25.4 Pigments and Paints
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548
25.4.1 Colour of opaque objects . . . . . . . . . . . . . . . . . . . . . . . . . . 548
25.4.2 Colour of transparent objects . . . . . . . . . . . . . . . . . . . . . . . . 548
25.4.3 Pigment primary colours . . . . . . . . . . . . . . . . . . . . . . . . . . 549
25.5 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 550
26 2D and 3D Wavefronts - Grade 12
553
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
26.2 Wavefronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
26.3 The Huygens Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
26.4 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556
26.5 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
26.5.1 Diffraction through a Slit . . . . . . . . . . . . . . . . . . . . . . . . . . 558
26.6 Shock Waves and Sonic Booms . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
26.6.1 Subsonic Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
26.6.2 Supersonic Flight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
26.6.3 Mach Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566
26.7 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
27 Wave Nature of Matter - Grade 12
571
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
27.2 de Broglie Wavelength
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571
27.3 The Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574
27.3.1 Disadvantages of an Electron Microscope . . . . . . . . . . . . . . . . . 577
xv
CONTENTS
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27.3.2 Uses of Electron Microscopes . . . . . . . . . . . . . . . . . . . . . . . . 577
27.4 End of Chapter Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578
28 Electrodynamics - Grade 12
579
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
28.2 Electrical machines - generators and motors . . . . . . . . . . . . . . . . . . . . 579
28.2.1 Electrical generators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
28.2.2 Electric motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
28.2.3 Real-life applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
28.2.4 Exercise - generators and motors . . . . . . . . . . . . . . . . . . . . . . 584
28.3 Alternating Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
28.3.1 Exercise - alternating current . . . . . . . . . . . . . . . . . . . . . . . . 586
28.4 Capacitance and inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
28.4.1 Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
28.4.2 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
28.4.3 Exercise - capacitance and inductance . . . . . . . . . . . . . . . . . . . 588
28.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588
28.6 End of chapter exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
29 Electronics - Grade 12
591
29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591
29.2 Capacitive and Inductive Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . 591
29.3 Filters and Signal Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596
29.3.1 Capacitors and Inductors as Filters . . . . . . . . . . . . . . . . . . . . . 596
29.3.2 LRC Circuits, Resonance and Signal Tuning . . . . . . . . . . . . . . . . 596
29.4 Active Circuit Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
29.4.1 The Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
29.4.2 The Light Emitting Diode (LED) . . . . . . . . . . . . . . . . . . . . . . 601
29.4.3 Transistor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
29.4.4 The Operational Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . 607
29.5 The Principles of Digital Electronics . . . . . . . . . . . . . . . . . . . . . . . . 609
29.5.1 Logic Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 610
29.6 Using and Storing Binary Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 616
29.6.1 Binary numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616
29.6.2 Counting circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
29.6.3 Storing binary numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
30 EM Radiation
625
30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
30.2 Particle/wave nature of electromagnetic radiation . . . . . . . . . . . . . . . . . 625
30.3 The wave nature of electromagnetic radiation . . . . . . . . . . . . . . . . . . . 626
30.4 Electromagnetic spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
30.5 The particle nature of electromagnetic radiation . . . . . . . . . . . . . . . . . . 629
xvi
CONTENTS
CONTENTS
30.5.1 Exercise - particle nature of EM waves . . . . . . . . . . . . . . . . . . . 630
30.6 Penetrating ability of electromagnetic radiation . . . . . . . . . . . . . . . . . . 631
30.6.1 Ultraviolet(UV) radiation and the skin . . . . . . . . . . . . . . . . . . . 631
30.6.2 Ultraviolet radiation and the eyes . . . . . . . . . . . . . . . . . . . . . . 632
30.6.3 X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
30.6.4 Gamma-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
30.6.5 Exercise - Penetrating ability of EM radiation . . . . . . . . . . . . . . . 633
30.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
30.8 End of chapter exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633
31 Optical Phenomena and Properties of Matter - Grade 12
635
31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
31.2 The transmission and scattering of light . . . . . . . . . . . . . . . . . . . . . . 635
31.2.1 Energy levels of an electron . . . . . . . . . . . . . . . . . . . . . . . . . 635
31.2.2 Interaction of light with metals . . . . . . . . . . . . . . . . . . . . . . . 636
31.2.3 Why is the sky blue? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
31.3 The photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
31.3.1 Applications of the photoelectric effect . . . . . . . . . . . . . . . . . . . 640
31.3.2 Real-life applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642
31.4 Emission and absorption spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 643
31.4.1 Emission Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643
31.4.2 Absorption spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
31.4.3 Colours and energies of electromagnetic radiation . . . . . . . . . . . . . 646
31.4.4 Applications of emission and absorption spectra . . . . . . . . . . . . . . 648
31.5 Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
31.5.1 How a laser works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 652
31.5.2 A simple laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654
31.5.3 Laser applications and safety . . . . . . . . . . . . . . . . . . . . . . . . 655
31.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656
31.7 End of chapter exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
V
Exercises
659
32 Exercises
VI
661
Essays
663
Essay 1: Energy and electricity. Why the fuss?
665
33 Essay: How a cell phone works
671
34 Essay: How a Physiotherapist uses the Concept of Levers
673
35 Essay: How a Pilot Uses Vectors
675
xvii
CONTENTS
CONTENTS
A GNU Free Documentation License
677
xviii
Chapter 3
Motion in One Dimension - Grade
10
3.1
Introduction
This chapter is about how things move in a straight line or more scientifically how things move in
one dimension. This is useful for learning how to describe the movement of cars along a straight
road or of trains along straight railway tracks. If you want to understand how any object moves,
for example a car on the freeway, a soccer ball being kicked towards the goal or your dog chasing
the neighbour’s cat, then you have to understand three basic ideas about what it means when
something is moving. These three ideas describe different parts of exactly how an object moves.
They are:
1. position or displacement which tells us exactly where the object is,
2. speed or velocity which tells us exactly how fast the object’s position is changing or more
familiarly, how fast the object is moving, and
3. acceleration which tells us exactly how fast the object’s velocity is changing.
You will also learn how to use position, displacement, speed, velocity and acceleration to describe
the motion of simple objects. You will learn how to read and draw graphs that summarise the
motion of a moving object. You will also learn about the equations that can be used to describe
motion and how to apply these equations to objects moving in one dimension.
3.2
Reference Point, Frame of Reference and Position
The most important idea when studying motion, is you have to know where you are. The
word position describes your location (where you are). However, saying that you are here is
meaningless, and you have to specify your position relative to a known reference point. For
example, if you are 2 m from the doorway, inside your classroom then your reference point is
the doorway. This defines your position inside the classroom. Notice that you need a reference
point (the doorway) and a direction (inside) to define your location.
3.2.1
Frames of Reference
Definition: Frame of Reference
A frame of reference is a reference point combined with a set of directions.
A frame of reference is similar to the idea of a reference point. A frame of reference is defined
as a reference point combined with a set of directions. For example, a boy is standing still inside
23
3.2
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
a train as it pulls out of a station. You are standing on the platform watching the train move
from left to right. To you it looks as if the boy is moving from left to right, because relative
to where you are standing (the platform), he is moving. According to the boy, and his frame of
reference (the train), he is not moving.
24
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
boy is standing still
3.2
train moving from left to right
b
From your frame of reference the boy is moving from left to right.
Figure 3.1: Frames of Reference
A frame of reference must have an origin (where you are standing on the platform) and at least
a positive direction. The train was moving from left to right, making to your right positive and
to your left negative. If someone else was looking at the same boy, his frame of reference will
be different. For example, if he was standing on the other side of the platform, the boy will be
moving from right to left.
For this chapter, we will only use frames of reference in the x-direction. Frames of reference will
be covered in more detail in Grade 12.
A boy inside a train which
is moving from left to right
negative direction (towards your left)
positive direction (towards your right)
Where you are standing
on the platform
(reference point or origin)
3.2.2
Position
Definition: Position
Position is a measurement of a location, with reference to an origin.
A position is a measurement of a location, with reference to an origin. Positions can therefore be
negative or positive. The symbol x is used to indicate position. x has units of length for example
cm, m or km. Figure 3.2.2 shows the position of a school. Depending on what reference point
we choose, we can say that the school is 300 m from Joan’s house (with Joan’s house as the
reference point or origin) or 500 m from Joel’s house (with Joel’s house as the reference point
or origin).
School
Jack
100 m
Joan
John
100 m
100 m
Jill
100 m
Shop
Joel
100 m
100 m
Figure 3.2: Illustration of position
The shop is also 300 m from Joan’s house, but in the opposite direction as the school. When
we choose a reference point, we have a positive direction and a negative direction. If we choose
25
3.2
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Joan’s house
(reference point)
School
Shop
x (m)
0
+300 +200 +100
-100
-200 -300
Figure 3.3: The origin is at Joan’s house and the position of the school is +300 m. Positions
towards the left are defined as positive and positions towards the right are defined as negative.
the direction towards the school as positive, then the direction towards the shop is negative. A
negative direction is always opposite to the direction chosen as positive.
Activity :: Discussion : Reference Points
Divide into groups of 5 for this activity. On a straight line, choose a reference point. Since position can have both positive and negative values, discuss the
advantages and disadvantages of choosing
1. either end of the line,
2. the middle of the line.
This reference point can also be called “the origin”.
Exercise: Position
1. Write down the positions for objects at A, B, D and E. Do not forget the units.
reference point
B
D
A
E
x (m)
-4
-3
-2
-1
0
1
2
3
4
2. Write down the positions for objects at F, G, H and J. Do not forget the units.
reference point
G
H
F
J
x (m)
4
3
2
1
0
-1
-2
-3
-4
3. There are 5 houses on Newton Street, A, B, C, D and E. For all cases, assume
that positions to the right are positive.
20 m
A
20 m
B
20 m
C
26
20 m
D
E
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
(a) Draw a frame of reference with house A as the origin and write down the
positions of houses B, C, D and E.
(b) You live in house C. What is your position relative to house E?
(c) What are the positions of houses A, B and D, if house B is taken as the
reference point?
27
3.2
3.3
3.3
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Displacement and Distance
Definition: Displacement
Displacement is the change in an object’s position.
The displacement of an object is defined as its change in position (final position minus initial
position). Displacement has a magnitude and direction and is therefore a vector. For example,
if the initial position of a car is xi and it moves to a final position of xf , then the displacement
is:
xf − xi
However, subtracting an initial quantity from a final quantity happens often in Physics, so we
use the shortcut ∆ to mean final - initial. Therefore, displacement can be written:
∆x = xf − xi
Important: The symbol ∆ is read out as delta. ∆ is a letter of the Greek alphabet and is
used in Mathematics and Science to indicate a change in a certain quantity, or a final value
minus an initial value. For example, ∆x means change in x while ∆t means change in t.
Important: The words initial and final will be used very often in Physics. Initial will always
refer to something that happened earlier in time and final will always refer to something
that happened later in time. It will often happen that the final value is smaller than the
initial value, such that the difference is negative. This is ok!
Finish
(Shop)
b
sp
Di
l ac
em
en
t
b
Start
(School)
Figure 3.4: Illustration of displacement
Displacement does not depend on the path travelled, but only on the initial and final positions
(Figure 3.4). We use the word distance to describe how far an object travels along a particular
path. Distance is the actual distance that was covered. Distance (symbol d) does not have a
direction, so it is a scalar. Displacement is the shortest distance from the starting point to the
endpoint – from the school to the shop in the figure. Displacement has direction and is therefore
a vector.
Figure 3.2.2 shows the five houses we discussed earlier. Jack walks to school, but instead of
walking straight to school, he decided to walk to his friend Joel’s house first to fetch him so that
they can walk to school together. Jack covers a distance of 400 m to Joel’s house and another
500 m to school. He covers a distance of 900 m. His displacement, however, is only 100 m
towards the school. This is because displacement only looks at the starting position (his house)
and the end position (the school). It does not depend on the path he travelled.
28
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.3
To calculate his distance and displacement, we need to choose a reference point and a direction.
Let’s choose Jack’s house as the reference point, and towards Joel’s house as the positive direction (which means that towards the school is negative). We would do the calculations as follows:
Distance(d)
= path travelled
= 400 m + 500 m
Displacement(∆x)
= 900 m
=
=
xf − xi
−100 m + 0 m
=
−100 m
Joel walks to school with Jack and after school walks back home. What is Joel’s displacement
and what distance did he cover? For this calculation we use Joel’s house as the reference point.
Let’s take towards the school as the positive direction.
Distance(d)
= path travelled
= 500 m + 500 m
Displacement(∆x)
= xf − xi
= 0 m+0 m
= 0m
= 1000 m
It is possible to have a displacement of 0 m and a distance that is not 0 m. This happens
when an object completes a round trip back to its original position, like an athlete running
around a track.
3.3.1
Interpreting Direction
Very often in calculations you will get a negative answer. For example, Jack’s displacement in
the example above, is calculated as -100 m. The minus sign in front of the answer means that
his displacement is 100 m in the opposite direction (opposite to the direction chosen as positive
in the beginning of the question). When we start a calculation we choose a frame of reference
and a positive direction. In the first example above, the reference point is Jack’s house and the
positive direction is towards Joel’s house. Therefore Jack’s displacement is 100 m towards the
school. Notice that distance has no direction, but displacement has.
3.3.2
Differences between Distance and Displacement
Definition: Vectors and Scalars
A vector is a physical quantity with magnitude (size) and direction. A scalar is a physical
quantity with magnitude (size) only.
The differences between distance and displacement can be summarised as:
Distance
1. depends on the path
2. always positive
3. is a scalar
Displacement
1. independent of path taken
2. can be positive or negative
3. is a vector
Exercise: Point of Reference
1. Use Figure 3.2.2 to answer the following questions.
(a) Jill walks to Joan’s house and then to school, what is her distance and
displacement?
(b) John walks to Joan’s house and then to school, what is his distance and
displacement?
29
3.3
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
(c) Jack walks to the shop and then to school, what is his distance and displacement?
(d) What reference point did you use for each of the above questions?
2. You stand at the front door of your house (displacement, ∆x = 0 m). The
street is 10 m away from the front door. You walk to the street and back again.
(a) What is the distance you have walked?
(b) What is your final displacement?
(c) Is displacement a vector or a scalar? Give a reason for your answer.
30
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.4
3.4
Speed, Average Velocity and Instantaneous Velocity
Definition: Velocity
Velocity is the rate of change of position.
Definition: Instantaneous velocity
Instantaneous velocity is the velocity of an accelerating body at a specific instant in time.
Definition: Average velocity
Average velocity is the total displacement of a body over a time interval.
Velocity is the rate of change of position. It tells us how much an object’s position changes in
time. This is the same as the displacement divided by the time taken. Since displacement is a
vector and time taken is a scalar, velocity is also a vector. We use the symbol v for velocity. If
we have a displacement of ∆x and a time taken of ∆t, v is then defined as:
velocity (in m · s−1 ) =
v
=
change in displacement (in m)
change in time (in s)
∆x
∆t
Velocity can be positive or negative. Positive values of velocity mean that the object is moving
away from the reference point or origin and negative values mean that the object is moving
towards the reference point or origin.
Important: An instant in time is different from the time taken or the time interval. It
is therefore useful to use the symbol t for an instant in time (for example during the 4th
second) and the symbol ∆t for the time taken (for example during the first 5 seconds of
the motion).
Average velocity (symbol v) is the displacement for the whole motion divided by the time taken
for the whole motion. Instantaneous velocity is the velocity at a specific instant in time.
(Average) Speed (symbol s) is the distance travelled (d) divided by the time taken (∆t) for
the journey. Distance and time are scalars and therefore speed will also be a scalar. Speed is
calculated as follows:
speed (in m · s−1 ) =
s=
distance (in m)
time (in s)
d
∆t
Instantaneous speed is the magnitude of instantaneous velocity. It has the same value, but no
direction.
Worked Example 5: Average speed and average velocity
31
3.4
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Question: James walks 2 km away from home in 30 minutes. He then turns around
and walks back home along the same path, also in 30 minutes. Calculate James’
average speed and average velocity.
2 km
Answer
Step 1 : Identify what information is given and what is asked for
The question explicitly gives
• the distance and time out (2 km in 30 minutes)
• the distance and time back (2 km in 30 minutes)
Step 2 : Check that all units are SI units.
The information is not in SI units and must therefore be converted.
To convert km to m, we know that:
1 km = 1 000 m
∴
2 km = 2 000 m (multiply both sides by 2, because we want to convert 2 km to m.)
Similarly, to convert 30 minutes to seconds,
∴
1 min
30 min
= 60s
= 1 800 s
(multiply both sides by 30)
Step 3 : Determine James’ displacement and distance.
James started at home and returned home, so his displacement is 0 m.
∆x = 0 m
James walked a total distance of 4 000 m (2 000 m out and 2 000 m back).
d = 4 000 m
Step 4 : Determine his total time.
James took 1 800 s to walk out and 1 800 s to walk back.
∆t = 3 600 s
Step 5 : Determine his average speed
=
s
=
=
d
∆t
4 000 m
3 600 s
1,11 m · s−1
Step 6 : Determine his average velocity
v
∆x
∆t
0m
=
3 600 s
= 0 m · s−1
=
Worked Example 6: Instantaneous Speed and Velocity
32
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.4
N
Question: A man runs around a circular track of radius 100 m. It takes him 120 s
to complete a revolution of the track. If he runs at constant speed, calculate:
1. his speed,
W
E
b
2. his instantaneous velocity at point A,
A
3. his instantaneous velocity at point B,
4. his average velocity between points A and B,
bB
S
100 m
5. his average speed during a revolution.
6. his average velocity during a revolution.
Answer
Step 1 : Decide how to approach the problem
To determine the man’s speed we need to know the distance he travels and how
long it takes. We know it takes 120 s to complete one revolution of the track.(A
revolution is to go around the track once.)
33
Direction the man runs
3.4
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Step 2 : Determine the distance travelled
What distance is one revolution of the track? We know the track is a circle and we
know its radius, so we can determine the distance around the circle. We start with
the equation for the circumference of a circle
C
= 2πr
= 2π(100 m)
= 628,32 m
Therefore, the distance the man covers in one revolution is 628,32 m.
Step 3 : Determine the speed
We know that speed is distance covered per unit
time. So if we divide the distance covered by the
time it took we will know how much distance was
covered for every unit of time. No direction is
used here because speed is a scalar.
s
=
=
=
d
∆t
628,32 m
120 s
5,24 m · s−1
Step 4 : Determine the instantaneous velocity at A
b
A
Consider the point A in the diagram.
We know which way the man is running around
the track and we know his speed. His velocity
at point A will be his speed (the magnitude of
the velocity) plus his direction of motion (the
direction of his velocity). The instant that he
arrives at A he is moving as indicated in the
diagram.
Direction the man runs
b
A
His velocity will be 5,24 m·s−1 West.
Step 5 : Determine the instantaneous velocity at B
Direction the man runs
Consider the point B in the diagram.
We know which way the man is running around
the track and we know his speed. His velocity
at point B will be his speed (the magnitude of
the velocity) plus his direction of motion (the
direction of his velocity). The instant that he
arrives at B he is moving as indicated in the
diagram.
His velocity will be 5,24 m·s−1 South.
bB
bB
34
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.4
Step 6 : Determine the average velocity between A and B
To determine the average velocity between A and B, we need the change in displacement between A and B and the change in time between A and B. The displacement
from A and B can be calculated by using the Theorem of Pythagoras:
(∆x)2
=
=
∆x =
1002 + 1002
20000
A
141,42135... m
∆x
The time for a full revolution is 120 s, therefore
the time for a 14 of a revolution is 30 s.
vAB
=
=
=
∆x
∆t
141,42...
30 s
4.71 m · s−1
100 m
B
100 m
O
Velocity is a vector and needs a direction.
Triangle AOB is isosceles and therefore angle BAO = 45◦ .
The direction is between west and south and is therefore southwest.
The final answer is: v = 4.71 m·s−1 , southwest.
Step 7 : Determine his average speed during a revolution
Because he runs at a constant rate, we know that his speed anywhere around the
track will be the same. His average speed is 5,24 m·s−1 .
Step 8 : Determine his average velocity over a complete revolution
Important: Remember - displacement can be zero even when distance travelled is not!
To calculate average velocity we need his total displacement and his total time. His
displacement is zero because he ends up where he started. His time is 120 s. Using
these we can calculate his average velocity:
v
=
=
=
3.4.1
∆x
∆t
0m
120 s
0s
Differences between Speed and Velocity
The differences between speed and velocity can be summarised as:
Speed
1. depends on the path taken
2. always positive
3. is a scalar
4. no dependence on direction and
so is only positive
Velocity
1. independent of path taken
2. can be positive or negative
3. is a vector
4. direction can be guessed from
the sign (i.e. positive or negative)
Additionally, an object that makes a round trip, i.e. travels away from its starting point and then
returns to the same point has zero velocity but travels a non-zero speed.
35
3.4
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Exercise: Displacement and related quantities
1. Theresa has to walk to the shop to buy some milk. After walking 100 m, she
realises that she does not have enough money, and goes back home. If it took
her two minutes to leave and come back, calculate the following:
(a)
(b)
(c)
(d)
(e)
How long was she out of the house (the time interval ∆t in seconds)?
How far did she walk (distance (d))?
What was her displacement (∆x)?
What was her average velocity (in m·s−1 )?
What was her average speed (in m·s−1 )?
b
shop
2 minute there and back
100 m
100 m
home
2. Desmond is watching a straight stretch of road from his classroom window.
He can see two poles which he earlier measured to be 50 m apart. Using his
stopwatch, Desmond notices that it takes 3 s for most cars to travel from the
one pole to the other.
(a) Using the equation for velocity (v = ∆x
∆t ), show all the working needed to
calculate the velocity of a car travelling from the left to the right.
(b) If Desmond measures the velocity of a red Golf to be -16,67 m·s−1 , in
which direction was the Gold travelling?
Desmond leaves his stopwatch running, and notices that at t = 5,0 s, a
taxi passes the left pole at the same time as a bus passes the right pole.
At time t = 7,5 s the taxi passes the right pole. At time t = 9,0 s, the
bus passes the left pole.
(c) How long did it take the taxi and the bus to travel the distance between
the poles? (Calculate the time interval (∆t) for both the taxi and the bus).
(d) What was the velocity of the taxi and the bus?
(e) What was the speed of the taxi and the bus?
(f) What was the speed of taxi and the bus in km·h−1 ?
50 m
3s
t=9s
t=5s
t=5s
t = 7,5 s
3. After a long day, a tired man decides not to use the pedestrian bridge to cross
over a freeway, and decides instead to run across. He sees a car 100 m away
travelling towards him, and is confident that he can cross in time.
36
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
(a)
(b)
(c)
(d)
If the car is travelling at 120 km·h−1 , what is the car’s speed in m·s−1 .
How long will it take the a car to travel 100 m?
If the man is running at 10 km·h−1 , what is his speed in m·s−1 ?
If the freeway has 3 lanes, and each lane is 3 m wide, how long will it take
for the man to cross all three lanes?
(e) If the car is travelling in the furthermost lane from the man, will he be able
to cross all 3 lanes of the freeway safely?
3m
car
3m
3m
100 m
Activity :: Investigation : An Exercise in Safety
Divide into groups of 4 and perform the following investigation. Each group will
be performing the same investigation, but the aim for each group will be different.
1. Choose an aim for your investigation from the following list and formulate a
hypothesis:
• Do cars travel at the correct speed limit?
• Is is safe to cross the road outside of a pedestrian crossing?
• Does the colour of your car determine the speed you are travelling at?
• Any other relevant question that you would like to investigate.
2. On a road that you often cross, measure out 50 m along a straight section, far
away from traffic lights or intersections.
3. Use a stopwatch to record the time each of 20 cars take to travel the 50 m
section you measured.
4. Design a table to represent your results. Use the results to answer the question posed in the aim of the investigation. You might need to do some more
measurements for your investigation. Plan in your group what else needs to be
done.
5. Complete any additional measurements and write up your investigation under
the following headings:
• Aim and Hypothesis
• Apparatus
• Method
• Results
• Discussion
• Conclusion
6. Answer the following questions:
(a) How many cars took less than 3 seconds to travel 50 m?
(b) What was the shortest time a car took to travel 50 m?
(c) What was the average time taken by the 20 cars?
(d) What was the average speed of the 20 cars?
(e) Convert the average speed to km·h−1 .
37
3.4
3.5
3.5
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Acceleration
Definition: Acceleration
Acceleration is the rate of change of velocity.
Acceleration (symbol a) is the rate of change of velocity. It is a measure of how fast the velocity
of an object changes in time. If we have a change in velocity (∆v) over a time interval (∆t),
then the acceleration (a) is defined as:
acceleration (in m · s−2 ) =
change in velocity (in m · s−1 )
change in time (in s)
a=
∆v
∆t
Since velocity is a vector, acceleration is also a vector. Acceleration does not provide any information about a motion, but only about how the motion changes. It is not possible to tell how
fast an object is moving or in which direction from the acceleration.
Like velocity, acceleration can be negative or positive. We see that when the sign of the acceleration and the velocity are the same, the object is speeding up. If both velocity and acceleration
are positive, the object is speeding up in a positive direction. If both velocity and acceleration
are negative, the object is speeding up in a negative direction. If velocity is positive and acceleration is negative, then the object is slowing down. Similarly, if the velocity is negative and the
acceleration is positive the object is slowing down. This is illustrated in the following worked
example.
Worked Example 7: Acceleration
Question: A car accelerates uniformly from and initial velocity of 2 m·s−1 to a final
velocity of 10 m·s1 in 8 seconds. It then slows down uniformly to a final velocity of 4
m·s−1 in 6 seconds. Calculate the acceleration of the car during the first 8 seconds
and during the last 6 seconds.
Answer
Step 9 : Identify what information is given and what is asked for:
Consider the motion of the car in two parts: the first 8 seconds and the last 6 seconds.
For the first 8 seconds:
For the last 6 seconds:
vi
=
2 m · s−1
vi
=
10 m · s−1
vf
ti
=
=
10 m · s
0s
vf
ti
=
=
4 m · s−1
8s
tf
=
8s
tf
=
14 s
−1
Step 10 : Calculate the acceleration.
For the first 8 seconds:
For the next 6 seconds:
∆v
∆v
a =
∆t
∆t
4 − 10
10 − 2
=
=
14 − 8
8−0
−2
=
−1
m · s−2
= 1 m·s
During the first 8 seconds the car had a positive acceleration. This means that its
velocity increased. The velocity is positive so the car is speeding up. During the
next 6 seconds the car had a negative acceleration. This means that its velocity
decreased. The velocity is positive so the car is slowing down.
a
=
38
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.6
Important: Acceleration does not tell us about the direction of the motion. Acceleration
only tells us how the velocity changes.
Important: Deceleration
Avoid the use of the word deceleration to refer to a negative acceleration. This word usually
means slowing down and it is possible for an object to slow down with both a positive and
negative acceleration, because the sign of the velocity of the object must also be taken into
account to determine whether the body is slowing down or not.
Exercise: Acceleration
1. An athlete is accelerating uniformly from an initial velocity of 0 m·s−1 to a final
velocity of 4 m·s−1 in 2 seconds. Calculate his acceleration. Let the direction
that the athlete is running in be the positive direction.
2. A bus accelerates uniformly from an initial velocity of 15 m·s−1 to a final velocity
of 7 m·s−1 in 4 seconds. Calculate the acceleration of the bus. Let the direction
of motion of the bus be the positive direction.
3. An aeroplane accelerates uniformly from an initial velocity of 200 m·s−1 to a
velocity of 100 m·s−1 in 10 seconds. It then accelerates uniformly to a final
velocity of 240 m·s−1 in 20 seconds. Let the direction of motion of the aeroplane
be the positive direction.
(a) Calculate the acceleration of the aeroplane during the first 10 seconds of
the motion.
(b) Calculate the acceleration of the aeroplane during the next 14 seconds of
its motion.
(c) Calculate the acceleration of the aeroplane during the whole 24 seconds of
its motion.
3.6
Description of Motion
The purpose of this chapter is to describe motion, and now that we understand the definitions of
displacement, distance, velocity, speed and acceleration, we are ready to start using these ideas
to describe how an object is moving. There are many ways of describing motion:
1. words
2. diagrams
3. graphs
These methods will be described in this section.
We will consider three types of motion: when the object is not moving (stationary object), when
the object is moving at a constant velocity (uniform motion) and when the object is moving at
a constant acceleration (motion at constant acceleration).
39
3.6
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.6.1
Stationary Object
The simplest motion that we can come across is that of a stationary object. A stationary object
does not move and so its position does not change, for as long as it is standing still. An example
of this situation is when someone is waiting for something without moving. The person remains
in the same position.
Lesedi is waiting for a taxi. He is standing two metres from a stop street at t = 0 s. After
one minute, at t = 60 s, he is still 2 metres from the stop street and after two minutes, at
t = 120 s, also 2 metres from the stop street. His position has not changed. His displacement
is zero (because his position is the same), his velocity is zero (because his displacement is zero)
and his acceleration is also zero (because his velocity is not changing).
displacement = 0 m
STOP
bb
b
t=0s
t = 60 s
t = 120 s
velocity = 0 m·s−1
acceleration = 0 m·s−2
2m
2
1
0
time (s)
60
(a)
120
0
time (s)
60
120
acceleration a (m·s−2 )
velocity v (m·s−1 )
position x (m)
We can now draw graphs of position vs.time (x vs. t), velocity vs.time (v vs. t) and acceleration
vs.time (a vs. t) for a stationary object. The graphs are shown in Figure 3.5. Lesedi’s position
is 2 metres from the stop street. If the stop street is taken as the reference point, his position
remains at 2 metres for 120 seconds. The graph is a horisontal line at 2 m. The velocity and
acceleration graphs are also shown. They are both horisontal lines on the x-axis. Since his
position is not changing, his velocity is 0 m·s−1 and since velocity is not changing acceleration is
0 m·s−2 .
0
(b)
time (s)
60
120
(c)
Figure 3.5: Graphs for a stationary object (a) position vs. time (b) velocity vs. time (c)
acceleration vs. time.
Definition: Gradient
The gradient of a line can be calculated by dividing the change in the y-value by the change
in the x-value.
∆y
m = ∆x
Since we know that velocity is the rate of change of position, we can confirm the value for the
velocity vs. time graph, by calculating the gradient of the x vs. t graph.
Important: The gradient of a position vs. time graph gives the velocity.
40
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.6
If we calculate the gradient of the x vs. t graph for a stationary object we get:
v
=
=
=
=
∆x
∆t
xf − xi
tf − ti
2 m−2 m
(initial position = final position)
120 s − 60 s
0 m · s−1 (for the time that Lesedi is stationary)
Similarly, we can confirm the value of the acceleration by calculating the gradient of the velocity
vs. time graph.
Important: The gradient of a velocity vs. time graph gives the acceleration.
If we calculate the gradient of the v vs. t graph for a stationary object we get:
a
=
=
=
=
∆v
∆t
vf − vi
tf − ti
0 m · s−1 − 0 m · s−1
120 s − 60 s
0 m · s−2
Additionally, because the velocity vs. time graph is related to the position vs. time graph, we
can use the area under the velocity vs. time graph to calculate the displacement of an object.
Important: The area under the velocity vs. time graph gives the displacement.
The displacement of the object is given by the area under the graph, which is 0 m. This is
obvious, because the object is not moving.
3.6.2
Motion at Constant Velocity
Motion at a constant velocity or uniform motion means that the position of the object is changing
at the same rate.
Assume that Lesedi takes 100 s to walk the 100 m to the taxi-stop every morning. If we assume
that Lesedi’s house is the origin, then Lesedi’s velocity is:
v
=
=
=
=
∆x
∆t
xf − xi
tf − ti
100 m − 0 m
100 s − 0 s
1 m · s−1
Lesedi’s velocity is 1 m·s−1 . This means that he walked 1 m in the first second, another metre
in the second second, and another in the third second, and so on. For example, after 50 s he
will be 50 m from home. His position increases by 1 m every 1 s. A diagram of Lesedi’s position
is shown in Figure 3.6.
We can now draw graphs of position vs.time (x vs. t), velocity vs.time (v vs. t) and acceleration
vs.time (a vs. t) for Lesedi moving at a constant velocity. The graphs are shown in Figure 3.7.
41
3.6
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
b
b
b
t=0s
x=0m
t = 50 s
x = 50 m
v = 1m·s−1
t = 100 s
x = 100 m
v = 1m·s−1
100
∆x
50
∆t
0
time (s)
50
1
0
100
(a)
time (s)
50
100
acceleration a (m·s−2 )
velocity v (m·s−1 )
position x (m)
Figure 3.6: Diagram showing Lesedi’s motion at a constant velocity of 1 m·s−1
0
(b)
time (s)
50
100
(c)
Figure 3.7: Graphs for motion at constant velocity (a) position vs. time (b) velocity vs. time
(c) acceleration vs. time. The area of the shaded portion in the v vs. t graph corresponds to
the object’s displacement.
100
∆t
50
0
∆x
0
50
100
-1
time (s)
50
time (s)
acceleration a (m·s−2 )
velocity v (m·s−1 )
position x (m)
In the evening Lesedi walks 100 m from the bus stop to his house in 100 s. Assume that Lesedi’s
house is the origin. The following graphs can be drawn to describe the motion.
0
100
(a)
(b)
time (s)
50
100
(c)
Figure 3.8: Graphs for motion with a constant negative velocity (a) position vs. time (b) velocity
vs. time (c) acceleration vs. time. The area of the shaded portion in the v vs.t graph corresponds
to the object’s displacement.
We see that the v vs. t graph is a horisontal line. If the velocity vs. time graph is a horisontal
line, it means that the velocity is constant (not changing). Motion at a constant velocity is
known as uniform motion.
We can use the x vs. t to calculate the velocity by finding the gradient of the line.
v
=
=
=
=
∆x
∆t
xf − xi
tf − ti
0 m − 100 m
100 s − 0 s
−1 m · s−1
42
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.6
Lesedi has a velocity of -1 m·s−1 , or 1 m·s−1 towards his house. You will notice that the v vs. t
graph is a horisontal line corresponding to a velocity of -1 m·s−1 . The horisontal line means that
the velocity stays the same (remains constant) during the motion. This is uniform velocity.
We can use the v vs. t to calculate the acceleration by finding the gradient of the line.
=
a
=
=
=
∆v
∆t
vf − vi
tf − ti
1 m · s−1 − 1 m · s−1
100 s − 0 s
0 m · s−2
Lesedi has an acceleration of 0 m·s−2 . You will notice that the graph of a vs.t is a horisontal line
corresponding to an acceleration value of 0 m·s−2 . There is no acceleration during the motion
because his velocity does not change.
We can use the v vs. t to calculate the displacement by finding the area under the graph.
v
=
Area under graph
=
=
ℓ× b
100 × (−1)
=
−100 m
This means that Lesedi has a displacement of 100 m towards his house.
43
3.6
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Exercise: Velocity and acceleration
1. Use the graphs in Figure 3.7 to calculate each of the following:
(a) Calculate Lesedi’s velocity between 50 s and 100 s using the x vs. t graph.
Hint: Find the gradient of the line.
(b) Calculate Lesedi’s acceleration during the whole motion using the v vs. t
graph.
(c) Calculate Lesedi’s displacement during the whole motion using the v vs. t
graph.
2. Thandi takes 200 s to walk 100 m to the bus stop every morning. Draw a graph
of Thandi’s position as a function of time (assuming that Thandi’s home is the
reference point). Use the gradient of the x vs. t graph to draw the graph of
velocity vs. time. Use the gradient of the v vs. t graph to draw the graph of
acceleration vs. time.
3. In the evening Thandi takes 200 s to walk 100 m from the bus stop to her
home. Draw a graph of Thandi’s position as a function of time (assuming that
Thandi’s home is the origin). Use the gradient of the x vs. t graph to draw
the graph of velocity vs. time. Use the gradient of the v vs. t graph to draw
the graph of acceleration vs. time.
4. Discuss the differences between the two sets of graphs in questions 2 and 3.
Activity :: Experiment : Motion at constant velocity
Aim:
To measure the position and time during motion at constant velocity and determine
the average velocity as the gradient of a “Position vs. Time” graph.
Apparatus:
A battery operated toy car, stopwatch, meter stick or measuring tape.
Method:
1. Work with a friend. Copy the table below into your workbook.
2. Complete the table by timing the car as it travels each distance.
3. Time the car twice for each distance and take the average value as your accepted
time.
4. Use the distance and average time values to plot a graph of “Distance vs. Time”
onto graph paper. Stick the graph paper into your workbook. (Remember
that “A vs. B” always means “y vs. x”).
5. Insert all axis labels and units onto your graph.
6. Draw the best straight line through your data points.
7. Find the gradient of the straight line. This is the average velocity.
Results:
Distance (m)
1
0
0,5
1,0
1,5
2,0
2,5
3,0
44
Time (s)
2
Ave.
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Conclusions:
Answer the following questions in your workbook.
Questions:
1. Did the car travel with a constant velocity?
2. How can you tell by looking at the “Distance vs. Time” graph if the velocity
is constant?
3. How would the “Distance vs. Time” look for a car with a faster velocity?
4. How would the “Distance vs. Time” look for a car with a slower velocity?
45
3.6
3.6
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.6.3
Motion at Constant Acceleration
The final situation we will be studying is motion at constant acceleration. We know that
acceleration is the rate of change of velocity. So, if we have a constant acceleration, this
means that the velocity changes at a constant rate.
Let’s look at our first example of Lesedi waiting at the taxi stop again. A taxi arrived and Lesedi
got in. The taxi stopped at the stop street and then accelerated as follows: After 1 s the taxi
covered a distance of 2,5 m, after 2 s it covered 10 m, after 3 seconds it covered 22,5 m and
after 4 s it covered 40 m. The taxi is covering a larger distance every second. This means that
it is accelerating.
STOP
2,5 m
t=1s
22,5 m
t=3s
10 m
t=2s
40 m
t=4s
To calculate the velocity of the taxi you need to calculate the gradient of the line at each second:
v1s
=
=
=
=
∆x
∆t
xf − xi
tf − ti
5m − 0m
1,5s − 0,5s
5 m · s−1
v2s
∆x
∆t
xf − xi
=
tf − ti
15m − 5m
=
2,5s − 1,5s
= 10 m · s−1
=
v3s
=
=
=
=
∆x
∆t
xf − xi
tf − ti
30m − 15m
3,5s − 2,5s
15 m · s−1
From these velocities, we can draw the velocity-time graph which forms a straight line.
The acceleration is the gradient of the v vs. t graph and can be calculated as follows:
a
∆v
∆t
vf − vi
=
tf − ti
15m · s−1 − 5m · s−1
=
3s − 1s
−2
= 5 m·s
=
The acceleration does not change during the motion (the gradient stays constant). This is
motion at constant or uniform acceleration.
The graphs for this situation are shown in Figure 3.9.
Velocity from Acceleration vs. Time Graphs
Just as we used velocity vs. time graphs to find displacement, we can use acceleration vs. time
graphs to find the velocity of an object at a given moment in time. We simply calculate the
area under the acceleration vs. time graph, at a given time. In the graph below, showing an
object at a constant positive acceleration, the increase in velocity of the object after 2 seconds
corresponds to the shaded portion.
v = area of rectangle = a × ∆t
= 5 m · s−2 × 2 s
= 10 m · s−1
46
3.6
acceleration a (m·s−2 )
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
b
22,5
velocity v (m·s−1 )
position x (m)
15
10
b
10
∆t
b
0
1
∆x
2
∆v
5
∆t
0
3 time (s)
(a)
1
2
3 time (s)
(b)
5
0
time (s)
1
2
(c)
Figure 3.9: Graphs for motion with a constant acceleration (a) position vs. time (b) velocity vs.
time (c) acceleration vs. time.
The velocity of the object at t = 2 s is therefore 10 m·s−1 . This corresponds with the values
obtained in Figure 3.9.
Exercise: Graphs
1. A car is parked 10 m from home for 10 minutes. Draw a displacement-time,
velocity-time and acceleration-time graphs for the motion. Label all the axes.
2. A bus travels at a constant velocity of 12 m·s−1 for 6 seconds. Draw the
displacement-time, velocity-time and acceleration-time graph for the motion.
Label all the axes.
3. An athlete runs with a constant acceleration of 1 m·s−2 for 4 s. Draw the
acceleration-time, velocity-time and displacement time graphs for the motion.
Accurate values are only needed for the acceleration-time and velocity-time
graphs.
4. The following velocity-time graph describes the motion of a car. Draw the
displacement-time graph and the acceleration-time graph and explain the motion of the car according to the three graphs.
v (m·s−1 )
6
t (s)
0
2
5. The following velocity-time graph describes the motion of a truck. Draw the
displacement-time graph and the acceleration-time graph and explain the motion of the truck according to the three graphs.
v (m·s−1 )
8
0
4
47
t (s)
3.7
3.7
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Summary of Graphs
The relation between graphs of position, velocity and acceleration as functions of time is summarised in Figure 3.10.
x (m)
v (m·s−1 )
t (s)
Stationary
Object
x (m)
v (m·s−1 )
x (m)
Motion with
constant acceleration
t (s)
t (s)
Uniform Motion
a (m·s−2 )
t (s)
a (m·s−2 )
t (s)
v (m·s−1 )
t (s)
t (s)
a (m·s−2 )
t (s)
Figure 3.10: Position-time, velocity-time and acceleration-time graphs.
Important: Often you will be required to describe the motion of an object that is presented
as a graph of either position, velocity or acceleration as functions of time. The description
of the motion represented by a graph should include the following (where possible):
1. whether the object is moving in the positive or negative direction
2. whether the object is at rest, moving at constant velocity or moving at constant
positive acceleration (speeding up) or constant negative acceleration (slowing down)
You will also often be required to draw graphs based on a description of the motion in words
or from a diagram. Remember that these are just different methods of presenting the same
information. If you keep in mind the general shapes of the graphs for the different types of
motion, there should not be any difficulty with explaining what is happening.
48
t (s)
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.8
3.8
Worked Examples
The worked examples in this section demonstrate the types of questions that can be asked about
graphs.
Worked Example 8: Description of motion based on a position-time graph
Question: The position vs. time graph for the motion of a car is given below.
Draw the corresponding velocity vs. time and acceleration vs. time graphs, and then
describe the motion of the car.
x (m)
5
4
3
2
1
t (s)
0
0
1
2
3
4
5
6
Answer
Step 1 : Identify what information is given and what is asked for
The question gives a position vs. time graph and the following three things are
required:
1. Draw a v vs. t graph.
2. Draw an a vs. t graph.
3. Describe the motion of the car.
To answer these questions, break the motion up into three sections: 0 - 2 seconds,
2 - 4 seconds and 4 - 6 seconds.
Step 2 : Velocity vs. time graph for 0-2 seconds
For the first 2 seconds we can see that the displacement remains constant - so the
object is not moving, thus it has zero velocity during this time. We can reach this
conclusion by another path too: remember that the gradient of a displacement vs.
time graph is the velocity. For the first 2 seconds we can see that the displacement
vs. time graph is a horizontal line, ie. it has a gradient of zero. Thus the velocity
during this time is zero and the object is stationary.
Step 3 : Velocity vs. time graph for 2-4 seconds
For the next 2 seconds, displacement is increasing with time so the object is moving. Looking at the gradient of the displacement graph we can see that it is not
constant. In fact, the slope is getting steeper (the gradient is increasing) as time
goes on. Thus, remembering that the gradient of a displacement vs. time graph is
the velocity, the velocity must be increasing with time during this phase.
Step 4 : Velocity vs. time graph for 4-6 seconds
For the final 2 seconds we see that displacement is still increasing with time, but
this time the gradient is constant, so we know that the object is now travelling at
a constant velocity, thus the velocity vs. time graph will be a horizontal line during
this stage. We can now draw the graphs:
49
3.8
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
So our velocity vs. time graph looks like this one below. Because we haven’t been
given any values on the vertical axis of the displacement vs. time graph, we cannot
figure out what the exact gradients are and therefore what the values of the velocities
are. In this type of question it is just important to show whether velocities are positive
or negative, increasing, decreasing or constant.
v (m·s−1 )
t (s)
0
1
2
3
4
5
6
Once we have the velocity vs. time graph its much easier to get the acceleration vs.
time graph as we know that the gradient of a velocity vs. time graph is the just the
acceleration.
Step 5 : Acceleration vs. time graph for 0-2 seconds
For the first 2 seconds the velocity vs. time graph is horisontal and has a value of
zero, thus it has a gradient of zero and there is no acceleration during this time.
(This makes sense because we know from the displacement time graph that the object is stationary during this time, so it can’t be accelerating).
Step 6 : Acceleration vs. time graph for 2-4 seconds
For the next 2 seconds the velocity vs. time graph has a positive gradient. This
gradient is not changing (i.e. its constant) throughout these 2 seconds so there must
be a constant positive acceleration.
Step 7 : Acceleration vs. time graph for 4-6 seconds
For the final 2 seconds the object is traveling with a constant velocity. During this
time the gradient of the velocity vs. time graph is once again zero, and thus the
object is not accelerating. The acceleration vs. time graph looks like this:
a (m·s−2 )
0
2
4
6
t (s)
Step 8 : A description of the object’s motion
A brief description of the motion of the object could read something like this: At
t = 0 s and object is stationary at some position and remains stationary until t = 2 s
when it begins accelerating. It accelerates in a positive direction for 2 seconds until
t = 4 s and then travels at a constant velocity for a further 2 seconds.
50
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.8
Worked Example 9: Calculations from a velocity vs. time graph
Question: The velocity vs. time graph of a truck is plotted below. Calculate the
distance and displacement of the truck after 15 seconds.
v (m·s−1 )
4
3
2
1
t (s)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
−1
−2
Answer
Step 1 : Decide how to tackle the problem
We are asked to calculate the distance and displacement of the car. All we need to
remember here is that we can use the area between the velocity vs. time graph and
the time axis to determine the distance and displacement.
Step 2 : Determine the area under the velocity vs. time graph
Break the motion up: 0 - 5 seconds, 5 - 12 seconds, 12 - 14 seconds and 14 - 15
seconds.
For 0 - 5 seconds: The displacement is
equal to the area of the triangle on the
left:
Area△
=
=
=
1
b×h
2
1
×5 ×4
2
10 m
For 12 - 14 seconds the displacement is
equal to the area of the triangle above
the time axis on the right:
Area△
=
=
=
1
b×h
2
1
×2 ×4
2
4m
For 5 - 12 seconds: The displacement
is equal to the area of the rectangle:
= ℓ×b
= 7 ×4
Area
= 28 m
For 14 - 15 seconds the displacement is
equal to the area of the triangle below
the time axis:
Area△
1
b×h
2
1
=
×1 ×2
2
= 1m
=
Step 3 : Determine the total distance of the car
Now the total distance of the car is the sum of all of these areas:
∆x =
=
10 + 28 + 4 + 1
43 m
51
15
3.8
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Step 4 : Determine the total displacement of the car
Now the total displacement of the car is just the sum of all of these areas. HOWEVER, because in the last second (from t = 14 s to t = 15 s) the velocity of the
car is negative, it means that the car was going in the opposite direction, i.e. back
where it came from! So, to find the total displacement, we have to add the first 3
areas (those with positive displacements) and subtract the last one (because it is a
displacement in the opposite direction).
∆x =
=
10 + 28 + 4 − 1
41 m in the positive direction
Worked Example 10: Velocity from a position vs. time graph
Question: The position vs. time graph below describes the motion of an athlete.
1. What is the velocity of the athlete during the first 4 seconds?
2. What is the velocity of the athlete from t = 4 s to t = 7 s?
x (m)
4
3
2
1
t (s)
0
0
1
2
3
4
5
6
7
Answer
Step 1 : The velocity during the first 4 seconds
The velocity is given by the gradient of a position vs. time graph. During the first
4 seconds, this is
v
∆x
∆t
4−0
=
4−0
= 1 m · s−1
=
Step 2 : The velocity during the last 3 seconds
For the last 3 seconds we can see that the displacement stays constant. The graph
shows a horisontal line and therefore the gradient is zero. Thus v = 0 m · s−1 .
52
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.8
Worked Example 11: Drawing a v vs. t graph from an a vs. t graph
Question: The acceleration vs. time graph for a car starting from rest, is given
below. Calculate the velocity of the car and hence draw the velocity vs. time graph.
a (m·s−2 )
2
1
t (s)
0
1
2
3
4
5
6
−1
−2
Answer
Step 1 : Calculate the velocity values by using the area under each part of
the graph.
The motion of the car can be divided into three time sections: 0 - 2 seconds; 2 - 4
seconds and 4 - 6 seconds. To be able to draw the velocity vs. time graph, the
velocity for each time section needs to be calculated. The velocity is equal to the
area of the square under the graph:
For 0 - 2 seconds:
Area
= ℓ×b
= 2 ×2
For 2 - 4 seconds:
Area
= 4 m · s−1
The velocity of the car is
4 m·s−1 at t = 2s.
For 4 - 6 seconds:
= ℓ×b
= 2 ×0
= ℓ×b
Area
= 2 × −2
= −4 m · s−1
= 0 m · s−1
The velocity of the car is
0 m·s−1 from t = 2 s to
t = 4 s.
The acceleration had a negative value, which means that
the velocity is decreasing.
It starts at a velocity of
4 m·s−1 and decreases to
0 m·s−1 .
Step 2 : Now use the values to draw the velocity vs. time graph.
v (m·s−1 )
4
3
The velocity vs. time graph
looks like this:
2
1
t (s)
0
0
53
1
2
3
4
5
6
3.9
3.9
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Equations of Motion
In this chapter we will look at the third way to describe motion. We have looked at describing
motion in terms of graphs and words. In this section we examine equations that can be used to
describe motion.
This section is about solving problems relating to uniformly accelerated motion. In other words,
motion at constant acceleration.
The following are the variables that will be used in this section:
vi
vf
=
=
initial velocity (m·s−1 ) at t = 0 s
final velocity (m·s−1 ) at time t
∆x =
t =
displacement (m)
time (s)
∆t
time interval (s)
=
a =
acceleration (m·s−2 )
vf
=
∆x =
∆x =
vf2
=
vi + at
(vi + vf )
t
2
1
vi t + at2
2
vi2 + 2a∆x
(3.1)
(3.2)
(3.3)
(3.4)
The questions can vary a lot, but the following method for answering them will always work.
Use this when attempting a question that involves motion with constant acceleration. You need
any three known quantities (vi , vf , ∆x, t or a) to be able to calculate the fourth one.
1. Read the question carefully to identify the quantities that are given. Write them down.
2. Identify the equation to use. Write it down!!!
3. Ensure that all the values are in the correct unit and fill them in your equation.
4. Calculate the answer and fill in its unit.
teresting Galileo Galilei of Pisa, Italy, was the first to determined the correct mathematical
Interesting
Fact
Fact
law for acceleration: the total distance covered, starting from rest, is proportional
to the square of the time. He also concluded that objects retain their velocity
unless a force – often friction – acts upon them, refuting the accepted Aristotelian
hypothesis that objects ”naturally” slow down and stop unless a force acts upon
them. This principle was incorporated into Newton’s laws of motion (1st law).
3.9.1
Finding the Equations of Motion
The following does not form part of the syllabus and can be considered additional information.
54
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.9
Derivation of Equation 3.1
According to the definition of acceleration:
a=
∆v
t
where ∆v is the change in velocity, i.e. ∆v = vf - vi . Thus we have
a
=
vf
=
vf − vi
t
vi + at
Derivation of Equation 3.2
We have seen that displacement can be calculated from the area under a velocity vs. time graph.
For uniformly accelerated motion the most complicated velocity vs. time graph we can have is
a straight line. Look at the graph below - it represents an object with a starting velocity of vi ,
accelerating to a final velocity vf over a total time t.
v (m·s−1 )
vf
vi
t
t (s)
To calculate the final displacement we must calculate the area under the graph - this is just
the area of the rectangle added to the area of the triangle. This portion of the graph has been
shaded for clarity.
Area△
=
=
=
Area
1
b×h
2
1
t × (vf − vi )
2
1
1
vf t − vi t
2
2
= ℓ×b
= t × vi
= vi t
Displacement = Area + Area△
1
1
∆x = vi t + vf t − vi t
2
2
(vi + vf )
t
∆x =
2
55
3.9
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Derivation of Equation 3.3
This equation is simply derived by eliminating the final velocity vf in equation 3.2. Remembering
from equation 3.1 that
vf = vi + at
then equation 3.2 becomes
∆x
=
=
∆x
=
vi + vi + at
t
2
2vi t + at2
2
1
vi t + at2
2
56
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.9
Derivation of Equation 3.4
This equation is just derived by eliminating the time variable in the above equation. From
Equation 3.1 we know
vf − vi
t=
a
Substituting this into Equation 3.3 gives
∆x
=
=
=
2a∆x =
vf2
=
vf − vi
1 vf − vi 2
) + a(
)
a
2
a
vi vf
v2
1 vf2 − 2vi vf + vi2
)
− i + a(
a
a
2
a2
vf2
v2
vi vf
v2
vi vf
− i +
−
+ i
a
a
2a
a
2a
−2vi2 + vf2 + vi2
vi (
vi2 + 2a∆x
(3.5)
This gives us the final velocity in terms of the initial velocity, acceleration and displacement and
is independent of the time variable.
Worked Example 12: Equations of motion
Question: A racing car is travelling north. It accelerates uniformly covering a
distance of 725 m in 10 s. If it has an initial velocity of 10 m·s−1 , find its acceleration.
Answer
Step 1 : Identify what information is given and what is asked for
We are given:
vi =
∆x =
t
a
=
=
10 m · s−1
725 m
10 s
?
Step 2 : Find an equation of motion relating the given information to the
acceleration
If you struggle to find the correct equation, find the quantity that is not given and
then look for an equation that does not have this quantity in it.
We can use equation 3.3
1
∆x = vi t + at2
2
Step 3 : Substitute your values in and find the answer
∆x
1
= vi t + at2
2
1
725 = (10 × 10) + a × (10)2
2
725 − 100 = 50 a
a = 12,5 m · s−2
Step 4 : Quote the final answer
The racing car is accelerating at 12,5 m·s−2 north.
57
3.9
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
Worked Example 13: Equations of motion
Question: A motorcycle, travelling east, starts from rest, moves in a straight line
with a constant acceleration and covers a distance of 64 m in 4 s. Calculate
•
•
•
•
its acceleration
its final velocity
at what time the motorcycle had covered half the total distance
what distance the motorcycle had covered in half the total time.
Answer
Step 1 : Identify what information is given and what is asked for
We are given:
vi
=
0 m · s−1 (because the object starts from rest.)
∆x
t
=
=
64 m
4s
a
vf
=
=
?
?
t
=
? at half the distance ∆x = 32 m.
∆x
=
? at half the time t = 2 s.
All quantities are in SI units.
Step 2 : Acceleration: Find a suitable equation to calculate the acceleration
We can use equations 3.3
1
∆x = vi t + at2
2
Step 3 : Substitute the values and calculate the acceleration
∆x
1
vi t + at2
2
1
(0 × 4) + a × (4)2
2
8a
=
64 =
64 =
=
a
8 m · s−2 east
Step 4 : Final velocity: Find a suitable equation to calculate the final velocity
We can use equation 3.1 - remember we now also know the acceleration of the
object.
vf = vi + at
Step 5 : Substitute the values and calculate the final velocity
vf
=
vi + at
vf
=
=
0 + (8)(4)
32 m · s−1 east
Step 6 : Time at half the distance: Find an equation to calculate the time
We can use equation 3.3:
1
= vi + at2
2
1
32 = (0)t + (8)(t)2
2
32 = 0 + 4t2
∆x
8
t
= t2
= 2,83 s
58
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.10
Step 7 : Distance at half the time: Find an equation to relate the distance
and time
Half the time is 2 s, thus we have vi , a and t - all in the correct units. We can use
equation 3.3 to get the distance:
∆x
=
=
=
1
vi t + at2
2
1
(0)(2) + (8)(2)2
2
16 m east
Exercise: Acceleration
1. A car starts off at 10 m·s−1 and accelerates at 1 m·s−2 for 10 s. What is its
final velocity?
2. A train starts from rest, and accelerates at 1 m·s−2 for 10 s. How far does it
move?
3. A bus is going 30 m·s−1 and stops in 5 s. What is its stopping distance for this
speed?
4. A racing car going at 20 m·s−1 stops in a distance of 20 m. What is its
acceleration?
5. A ball has a uniform acceleration of 4 m·s−1 . Assume the ball starts from rest.
Determine the velocity and displacement at the end of 10 s.
6. A motorcycle has a uniform acceleration of 4 m·s−1 . Assume the motorcycle
has an initial velocity of 20 m·s−1 . Determine the velocity and displacement at
the end of 12 s.
7. An aeroplane accelerates uniformly such that it goes from rest to 144 km·hr−1 in
8 s. Calculate the acceleration required and the total distance that it has
traveled in this time.
3.10
Applications in the Real-World
What we have learnt in this chapter can be directly applied to road safety. We can analyse the
relationship between speed and stopping distance. The following worked example illustrates this
application.
Worked Example 14: Stopping distance
Question: A truck is travelling at a constant velocity of 10 m·s−1 when the driver
sees a child 50 m in front of him in the road. He hits the brakes to stop the truck.
The truck accelerates at a rate of -1.25 m·s−2 . His reaction time to hit the brakes
is 0,5 seconds. Will the truck hit the child?
Answer
Step 1 : Analyse the problem and identify what information is given
It is useful to draw a timeline like this one:
59
3.10
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
child is here
driver hits brakes
driver sees child
v = 10 m·s−1
0,5 s
Bb
Abb
Cb
negative acceleration
constant v
50 m
We need to know the following:
• What distance the driver covers before hitting the brakes.
• How long it takes the truck to stop after hitting the brakes.
• What total distance the truck covers to stop.
Step 2 : Calculate the distance AB
Before the driver hits the brakes, the truck is travelling at constant velocity. There
is no acceleration and therefore the equations of motion are not used. To find the
distance traveled, we use:
v
=
10 =
d =
d
t
d
0,5
5m
The truck covers 5 m before the driver hits the brakes.
Step 3 : Calculate the time BC
We have the following for the motion between B and C:
vi
vf
=
=
10 m · s−1
0 m · s−1
a
t
=
=
−1,25 m · s−2
?
We can use equation 3.1
vf =
0 =
−10 =
t =
vi + at
10 + (−1,25)t
−1,25t
8s
Step 4 : Calculate the distance BC
For the distance we can use equation 3.2 or equation 3.3. We will use equation 3.2:
∆x
∆x
∆x
(vi + vf )
t
2
10 + 0
=
(8)
s
= 40 m
=
Step 5 : Write the final answer
The total distance that the truck covers is dAB + dBC = 5 + 40 = 45 meters. The
child is 50 meters ahead. The truck will not hit the child.
60
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.11
3.11
Summary
• A reference point is a point from where you take your measurements.
• A frame of reference is a reference point with a set of directions.
• Your position is where you are located with respect to your reference point.
• The displacement of an object is how far it is from the reference point. It is the shortest
distance between the object and the reference point. It has magnitude and direction
because it is a vector.
• The distance of an object is the length of the path travelled from the starting point to the
end point. It has magnitude only because it is a scalar.
• A vector is a physical quantity with magnitude and direction.
• A scalar is a physical quantity with magnitude only.
• Speed (s) is the distance covered (d) divided by the time taken (∆t):
s=
d
∆t
• Average velocity (v) is the displacement (∆x) divided by the time taken (∆t):
v=
∆x
∆t
• Instantaneous speed is the speed at a specific instant in time.
• Instantaneous velocity is the velocity at a specific instant in time.
• Acceleration (a) is the change in velocity (∆x) over a time interval (∆t):
a=
∆v
∆t
• The gradient of a position - time graph (x vs. t) give the velocity.
• The gradient of a velocity - time graph (v vs. t) give the acceleration.
• The area under a velocity - time graph (v vs. t) give the displacement.
• The area under an acceleration - time graph (a vs. t) gives the velocity.
• The graphs of motion are summarised in figure 3.10.
• The equations of motion are used where constant acceleration takes place:
vf
=
∆x
=
∆x
=
vf2
=
61
vi + at
(vi + vf )
t
2
1
vi t + at2
2
vi2 + 2a∆x
3.12
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.12
End of Chapter Exercises: Motion in One Dimension
1. Give one word/term for the following descriptions.
(a) The shortest path from start to finish.
(b) A physical quantity with magnitude and direction.
(c) The quantity defined as a change in velocity over a time period.
(d) The point from where you take measurements.
(e) The distance covered in a time interval.
(f) The velocity at a specific instant in time.
2. Choose an item from column B that match the description in column A. Write down only
the letter next to the question number. You may use an item from column B more than
once.
Column A
a. The area under a velocity - time graph
b. The gradient of a velocity - time graph
c. The area under an acceleration - time graph
d. The gradient of a displacement - time graph
Column B
gradient
area
velocity
displacement
acceleration
slope
3. Indicate whether the following statements are TRUE or FALSE. Write only ’true’ or ’false’.
If the statement is false, write down the correct statement.
(a) A scalar is the displacement of an object over a time interval.
(b) The position of an object is where it is located.
(c) The sign of the velocity of an object tells us in which direction it is travelling.
(d) The acceleration of an object is the change of its displacement over a period in time.
4. [SC 2003/11] A body accelerates uniformly from rest for t0 seconds after which it continues
with a constant velocity. Which graph is the correct representation of the body’s motion?
x
x
t0
(a)
t
x
t0
x
t
t0
(b)
(c)
t
t0
t
(d)
5. [SC 2003/11] The velocity-time graphs of two cars are represented by P and Q as shown
v (m·s−1 )
6
5
4
3
2
1
0
P
Q
t (s)
0 1 2 3 4
The difference in the distance travelled by the two cars (in m) after 4 s is . . .
62
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.12
(a) 12
(b) 6
(c) 2
(d) 0
6. [IEB 2005/11 HG] The graph that follows shows how the speed of an athlete varies with
time as he sprints for 100 m.
speed (m·s−1 )
10
t
11
time (s)
Which of the following equations can be used to correctly determine the time t for which
he accelerates?
(a) 100 = (10)(11) − 21 (10)t
(b) 100 = (10)(11) + 21 (10)t
(c) 100 = 10t + 21 (10)t2
(d) 100 = 21 (0)t + 12 (10)t2
7. [SC 2002/03 HG1] In which one of the following cases will the distance covered and the
magnitude of the displacement be the same?
(a) A girl climbs a spiral staircase.
(b) An athlete completes one lap in a race.
(c) A raindrop falls in still air.
(d) A passenger in a train travels from Cape Town to Johannesburg.
8. [SC 2003/11] A car, travelling at constant velocity, passes a stationary motor cycle at a
traffic light. As the car overtakes the motorcycle, the motorcycle accelerates uniformly
from rest for 10 s. The following displacement-time graph represents the motions of both
vehicles from the traffic light onwards.
x (m)
motorcycle
car
375
300
0
5
X 10
15
t (s)
(a) Use the graph to find the magnitude of the constant velocity of the car.
(b) Use the information from the graph to show by means of calculation that the magnitude of the acceleration of the motorcycle, for the first 10 s of its motion is 7,5
m·s−2 .
(c) Calculate how long (in seconds) it will take the motorcycle to catch up with the car
(point X on the time axis).
(d) How far behind the motorcycle will the car be after 15 seconds?
63
3.12
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
9. [IEB 2005/11 HG] Which of the following statements is true of a body that accelerates
uniformly?
(a) Its rate of change of position with time remains constant.
(b) Its position changes by the same amount in equal time intervals.
(c) Its velocity increases by increasing amounts in equal time intervals.
(d) Its rate of change of velocity with time remains constant.
10. [IEB 2003/11 HG1] The velocity-time graph for a car moving along a straight horizontal
road is shown below.
v (m·s−1 )
20
Area A
12
Area B
0
t
t (s)
Which of the following expressions gives the magnitude of the average velocity of the car?
(a)
AreaA
t
(b)
AreaA + AreaB
t
(c)
AreaB
t
(d)
AreaA − AreaB
t
11. [SC 2002/11 SG] A car is driven at 25 m·s−1 in a municipal area. When the driver sees a
traffic officer at a speed trap, he realises he is travelling too fast. He immediately applies
the brakes of the car while still 100 m away from the speed trap.
(a) Calculate the magnitude of the minimum acceleration which the car must have to
avoid exceeding the speed limit, if the municipal speed limit is 16.6 m·s−1 .
(b) Calculate the time from the instant the driver applied the brakes until he reaches the
speed trap. Assume that the car’s velocity, when reaching the trap, is 16.6 m·s−1 .
12. A traffic officer is watching his speed trap equipment at the bottom of a valley. He can
see cars as they enter the valley 1 km to his left until they leave the valley 1 km to his
right. Nelson is recording the times of cars entering and leaving the valley for a school
project. Nelson notices a white Toyota enter the valley at 11:01:30 and leave the valley at
11:02:42. Afterwards, Nelson hears that the traffic officer recorded the Toyota doing 140
km·hr−1 .
(a) What was the time interval (∆t) for the Toyota to travel through the valley?
(b) What was the average speed of the Toyota?
(c) Convert this speed to km·hr−1 .
(d) Discuss whether the Toyota could have been travelling at 140km·hr−1 at the bottom
of the valley.
(e) Discuss the differences between the instantaneous speed (as measured by the speed
trap) and average speed (as measured by Nelson).
64
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
3.12
13. [IEB 2003/11HG] A velocity-time graph for a ball rolling along a track is shown below.
The graph has been divided up into 3 sections, A, B and C for easy reference. (Disregard
any effects of friction.)
velocity (m·s−1 )
0,6
0
A
B
5
C
10
t1
12
-0,2
(a) Use the graph to determine the following:
i. the speed 5 s after the start
ii. the distance travelled in Section A
iii. the acceleration in Section C
(b) At time t1 the velocity-time graph intersects the time axis. Use an appropriate
equation of motion to calculate the value of time t1 (in s).
(c) Sketch a displacement-time graph for the motion of the ball for these 12 s. (You do
not need to calculate the actual values of the displacement for each time interval,
but do pay attention to the general shape of this graph during each time interval.)
14. In towns and cities, the speed limit is 60 km·hr−1 . The length of the average car is 3.5
m, and the width of the average car is 2 m. In order to cross the road, you need to be
able to walk further than the width of a car, before that car reaches you. To cross safely,
you should be able to walk at least 2 m further than the width of the car (4 m in total),
before the car reaches you.
(a) If your walking speed is 4 km·hr−1 , what is your walking speed in m·s−1 ?
(b) How long does it take you to walk a distance equal to the width of the average car?
(c) What is the speed in m·s−1 of a car travelling at the speed limit in a town?
(d) How many metres does a car travelling at the speed limit travel, in the same time
that it takes you to walk a distance equal to the width of car?
(e) Why is the answer to the previous question important?
(f) If you see a car driving toward you, and it is 28 m away (the same as the length of 8
cars), is it safe to walk across the road?
(g) How far away must a car be, before you think it might be safe to cross? How many
car-lengths is this distance?
15. A bus on a straight road starts from rest at a bus stop and accelerates at 2 m·s−2 until it
reaches a speed of 20 m·s−1 . Then the bus travels for 20 s at a constant speed until the
driver sees the next bus stop in the distance. The driver applies the brakes, stopping the
bus in a uniform manner in 5 s.
(a) How long does the bus take to travel from the first bus stop to the second bus stop?
(b) What is the average velocity of the bus during the trip?
65
time (s)
3.12
CHAPTER 3. MOTION IN ONE DIMENSION - GRADE 10
66
Appendix A
GNU Free Documentation License
Version 1.2, November 2002
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APPENDIX A. GNU FREE DOCUMENTATION LICENSE
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APPENDIX A. GNU FREE DOCUMENTATION LICENSE
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APPENDIX A. GNU FREE DOCUMENTATION LICENSE
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APPENDIX A. GNU FREE DOCUMENTATION LICENSE
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