Physics for GCE 'O' Level

All YOU NEED TO KNOW:

PHYSICS FOR GCE ‘O’ LEVEL

Lawrence Lau • Josephine Fong

Published by Alston Publishing House Pte Ltd 745 Lorong 5 Toa Payoh, #01-07, Singapore 319455 enquiry@alstonpublishinghouse.com www.alstonpublishinghouse.com Š 2013 Alston Publishing House Pte Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the copyright owner.

First published 2013 ISBN 978-981-4370-90-5 Publisher: Sim Wee Chee Editor: Melanie Sim Design Team: Anna Gallenero, Eric Sto Domingo, Melissa Lee, Raymond Tan Printed in Singapore

CONTENTS Read Me First!: How to Study Physics 5058

v

How to Use This Study Guide

vii

Practical Skills

viii

General Tips on Answering Questions

xvi

CHAPTER 01 Physical Quantities, Units and Measurements

1

CHAPTER 02 Kinematics

16

CHAPTER 03 Dynamics

31

CHAPTER 04 Mass, Weight and Density

43

CHAPTER 05 Turning Effects of a Force

50

CHAPTER 06 Pressure

62

CHAPTER 07 Energy, Work and Power

74

CHAPTER 08 Kinetic Model of Matter

85

CHAPTER 09 Transfer of Thermal Energy

95

CHAPTER 10 Temperature

108

CHAPTER 11 Thermal Properties of Matter

118

Contents

iii

iv

CHAPTER 12 General Wave Properties

131

CHAPTER 13 Light

143

CHAPTER 14 Electromagnetic (EM) Spectrum

165

CHAPTER 15 Sound

173

CHAPTER 16 Static Electricity

187

CHAPTER 17 Current Electricity

200

CHAPTER 18 D.C. Circuits

215

CHAPTER 19 Practical Electricity

238

CHAPTER 20 Magnetism

249

CHAPTER 21 Electromagnetism

260

CHAPTER 22 Electromagnetic Induction

275

Essential Formula List

295

Glossary

298

Answers

309

All You Need to Know: Physics for GCE ‘O’ Level

Read Me First!: How to Study Physics 5058 Congratulations on taking up Physics! You will find that Physics 5058 is a demanding subject – it requires you to learn in new ways, but it is very rewarding when you ‘get it’.

First things first: You MUST get an A or at least a B3 for this!

There is no point scoring below a B3 for Physics, or any pure Science subject for that matter. The reason is simple: you could take Combined Science instead, and easily get a distinction. In fact, many teachers believe that students who score a B4 for pure Sciences can easily score a distinction if they took Combined Science instead. Taking a pure Science subject is supposed to demonstrate your strength, not expose your limits.

Physics is NOT Mathematics!

Rather, Physics studies the known universe – from the smallest subatomic particles to the universe itself, and looks for the simplest laws that explain their behavior. These laws are often (but not always) expressed in mathematical formulae. So Physics uses Mathematics as a tool, but it is not the same as Mathematics. (An analogy: You use a calculator as a tool, but you are not a calculator).

Physics is NOT playing the formula game!

Some students think that they can play the formula game: read the question, identify the variables given, recall the formula that contains these variables, substitute and calculate the value of the missing variable. While this can get you marks, you will need much more than that to answer questions that require you to give explanations or apply knowledge.

Physics can NOT be learnt by memorisation alone!

Among the subjects you are taking, Physics is probably the subject that requires the least amount of memorisation, but the greatest amount of understanding. You cannot learn Physics by memorising solutions to worked examples! Physics requires you to make sense of what you are learning, not just be able to reproduce all the examples you encounter. The study approach you adopt for Physics will determine the grades you will achieve: Common study approaches and their usual outcome: Study approach

Usual outcome

Read the textbook and notes like a storybook

Fail the subject

Memorise every definition, formula and solutions to worked examples

Pass, but difficult to get a distinction

Understand the concepts: see the common idea behind the worked examples in the same topic, and know how these ideas are interconnected with one another

Distinction!

How to Study Physics 5058

v

Learning Physics is about MAKING SENSE.

Each chapter in Physics is made up of a number of main ideas or concepts. These concepts are usually very fundamental (basic) ways that describe how the world works – they apply to every situation. This is different from simpler ideas you may already have (eg, “All moving things will eventually stop” – not true). These concepts are usually illustrated by a few worked examples, and it is the common idea behind each example. From these worked examples, you MUST find the same concept, or the ‘common idea’. In other words, you must make sense of the concept.

One way to know if you have understood a concept, is to reiterate the concept/a related observation in your own words, and then check with your teacher.

Studying Physics requires you to BELIEVE what you are learning.

You cannot learn a concept in Physics but believe that the world actually works in some other way.

Trying to do so will cause you to apply wrong ideas when stressed, such as during a national examination.

You must be prepared to replace your beliefs in many concepts that are inter-related, and the process is difficult. Once you start learning the correct concepts, you will become more confused at first, as they conflict with your original beliefs. Persevere – your confusion will start to clear up as you replace more and more of the wrong ideas with the correct ones. (A good example is again, “All moving things will eventually stop”. You will find that it contradicts the idea of inertia and the formula F = ma.)

Brush up on your English!

Physics depends a lot on whether you can comprehend what the teacher and the textbook are saying, and on whether you can express your ideas across to the marker. If you have been failing your English or have trouble understanding what your teacher is saying – work on it AS MUCH AS YOU CAN.

vi

All You Need to Know: Physics for GCE ‘O’ Level

How to Use This Study Guide This study guide is not a textbook. So it will take the shortest path to get all the concepts across. It will not be as interesting as a textbook which can develop ideas more fully and clearly, and show how the concepts apply to everyday life.

Below are the features of this book and how you can make full use of it: 1) The important concepts are listed in bulleted format for an uncluttered view. This includes all the necessary definitions and formulae.

2) We have included loads of worked examples – they are set such that you can see how the concept is being applied. Not only should you study how the solutions are arrived at, you should also take note of how the solutions are presented!

3) Each chapter is peppered with notes that we feel are important to you in your learning. These notes include the Introduction, which gives you important examination pointers such as the type of questions likely to appear from a particular chapter. Clarify Your Concept! points out possible misconceptions or common mistakes students make, while Tip gives tips on steps to answer a particular type of question.

4) Questions at the end of each chapter are for you to test your understanding. The questions are modelled after past year papers.

5) We also have a list of formulae for your quick reference.

6) Lastly, a glossary is included for those who need to look up the terms used in this book.

Not everything in this guide is equally important: Concepts are more important than formulae (if any), which are more important than worked examples.

Make sense of the concepts from the examples given. Test your understanding of the concepts by trying the exercises at the end of each chapter.

If you: • don’t know how to answer them…

• get some questions correct and some wrong (not caused by miscalculation)…

• get all questions correct (aside from miscalculations):

…you didn’t understand the concept, or your concept is inaccurate. Go back to the concept again and ask your teacher about how it works. …you probably have the correct concept!

Once you are sure that you get the concept, try out the questions from past year papers without referring to any material. You should get all of them correct (aside from misspellings and miscalculations). Any further mistakes are caused by an incomplete understanding of the concept, or you may have switched back to your old, wrong ideas. FIX THEM. Good luck!

How to Use This Study Guide

vii

Practical Skills Students often lose marks in practicals due to poor revision – for all Physics experiments, there is a set format for how the data is to be presented, and so this should be followed. Students usually find planning for experiments very challenging, and that depends on how thoughtful the students are during day-to-day experiments: Why are the procedures this way? Can they be improved?

AIM OF THE EXPERIMENT • Every experiment is about finding the relationship between two measurable variables: 1) Independent variable – a physical quantity that you control and change 2) Dependent variable – the physical quantity that you measure as a result of changing the

independent variable

You can identify the two types of variables from the question in the experiment.

Example In an experiment, a student is asked to find out how the length l of a pendulum affects its period T. So the independent variable is the length l and the dependent variable is the period T. • If you want to plan for an experiment, but the given aim is not clear on what the variables are, you should re-state the aim of the experiment such that the variables are clear.

Example You are asked to design an experiment based on the given aim: “Investigate how the electrical properties of a conductor change with temperature”. How would you re-state the aim of the experiment?

Solution There are three different physical quantities related to the ‘electrical properties of a conductor’: 1) the current through it 2) the potential difference across it 3) its resistance

We also know that, from Ohm’s Law, the resistance of a conductor may change with temperature.

We know that we cannot measure the temperature of a resistance wire, so we need to improvise.

Type of variable

How to obtain its measurements

The temperature of the conductor

The temperature of the oil bath it is immersed in

Dependent

The resistance of the resistance wire

The voltmeter reading across it and the ammeter reading in series with it

Controlled

The length of the resistance wire, the e.m.f. of the dry cell

—

Independent

Table A

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What they are

Isolating the variables

All You Need to Know: Physics for GCE ‘O’ Level

So the aim of the experiment can be re-stated as: “Find how the resistance of a fixed length of resistance wire changes with the temperature of the wire”.

TIP

In this experiment, the method of keeping the temperature constant should be explicitly stated. If you do not have to actually carry it out, you can use apparatus that your school laboratory does not have, eg, a constant temperature oil bath. If you need to carry it out, you will need to state how the temperature of the oil bath is maintained and monitored.

MAKING MEASUREMENTS When making measurements, two questions will usually surface: Question 1. How many decimal places should I record?

Answer: This is usually determined by the size of the smallest division on the instrument.

Example (a)

(b) Reading is 18.0 cm.

Reason: The smallest division is 0.1 cm.

Figure A

Reading is 18 cm.

Reason: The smallest division is 1 cm.

The ‘.0’ shows that ruler (a) is more precise than ruler (b).

Table B shows the precision for common instruments you would usually encounter: Instrument

Units

Number of decimal places

Example

Metre rule

mm cm m

0 1 3

150 mm 15.0 cm 0.150 m

mm cm

2 3

6.48 mm 0.648 cm

Vernier calipers Micrometer screw gauge

mm cm

1 2

32.1 mm 3.21 cm

Stopwatch

s

2

10.48 s

Laboratory thermometer*

°C

1

48.5 °C

Ammeter* Voltmeter*

A V

Depends on the scale

—

Protractor

°

0

29°

Table B

Precision of common measuring instruments

(Instruments marked with an asterisk (*) usually have physically large divisions (> 1 mm), so you are required to estimate to half of the smallest division.)

Practical Skills

ix

Question 2. What is the unit?

Answer: You need to note: –– the unit the instrument is calibrated for –– the prefixes, so that you can convert the measurements correctly Many students lose marks when they measure, say, in cm but do the wrong calculation to record in m or mm.

Example

18.0 cm = 0.180 m (not 0.018 m or 1.8 m)

PRESENTING DATA In Physics experiments, all data must be presented clearly in a table. Do note the following: 1) All data is presented in columns to allow quick checking of the decimal places of each reading.

2) Each column heading should have the symbol for the variable and the unit.

3) The independent variable should be the first column (on the left).

4) There must be at least 5 sets of data (6 rows, including the heading).

5) Try to use the widest practical range of values for your independent variable.

Example A student investigated how the period of a pendulum changes with the length of the pendulum. He was given a 1.2 m thread, a retort stand, and a pendulum bob. He is also given a metre rule and a stopwatch like the one on the right.

He presented the results as follows: Length = 50 cm, period = 1.4 s Length = 60 cm, period = 1.54 s

(a) What mistakes did he make? (b) How would you present the data?

Solution 1. The data was not presented in a table.

2. The minimum number of sets of data is 5. This student had only 4.

3. The recorded measurements do not reflect the precision of the instruments.

Length = 55 cm, period = 1.47 s Length = 65 cm, period = 1.6 s

Recorded measurements

x

What it should have been…

• No decimal places for lengths

They should be recorded to 1 decimal place in cm as they are measured with a metre rule.

• The number of decimal places for the times are not uniform

The stopwatch can measure up to 0.01 s, so all measurements should have 2 decimal places.

All You Need to Know: Physics for GCE ‘O’ Level

4. The range of lengths (50.0 – 65.0 cm) used was too narrow. The graph should have been a curve.

According to just this data, the graph would look like a straight line instead.

(b)

Length / cm

Period / s

50.0

1.40

55.0

1.47

60.0

1.54

65.0

1.60

CALCULATIONS AND DECIMAL PLACES If you are required to perform calculations, the number of decimal places that should be recorded depends on the situation, NOT be the number of decimals your calculator can display!

Note that these are required for your theory papers too. Marks may be deducted whenever a student makes a mistake in the number of significant figures.

1) Addition and Subtraction: The answer follows the smallest number of decimal places from the values used in the calculation.

Example (a) 18.5 N + 9.51 N = 28.0 N (1 d.p) (b) 0.563 kg − 0.47 kg = 0.09 kg (2 d.p) 2) Multiplication and Division The answer follows the lowest number of significant figures from the values used in the calculation.

Example (a) 10 N × 0.250 m = 2.5 Nm (2 s.f.)

450 g (b) = 8.93 g/cm3 (3 s.f.)

50. 4 cm3 Exceptions!

When a measurement is divided by a numeral, then the number of decimal places should be followed.

Example A stack of 5 sheets of paper has a measured thickness of 4.23 mm on a micrometer screw gauge.

So, the thickness of 1 sheet of paper =

4. 23 mm 5

= 0.85 mm (2 d.p)

Practical Skills

xi

PLOTTING A GRAPH In most cases, the relationship between the independent and dependent variables are best shown in the form of a graph. These are the things you need to note: Feature

Things to note

Axes

• The independent variable is usually represented by the horizontal axis.

• The dependent variable is usually represented by the vertical axis.

Range of points

The range on each axis depends on the question: • if you are required to find the horizontal (vertical) intercept, then you need to start the vertical (horizontal) axis from 0; • if you are required to find the gradient only, you don’t have to start the axes from the origin (0,0) at all.

Scale

• You should choose a scale such that your y y y actual graph occupies at least half of the graph paper vertically and horizontally (Figure 3). x x x • You should note that Too small Too small on Ok! (a) each axis can have a different on both horizontal axes axis scale, and Figure 1 Figure 2 Figure 3 (b) the scale should not be an awkward one (eg multiples of prime numbers).

Lines

• All lines drawn on the graph (including y y y the axes) should be in pencil, sharp and thin.

• Draw a best-fit line/curve through the x x x plotted points. Note the following: Dots Best-fit line Ok! (a) NEVER ‘join the dots’! (Figure 4) should not is not be joined balanced (b) The best-fit line does not Figure 4 Figure 5 Figure 6 necessarily pass through all the plotted points (Figure 6). (c) The best-fit line should look ‘balanced’, with a roughly equal number of plotted points on either side of the line. Eg Figure 6 looks more balanced than Figure 5.

Gradient of the best-fit line

• You need to (a) draw a large (dotted) triangle (Figure 8) (b) label the co-ordinates used to show how you determine the rise and the run of the line.

y

y

Triangle too small Figure 7

Table C

xii

Features of a graph

All You Need to Know: Physics for GCE ‘O’ Level

x

Ok! Figure 8

x

INTERPRETING THE GRAPH Based on the graph you have plotted, you may be required to state the relationship between the two variables: Type of graph

Relationship between x and y

y

y is directly proportional to x x y

y and x have a linear relationship with a positive gradient x y

y and x have a linear relationship with a negative gradient x y

y increases with x x

Table D

Deducing the relationship between x and y from a graph

TIP

If your best-fit line passes very close to the origin, but does not actually pass through it (which is almost always the case), is the relationship ‘directly proportional’, or is it a ‘linear relationship’? The answer is simple – if the relationship should have been directly proportional, the perpendicular distance between your best-fit line and the origin should be about the same as the perpendicular distance between your best-fit line and your other data points. In other words, the origin looks like it could have been one of your data points. It is therefore likely that the origin was missed due to experimental error:

Type of graph y

Relationship between x and y The origin is too far from the best-fit line compared to other data points.

Hence, the relationship is ‘linear with a positive gradient’.

x y

x

Table 1

The origin is as close to the best-fit line as the other data points.

It is very probable that the best-fit line missed the origin due to experimental error.

Hence, this relationship is ‘directly proportional’.

Best-fit lines

Practical Skills

xiii

SOURCES OF ERROR Almost every experiment has some sources of error – these errors are NOT caused by careless students (eg ignoring zero errors, reading from the wrong instrument).

Rather, they are caused by the limitations of the apparatus used, or limitations caused by the design of the experiment. Some sources of error have significant impacts on the accuracy of an experiment, other only have minor impacts. The student must therefore prioritise on which ones to list first.

Example Aim To determine the specific heat capacity of a solid by the method of mixtures.

Apparatus Styrofoam cup brass mass glass rod Thermometer, 0–100 ºC

Diagram

water beaker balance thread

0—100 °C

xiv

Rod

Procedure 1. Weigh the brass mass (m0) on the balance.

2. Weigh the Styrofoam cup (m1) on the balance.

3. Half-fill the Styrofoam cup with tap water and weigh it again (m2).

4. Record the temperature (t0) of the water in the Styrofoam cup.

5. Boil some water in a beaker. Record the temperature (t1) of the boiling water.

6. Use the thread to tie a loop round the mass. Suspend it in the boiling water by means of

the rod. Remove the brass mass; shake it briskly, and quickly transfer it into the Styrofoam cup.

Gently stir the water continuously with the brass mass. Record the highest steady temperature (t2) reached.

All You Need to Know: Physics for GCE ‘O’ Level

Question What are the sources of error in the above experiment?

Solution Some possible errors are listed below: Description

Impact on accuracy

1. The thread will release heat energy to the water in the Styrofoam cup.

Low – The mass of the thread is almost negligible, hence its heat capacity also is negligible.

2. There is some boiling water left adhering to the bob and thread that is difficult to remove quickly.

High – Water has a specific heat capacity 10 times more than most metals. Given the mass of the bobs used are usually 20 g or thereabouts, 0.2 g of water at 100°C is equivalent to 2 g of extra metal. This causes a 10% error!

3. The hot metal bob needs to be shaken in air and then transferred to the water.

Medium – The metal bob at 100°C cools down very quickly in room temperature, especially with water evaporating from its surface. Given that the usual drop in temperature is about 70°C, a 3°C drop causes almost 5% error.

4. Some heat energy is lost from the watermetal mixture to the Styrofoam container and surroundings.

Very low – The temperature rise in an experiment like this is only about 1–3°C above room temperature. The small difference causes little heat loss to the Styrofoam container and surroundings.

5. Some energy from the hot metal bob is not passed to the water but absorbed by the thermometer instead.

Very low – The heat capacity of the bulb and glass is small compared to the heat capacity of the water.

Based on the points listed above, the main sources of error in this experiment should be points 2 and 3.

Practical Skills

xv

General Tips on Answering Questions KNOWING THE SYLLABUS Before you even start studying for the examinations, have you read the syllabus? You need to know what is expected of you before you even attend the course!

For a start, you can go to the Singapore Examination and Assessment Board (SEAB) website and download the Physics syllabus you have signed up for. The syllabus will give you details like: a) the assessment objectives b) the structure of the paper – marks, weighting, duration, etc. c) the proportion of marks allocated to the different types of questions d) the quantities, symbols and units that you need to know e) the practical work that you should be familiar with f) glossary of terms used in Physics

THE ASSESSMENT OBJECTIVES AND THEIR WEIGHTING The assessment objectives for the theory paper are the skills and knowledge that are expected of you.

These are broadly classified into two types: a) Knowledge with Understanding (55 %) Questions under this category require you to know the facts of the subject – you need to recall facts and explain observations. They usually start with these words: state, define, describe, explain, outline, etc.

20 % of the marks are allocated to questions that need you to recall facts (definitions, laws, principles, etc). Do refer to the Summary of Key Quantities, Symbols and Units – they list the quantities and units whose definitions you need to recall.

The rest of the marks (35 %) is allocated to questions that need you to describe or explain (eg how an instrument such as a DC motor works, how earthing helps in preventing hazards, etc).

b) Handling Information and Solving Problems (45 %) Questions under this category require you to apply the knowledge you have learnt. The settings in these questions are usually not familiar to you, and you have to use the concepts you have learnt to solve problems. Such questions usually start with words such as: determine, predict, suggest, calculate, etc.

GLOSSARY OF TERMS USED This glossary is very useful if you do not know how much you need to include in your answer. For example, the terms ‘define’ and ‘what is meant by’ require you to give slightly different amounts of answer.

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All You Need to Know: Physics for GCE ‘O’ Level

ANSWERING PAPER 1 There are 40 Multiple Choice Questions (MCQ) in this paper. There are different ways to approach answering the questions. The best way, of course, is to work out the answer and select the correct option.

Here are some tips: 1. The proportions of the correct options are usually roughly equal. Thus, if you have an abnormally large proportion of A’s (eg 15 A’s), the chance of some of these answers being wrong is relatively high – and you should re-check your answers.

2. The same correct options rarely appear more than three times consecutively. This means that it is highly unlikely that, for example, there would be 4 consecutive B’s.

3. If you do not know the correct option right after reading the question, you can work it out. Based on what you know, eliminate as many options as you can, so that the chance of the correct option being left is higher.

4. Attempt the questions you know first, and skip the ones you do not know. Do not spend too much time (> 1 min) on one question at the expense of the rest. You can always go back to the ones you have skipped.

ANSWERING PAPER 2 Paper 2 consists of two sections: Section A (50 marks) and Section B (30 marks).

There is a mix of recall, explanation and application questions according to the weighting. The marks indicated are a good guide of how much you need to answer: [1] for one concept point.

The following, though not exhaustive, contains some tips on answering questions in this paper.

1. For explanation-type questions, the marker would be looking for detailed and clear explanations.

–– Do not repeat a statement in the question – it takes up your writing space and will not earn you any marks.

–– Adapt a general explanation so that it will be relevant to the actual situation set in the question.

–– Use and include any specific information given in the question, rather than giving general statements.

–– To help in your explanation, you can include calculations, or simple labelled diagrams/sketches (labels in the diagram help in your explanation too).

–– You may also express your answers in bulleted form, if you are running out of time or find it difficult to express your ideas. Remember: some answer is better than no answer.

–– The space provided should serve as a good guide as to how much you should answer.

General Tips on Answering Questions

xvii

2. For calculation-type questions, the marker would be looking for clear working.

–– All calculations based on formulae should be presented in this format: –– Quote the formula –– Show substitution of values –– Answer with the appropriate unit If it takes a few stages to reach the answer, explain what you are calculating at each stage by using statements.

–– All numerical answers should be between 2–3 significant figures (s.f.). There are some strict rules regarding expressing answers with the correct number of s.f.: a. When the values in the question are multiplied or divided, they should be expressed to the lower s.f. e.g. 2.130 / 3.0 = 0.71 (2 s.f.) b. When the values are added or subtracted, they should be expressed to the smaller decimal place (d.p.) e.g. (52.14 + 56.0) / 2 = 54.1 (1 d.p.)

–– Answers from calculations should not be expressed as fractions (unless instructed). Convert them into decimals.

–– Appropriate units must be given.

3. For sketching/drawing-type questions, the marker would be looking for clear and neat sketches.

–– All sketches must be made with a sharp pencil.

–– All scales must be appropriate and stated if possible.

–– The line for the graph must be sharp and clear, not ‘feathery’.

–– If you need to extract information from the graph, show how it is done. For example, to obtain the gradient, you need to draw a large triangle or mark out two sets of coordinates, or to obtain a reading off the graph, indicate how you obtain it by using dotted lines.

–– For graphical method of vector addition, make sure the scale is large enough. All arrows should be drawn and labelled.

–– For diagrams of machines such as DC motors and AC generators, make sure they are clearly labelled.

–– When drawing forces, make sure the force starts from the point of application, and is not hanging in the air!

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All You Need to Know: Physics for GCE ‘O’ Level

02

KINEMATICS

INTRODUCTION For this chapter, examination questions mostly involve the interpretation of the speed-time graph. Although there are equations of motion you can memorise and apply, all questions in kinematics can be solved using the graphical method alone. In examinations, questions almost always involve motion along a straight line and in one direction. Hence, you don’t need to learn how to solve questions involving an object turning back.

The greatest difficulty encountered by students is to UNLEARN the Distance = Speed × Time formula! You will learn that this formula does not apply all the time. In fact, it is seldom used in this chapter.

SPEED, VELOCITY AND ACCELERATION • Definitions:

Physical quantity

Definition

SI unit

Distance d

Total length travelled by a moving object (along whatever path it takes)

m

Displacement s

Straight line distance measured from the starting point to the ending point for a journey

m

Speed v

Change in distance per unit time, OR rate of change of distance

m/s

Instantaneous speed v

Speed of an object at any given moment

m/s

Average speed <v>

Total distance travelled by an object over a period of time <v> = total distance / total time

m/s

Velocity v

Displacement of an object per unit time, OR rate of change of displacement

m/s

Change of velocity per unit time, OR rate of change of velocity of an object

m/s2

v=

v= Acceleration a

a=

d t

s t

change in velocity time

=

(v − u) t

where v: final velocity; u: initial velocity Table 2. 1 16

Definitions of speed, velocity and acceleration

All You Need to Know: Physics for GCE ‘O’ Level

Clarify Your Concept!

‘Rate of change of X’ is not ‘rate of change of X per unit time’!

Worked Example 2. 1 A man walked from his house 50 m due North at 1 m/s and then 40 m due West at 0. 5 m/s to reach the bus stop.

0.5 m/s 40 m

Schedule Bus No. Arrival Departure

1 m/s

50 m

(a) What was his displacement from his house when he reached the bus stop? [2] (b) What was his average velocity for the whole journey? [3]

0.4 m/s

Solution

(a) Using a scaled diagram,

s = 64 m [1]

0.5 m/s

s

θ = 38.7° [1]

θ

So, his displacement is 64 m at an angle of 38.7° from the North.

50 1 = 50 s 40 = 80 s 0. 5

(b) Time taken to walk 50 m North =

Time taken to walk 40 m West =

Total displacement = 64 m [1]

Total time taken for the whole journey = 50 + 80 = 130 s

64 130

Average velocity = Total displacement travelled =

His average velocity for the whole journey is 0. 49 m/s.

Total time taken

[1]

= 0. 49 m/s [1]

TIP

1) How to answer a question involving calculations: 1. List the given quantities, and convert the necessary units.

2. Write the quantity to be found.

3. Write down the formula.

4. Substitute the values into the formula.

5. Calculate and express the answer with the appropriate units.

2) Remember to show the directions of the vectors by indicating them with arrows!

Chapter 2 | Kinematics

17

Worked Example 2. 2 A car is travelling along a road. Which of the following will not change when it is accelerating?

A The direction of travel of the car

TIP

B The mass of the car

The question in this worked example is a negative question, which can be quite difficult. You can try converting the negative question into a positive one, such as “Which of the following will change when it is accelerating?”. Then, work out the answers and eliminate them from the given options.

The option left is the answer to the original (negative) question.

C The speed of the car

D The velocity of the car Solution B

Clarify Your Concept!

The amount of fuel is not considered as a part of the car. So, even as the car accelerates and uses up fuel (mass of fuel decreases), the mass of the car still remains constant.

Worked Example 2. 3 A boy cruising on his skateboard at 11 m/s increases his speed to 12. 5 m/s in 6 s. Calculate his acceleration during this time interval. [3] Solution

Change in speed = 12. 5 − 11. 0 = 1. 5 m/s [1] Acceleration

=

Change in speed 1. 5 = 6 Time taken

[1]

= 0. 25 m/s2 [1]

The acceleration of the boy is 0. 25 m/s2.

• Uniform and non-uniform acceleration: Acceleration

Change of velocity per unit time

Non-uniform

Not constant

Uniform

Table 2. 2

18

Constant

Uniform and non-uniform acceleration

All You Need to Know: Physics for GCE ‘O’ Level

Clarify Your Concept!

1) When an object is moving at 2 m/s2, it means that it is increasing its speed by 2 m/s

every second.

Unit of acceleration =

2) Acceleration ≠

Unit of velocity Unit of time

=

m/s s

= m/s2

Velocity Time

Example:

If an object has an initial velocity of 2 m/s and accelerates at 1 m/s2, then its velocity increases v by 1 m/s as shown in the third column of Table 2.3. But if we use a = t , then the acceleration of the object is not uniform, as shown in the last column (which is wrong)!

t/s

v / m/s

0

2

1

1

3

3−2 1−0

2

4

3

5

Correct acceleration a =

change in v change in t

/ m/s2 Incorrect acceleration a =



2 0 =α



=1



3 =3 1



4−3 2−1

=1



4 2 =2



5−4 3−2

=1



5 3 = 1.7

v t

/ m/s2



Table 2.3 Correct and incorrect calculation of acceleration 3) Decreasing acceleration ≠ Deceleration

Remember that • Acceleration ➞ increase in speed • Deceleration ➞ decrease in speed

For an object with decreasing acceleration, its speed never decreases — its speed will increase less over time.

Chapter 2 | Kinematics

19

GRAPHICAL ANALYSIS OF MOTION • Distance-time graphs:

Gradient of graph = Speed of object

General information from the graph

Distance-time graph t

Note the following features of the graph

• Object is at rest • Speed = 0 m/s

• The value of d at t = 0 s is the initial distance of the object from the starting point.

• Object is moving with a uniform speed • Speed = Gradient of line

• The value of d at t = 0 s is the initial distance of the object from the starting point. In this case, d = 0 m.

• Object is moving with non-uniform speed • Speed = Gradient of line

• The value of d at t = 0 s is the initial distance of the object from the starting point.

• The gradient of the line at t = 0 s is the initial speed of the object.

• How the gradient changes with time will indicate if the speed of the object is increasing (acceleration) or decreasing (deceleration).

d

t

d

t

d

Table 2. 4

Different distance-time graphs

Clarify Your Concept!

Other graphs with non-uniform speed: d

d

t

d

t

(a) Increasing speed (acceleration)

(numerical value of the gradient increases with time) Figure 2. 1 d-t graphs with non-uniform speed

20

All You Need to Know: Physics for GCE ‘O’ Level

d

t

(b) Decreasing speed (deceleration)

(numerical value of the gradient decreases with time)

t

Worked Example 2. 4 Consider a man riding a bike along a straight track, and the distance from his starting point is shown in the table below: 0

t/s

1

0

d/m

10

2

3

20

4

20

5

20

6

26

40

Plot and explain which part of the graph shows uniform and non-uniform acceleration. [7] Solution The d-t graph of the man: d/m The flat line shows the cyclist at rest (not moving). The speed of the cyclist is 0 m/s (gradient = 0). [1]

A straight line means the cyclist is moving with uniform speed [1] 20 − 0

40

Speed = gradient of line = 2 − 0

The curved line means the cyclist is moving with non-uniform speed. [1] The speed (gradient) is increasing. [1]

×

20

= 2

= 10 m/s

20

×

×

×

×

2

3

4

× 0

×

1

5

6

t/s

Correctly labelled axes – [1] Correctly plotted points – [1] Correctly drawn line – [1]

Chapter 2 | Kinematics

21

• Speed-time graphs: –– Gradient of line = Acceleration of object –– Area under graph = Distance travelled by the object

General information from the graph

Speed-time graph v

0

Notes

• Object is at rest • Speed = 0 m/s

• The object is not moving.

• Object is moving at a uniform speed • Speed = v m/s • Acceleration = 0 m/s2

• The value of v at t = 0 s is the initial speed of the object.

• Object is moving with uniform acceleration • Speed = v m/s • Acceleration = gradient of line

• See above • Check the gradient of line –– if it is positive, the object is accelerating –– if it is negative, the object is decelerating –– the gradient of the line at t = 0 s is the initial acceleration of the object

t

v

t

0

v

0

t

• Object is moving with • See above non-uniform acceleration • How the gradient changes with time • Acceleration = gradient will indicate if the object having of line increasing or decreasing acceleration

v

0

Table 2.5

t

Different speed-time graphs

Clarify Your Concept!

Other graphs with non-uniform acceleration (follow how the gradient changes with time): (Gradient increasing with time) v

(a) Increasing

acceleration

(Gradient decreasing with time)

v

t

v

(b) Increasing deceleration

t

Figure 2. 2 v-t graphs with non-uniform speed 22

All You Need to Know: Physics for GCE ‘O’ Level

v

(c) Decreasing acceleration

t

(d) Decreasing deceleration

t

Worked Example 2. 5 A man jogged along a straight path and records how his speed changes with time in the following table: t/s

d/m

0

10

0

1

20

30

2

40

3

50

3

60

3

70

2

1. 5

a) Plot a speed-time graph of the man. [3]

b) Determine the average speed of the man in the first 50 s. [3] Solution

a) The speed-time graph of the man:

v / m/s

A straight line means the man is jogging with uniform acceleration.

8 7

This flat line shows the man is moving with a uniform speed of 3 m/s.

The curved line means the man is slowing down (decelerating and moving with non-uniform (negative) acceleration.

Acceleration = gradient of line

6

5

4

3−0

= 30 − 0 = 0.1 m/s2

3 2

×

1

×

×

0 10 20

Correctly labelled axes – [1] Correctly plotted points – [1] Correctly drawn line – [1]

×

×

×

The acceleration (gradient) is decreasing. Instantaneous acceleration can be found by determining the gradient of the tangent.

×

× t/s

30

40

50

60

70

b) Total distance the man covered in the first 50 s = Shaded area under the graph

= Shaded area of trapezium =

Average speed =

Total distance travelled in first 50 s = 105 50 Time taken

1 2

× 3 (20 + 50)

= 105 m [1]

[1]

= 2.1 m/s [1]

The average speed of the man is 2.1 m/s.

Chapter 2 | Kinematics

23

Clarify Your Concept!

Distance covered by an accelerating object cannot be calculated by using fixed speed (d = v × t).

This is because when an object is accelerating, its speed is never constant!

TIP

To describe the motion of the moving object (eg man), separate the motion of the object into different sections according to the graph (type of movement), then check the following: • What is the time interval for this part of the motion?

• What are the initial and final speeds of the object?

• Is it accelerating/decelerating/having constant speed?

• Is the acceleration/deceleration uniform?

Then describe the motion in the format accordingly.

So, we can describe the motion of the jogging man as follows:

From t = 0 s to 3 s, the man accelerates uniformly from rest (0 m/s) to 3 m/s. Then he jogs at a uniform speed of 3 m/s for the next 20 s. Finally, he decelerates from 3 m/s to 1. 5 m/s in 20 s.

Worked Example 2. 6 A car accelerates uniformly from 20 m/s to 28 m/s at 1. 6 m/s². Calculate the distance it covers during this acceleration. [3] Solution

v

u = 20 m/s

28

v = 28 m/s

a = 1.6 m/s2

20

From the v-t graph of the car,

t

Gradient of the line = 1. 6 = (28 −t 20)

t=

8 1. 6

Distance = Area under graph =

= 5 s [1]

1 2 (5)(20

+ 28) [1]

= 120 m [1]

It covers a distance of 120 m during this acceleration.

24

t

All You Need to Know: Physics for GCE ‘O’ Level

TIP

Students often make the mistake of calculating the area of the triangle, rather than the area of the trapezium.

Note that the area under the graph is the total area sandwiched between the line and the time axis (as shaded).

Worked Example 2. 7 The graph shows how the speed of a car changes in the first 5 seconds of its journey.

Speed / m/s 6 4 2 0

1

2

3

4

5

Time / s

a) The car is travelling at increasing deceleration during the period ____________.

A 0 s < t < 2 s

C 3 s < t < 4 s

B 2 s < t < 3 s D 4 s < t < 5 s

b) What is the distance covered by the car when it is travelling at constant acceleration?

A 5 m

C 12 m

B 6 m

D 18 m

Solution a) C b) B

1) Although the answer to (a) can easily be recalled from Figure 2.2, there is actually no need to memorise each of the graphs. There is an easy way identify the type of v-t graph. a) Note whether the line shows acceleration or deceleration.

Gradient is always positive: The line shows acceleration.

Gradient is always negative: The line shows deceleration. b) Note how the gradient of the line changes.

2) For (a), there are several ways to help you nail the correct option. a) Use a pencil to separate the graph into

sections as indicated by the options. b) Eliminate some obvious options, such as

option B (constant zero acceleration), or

even option A (acceleration). c) Identify how the gradient changes for each

of the remaining lines.

TIP

Speed / m/s 6 4 2 0

1

2

3

4

5

Time / s

Chapter 2 | Kinematics

25

TIP

3) The distance travelled is the area under the graph.

Speed / m/s

Speed / m/s

6

6

4

4

2

2

0

1

2

3

4

5

Time / s

This is quite straightforward as you just need to use the formula: Area = l × b

0

1

2

3

4

5

This is a bit tedious — you can find the area this way: 1) Count the number of squares N under the curve (shaded); 2) Find out the number of squares n that represents 1 m (32 squares represent 1 m) 3) Area = N (in metres) n

FREE FALL • The acceleration due to free fall, g, for an object near Earth: g = 10 m/s2 • The speed-time graph of a free falling object: No air resistance g = 10 m/s2

v

Terminal velocity

With air resistance

0

t

Figure 2. 3 v-t graph of an object falling under gravity –– Without air resistance, the object will free fall with an acceleration of 10 m/s2.

–– With air resistance, the object will eventually fall with a constant terminal velocity.

(See page 37 for more information.)

Clarify Your Concept!

The initial gradient of the two lines (at time t = 0 s) in the speed-time graph are the same (10 m/s2).

26

All You Need to Know: Physics for GCE ‘O’ Level

Time / s

QUESTIONS Section A

1. Which of the following is not true if an object is accelerating?

A B C

D

Speed of object

Direction of motion

Changing

Constant

Changing

Changing

Constant

Changing

Constant

Constant

2. A trolley has an initial velocity of 5. 4 cm/s and accelerates uniformly at â&#x2C6;&#x2019;0. 2 cm/s2.

What is its velocity after 20 s?

A 1. 4 cm/s

C 9. 4 cm/s

B 4. 0 cm/s

D 42 cm/s

3. The graph shows how the speeds of two cars in motion, P and Q, change with time.

Speed

P Q Time

Which of the following statements is true?

B The acceleration of P is always higher than the acceleration of Q.

A Both cars cover the same distance at time t.

C The acceleration of P is higher than the acceleration of Q at time t.

D The average speed of P is higher than the average speed of Q.

4. Below is the speed-time graph of a car.

At which point is the car having the highest deceleration?

Speed A B

D C Time Chapter 2 | Kinematics

27

5. A man is cycling round a circular cycling track at 15 m/s.

Cycling track

Man

Which of the following statements is correct?

A The velocity is constant.

B The speed is constant.

C There is no acceleration.

D None of the above

28

All You Need to Know: Physics for GCE â&#x20AC;&#x2DC;Oâ&#x20AC;&#x2122; Level

Section B

6. Below is the speed-time graph of a trolley X for 5 s.

Speed / m/s

6 5 4 3 2 1 Time / s

0

1

2

3

4

5

6

a) Describe the motion of the trolley in the first 5 s. [2]

b) Calculate the acceleration of the trolley in the first 3 s. [2] c) What is the total distance travelled by the trolley X? [2]

d) Another trolley Y is travelling at a constant speed v m/s. It covers the same distance as X from t = 0 s to t = 5 s. Determine the value of v and, on the axes above, draw a line to show

how the speed of Y changes with time. [3] 7. A motorist X starts his journey from Town A to Town B while another motorist Y starts driving from Town B towards Town A 10 minutes later than X. The graph below shows the distance-time graph of the two motorists X and Y. Distance / km

60 50 40

Y

X

30 20 10 0

10

20

30

40

50

60

Time / min

The motorists meet each other at a pit-stop midway between Towns A and B.

a) Determine

(i) the time taken for motorist Y to reach the pit-stop. (ii) the distance between Town A and Town B. [2]

b) Calculate the speed of motorist Y. Express your answer in km/h. [2]

c) Given that the motorists X and Y stop driving once they reach their destinations Town B and Town A respectively, complete the graph to show the distance travelled by motorists X and Y. [2] Chapter 2 | Kinematics

29

8. A skydiver exits an aircraft and falls towards the earth.

a) State what is meant by acceleration. [2]

c) A stone is dropped off from an aircraft. If it falls under gravity for 30 s, determine its speed as it hits the ground (ignore air resistance). [2]

b) State the acceleration due to free-fall. [1]

9. The graph below shows the vertical velocity of a helicopter over a period of time.

v / m/s

A B

10 9 8 7 6 5 4 3 2 1 C 0

20

30

40

50

60

t/s

a) Determine the initial vertical acceleration of the helicopter. [2]

c) Find the total vertical distance travelled by the helicopter over the period of 60 s? [2]

30

10

b) What is the vertical deceleration of the helicopter between B and C? [2]

d) What is the average vertical velocity of the helicopter over the period of 60 s? [2]

All You Need to Know: Physics for GCE â&#x20AC;&#x2DC;Oâ&#x20AC;&#x2122; Level