Mass in Motion (Longman Physics Topics)

LONGMAN PHYSICS TOPICS

LONGMAN PHYSICS TOPICS General Editor J . L. Lewis. Malvern College; formerly Associate Organiser. Nuffield O-Ievel Physics Project This series provides background material for modern courses in physics. The authors were closely associated w ith the Nuffield Foundation Physics Project. and thus have an intimate knowledge of its spirit. These books are not textbooks in the conventio nal sense. nor do they give the answers to investigatio ns that pupils wi ll be carrying out in the laboratory. Instead they show the relevance and application in the outside world of the principles studied in school.

This series provides background material for modern courses in physics. The authors were closely associated w ith the Nuffield Foundation Physics Project. and thus have an int imate knowledge of its spirit. These books are not textbooks in the conventio nal sense. nor do they give

LONGM AN PHYSICS TOPI CS

I

General Editor: John L. Lewis

MASS IN MOTION Jim Jardine Head ofthe Physics Department George Watson's College , Edinburgh fo rmerly Scottish Team , Nuffield Phys ics Project

Illustrated by Geoffrey Salter

LONG MAN

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LONGMAN GROUP LIMITED

London Associated companies. branches and representatives throughout the world © Longman Group Ltd 1970 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 1970

ISBN 0 582 32202 2

Printed in Great Britain by Butler and Tanner Ltd. Frome and London

I ACKNOWLEDGEMENTS

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The author and publisher are grateful to the following for permission to reproduce copyright photographs: front cover Ford Motor Company Limited; back cover Picturepoint Limited; page 4 Teltron Limited; page 5 (left) Dr. Harold E. Edgerton, Massachusetts Institute of Technology; page 5 (right) Stanley Rosenthal, Syndication International; page 6 ICI; page 7 (above) British Nylon Spinners Limited and G. Q. Parachute Company Limited; page 7 (below) Dunlop Company Limited; page 8 (above) British Hovercraft Corporation Limited; page 8 (below left) Associated Press Limited; pages 8 (below right) and 24 (above) Esso Petroleum Company Limited; page 9 National Physical Laboratory, Hovercraft Laboratory (Crown copyright reserved); page 10 UKAEA; page 13 (above left) Philip Harris Limited; page 14 (left) Smiths Industries Limited; page 14 (right) John Emery, Glenalmond; page 14 (below) Venner Limited; page 15 Panax Equipment Limited; page 17 Morris Laboratory Instruments Limited; page 18 (left) Strobe Automation Limited; page 19 (above) BBC; pages 19 (below), 22, 23, 24 (below), 26 (below), 27, 34 (right), 35, 36 (left), 44, 49, 50 and 51 Heinemann Educational Books Limited, from Jardine Physics is Fun L, 2, 3; pages 30, 33, 36 (left), 38, 39,40 and 41 Kodansha Limited, from Stroboscope and Photographs ofPhysical Phenomena; pages 25 (left), 36 (right) and 37 Kodansha Limited, from Colour Slides ofPhysical Phenomena, distributed in the UK by Philip Harris Limited; page 26 (above) British Leyland Corporation Limited; page45 USIS: page 46 Kiekhaefer Mercury; Pages 48 and 58 (below) Science Journal; page 52 (above) Professor Lord Blackett FRS and the Royal Society; page 52 (below) CERN; pages 57 and 58 Road Research Laboratory, Crowthorne (Crown copyright reserved). We are particularly grateful to Heinemann Educational Books Limited and to Kodansha Limited for their cooperation.

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This book is one in the series of Physics background books intended primarily for use with the Nuffield O-level Physics Project. Most of the team of writers who have contributed to the series were associated with that project. It was always intended that the Nuffield teachers' materials should be accompanied by background books for pupils to read, and a number of such books are being produced under the Foundation's auspices. This series is intended as a supplement to the Nuffield materials - not books giving the answers to all the investigations pupils will be doing in the laboratory, certainly not textbooks in the conventional sense, but books, easy to read and copiously illustrated, which show how the principles studied in school are applied in the outside world. The books are such that they can be used with conventional courses as well as with the new programmes. Whatever course the pupils are following, they often need straightforward books to help clarify their knowledge, sometimes to help them catch up on any topic they missed in their school course. It is hoped that this series will meet that need. This background series will provide suitable material for reading in homework. This volume is divided into sections, and a teacher may feel that one section at a time is suitable for each homework session.

NOTE TO THE ,TEACHER

I CONTENTS

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Forces Measuring motion Newton's first and second laws Inertia Projectiles Explosions and collisions Summary Answers to questions in the text

4 14

22 30 34

44 59 61

I

FORCES

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To start a ball rolling you throw it or kick it. In each case you are exerting a force on the ball. To make it change direction you can head the ball. Again you are exerting a force on the ball. When you catch a cricket ball you stop its motion by exerting a force on it.

In all the above cases forces are being used to change the motion of a ball. Does the ball exert a force on you in each case?' (NB: the answer to this and to the other numbered questions in this book will be found on pp. 61-4.) In the tube illustrated here, electrons are given off by a hot filament and speeded up by an electric force. The beam is then bent by a magnetic force produced by two large coils. Of course it is possible to exert a force on something without moving it. You can lean against a wall, squeeze a rubber ball or twist a piece of plasticine, but even then part of the object moves with respect to the rest. The next photograph shows a tennis ball which has been squashed as it strikes a tennis racket. What is the important difference in the behaviour of rubber and of plasticine after they have been squashedr? 4

This photograph shows the titanium boom on the America's Cup winner Intrepid bent under the action of tremendous forces. Titanium was chosen since it combines strength with flexibility.

5

When a body is stationary, either no forces act on it - an unlikely state of affairs - or the forces are balanced. In the diagram left, two balanced forces keep the television set at rest. If we consider the forces acting on the linesman's feet in the photograph as a single force, we can say that he is in equilibrium under the action ofthree balanced forces. These forces are represented by three arrows in the diagram. Do you notice anything special about the directions of these three forcesr' .

lable exe,ts an upwa'd force on televis,ooset

FRICTION When a car runs out of petrol on a perfectly level road, it eventually stops. Its motion has been altered. A force, or forces, must have been acting on the car. What are some of these forces?' The following pictures show how motion can be arrested by solid to solid friction in a disc brake (left) or by air resistance in a parachute. 6

Something to do Examine the frictional forces between two flat pieces of wood, metal, glass etc. Can you find more than one way of reducing the friction? If you can find an old dry wheel bearing (for example, in a bicycle or roller skate), put a drop of oil on it and see the effect produced.

If forces are necessary to change the shape or motion of a body, it might be interesting to see what happens if one of these forces - friction - is reduced.

Something to do Here is a simple balloon puck you can build at home. Glue a cork in the centre of the rough side of a piece of hardboard, and then drill a 3-mm hole through the centre of the cork and board. Fit an inflated balloon on the cork so that the air escapes through the hole, and place the puck on a smooth level surface such as a polished table. How does the puck move when you give it a push?

7

I FORCES

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The photograph shows a hovercraft moving on a cushion of air. As air friction is very much less than the friction between solid and solid, or solid and liquid, the hovercraft's driving engine will not need to exert a large force to keep it going.

A huge oil storage tank was recently floated on a cushion of air and then moved 350 metres by a small tractor.

The French Aerotrain is supported and guided by airbearing pads, and is capable of speeds greater than 300 km/h, Ifit were possible to reduce thefrictionalforces completely, how much force would be needed to keep the Aerotrain movingr' Can you give an example ofa body moving without friction?" 8

I FORCES I

FIELD FORCES If we are going to think of a force as something which changes shape or motion, we will have to admit that some forces act through empty space. For example, you can push a trolley without touching it, using two horse-shoe magnets as shown.

This photograph shows a vehicle propelled by a linear motor which depends for its operation on powerful electromagnetic forces. 9

You have no doubt charged a plastic rod or pen and used it to repel or attract other rods or to pick up pieces of paper. The diagram left shows the dome of a Van de Graaff generator attracting soap bubbles. The much bigger Van de Graaff generator in the photograph is used by nuclear physicists to accelerate atomic particles. In each case electric forces are exerted.

. - .. ~

--~--

Very carefully designed experiments have shown that there is another force which always tends to draw all bits of matter together-gravitational attraction. The experiment illustrated above can be used to measure this force. The heavier the bits of matter are, the greater is the attraction; and the nearer they are together, the greater is the attraction. As this force is extremely small, a very fine suspension wire is essential. The force between the adjacent spheres is measured by the twist of the wire. A beam of light reflected from a small mirror fixed to the suspension wire indicates the amount of twist. If the force between two chunks of stone is F when they are 1 metre apart, it will be F 4 when they are 2 metres apart and

F "9 when

they are 3 metres apart

What do you think the force would be if they were 4 metres apart?' This kind of change of force, with the square ofthe distance, is called an 'inverse square law' relationship. 10

I FORCES

It is this force which holds you on the Earth's surface and causes things to fall. As in the case of magnetic force and electric force, this gravitational force acts through empty space. These three forces are sometimes calledfieldforces, and we refer to the regions in which the forces act as magnetic fields, electric fields and gravitational fields. Gravitational force differs from the other two in that it is always a force of attraction and never of repulsion.

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MEASURING FORCES Science is concerned with measurement. Lord Kelvin once said that unless you can measure something and express the result in numbers you have not advanced to the stage of science. To study forces, then, we must find some way of measuring them.

Adding forces When several forces (e.g. weights) act side by side, the combined force is the sum of these forces.

!

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I FORCES

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If one spring stretched by a certain amount supports a weight W, then two identical springs stretched by the same amount will support 2 W, and so on.

Instead of springs we might use a number of identical elastic bands each, say, 10 centimetres long. One way of defining a unit of force might be to say that one unit of force is needed to stretch one band until it was 15 centimetres long. Here we are using the idea that forces change the shape of a body.

F=2 units

-.---I111

]!---.. .--F~2

units

If two such bands were placed side by side, two units of force would then be needed to stretch both bands to 15 centimetres. Three bands side by side would exert three units offorce when stretched to 15 centimetres and so on. A number of identical elastic bands could therefore be used to calibrate a spring in 'units of force'. In this way a simple spring balance could be constructed. 12

Some examples of commercial spring balances, calibrated in newtons, are shown in the photographs. It is important to remember that spring balances measure force even if they are calibrated in mass units such as kilogrammes. About 1660 the British scientist Robert Hooke discovered that, when a spring was stretched, the increase in length was related to the force in a simple way. Twice the force produced twice the increase in length, three times the force produced three times the increase in length, and so on. We could say that the increase in length of the spring is directly proportional to the force applied. This statement is known as Hooke's Law. There is, however, a limit to Hooke's Law. When do you think it ceases to be true?" Something to do See how the strength of an elastic band varies with the force applied to it. Does it behave in the way described by Hooke's Law?

13 L

MEASURING MOTION

The measurement of motion is much more difficult than the measurement of force, since it involves speed (distance per unit time) and direction. Even if we restrict our present studies to the measurement of motion in a straight line, we still have to measure the distance travelled in a particular time interval. If it were always possible to fit a speedometer to the object we were studying, measuring speed would be simplified. This can be done with cars and even with expensive trolleys but it becomes rather more difficult when dealing with bouncing balls or atomic particles!

Fortunately there are several techniques which enable us to measure small time intervals fairly accurately, and from these speed can be deduced. Here are some of them.

14

MEASURING MOTION

STOP CLOCKS As hand-operated stop watches are not suitable for measuring intervals ofless than a second, we use electricallyoperated clocks such as the one illustrated at the foot of p. 14 or the scaler shown below.

The scaler has a 1000 Hz oscillator which operates a set of dials. If it runs for one second the dials read 1000, and so we can use this clock to read time intervals accurately to one thousandth of a second. If we operate any of these devices ourselves (e.g. by pressing a switch at the beginning and end of a certain interval of time) the result obtained is not very accurate, as our own reactions are quite slow. The time between our seeing an event and responding to it by pressing the switch is called our reaction time. Something to do Devise an experiment to measure your reaction time, using a stop watch or other timing device.

15

MEASURING MOTION

Fortunately automatic methods of timing which do not involve human reaction time can be arranged. This diagram shows a method of using the scaler as an electronic clock to find the speed of a trolley. When the card, which is 10 centimetres long, interrupts a beam of light, the clock is switched on. The clock then runs until the card passes out of the beam of light. If the clock reads 50 milliseconds, the trolley has travelled 10 centimetres in 50 milliseconds, which is 20 centimetres in 100 milliseconds (O.ls) and therefore 200 centimetres in 1 second. Its average speed is therefore 2 metres/ second. Why do we say average speed?"

.

16

MEASURING MOTION

The diagram at the foot of p. 16 shows the scaler being used as an electronic clock to find the speed of a rifle bullet. As it shoots through a thin aluminium foil, the bullet breaks one circuit and starts the clock. After travelling one metre it breaks another circuit, in a similar way, and stops the clock.

TICKER TAPE When a ticker timer is wired to a 50 Hz supply the arm vibrates up and down fifty times a second. This vibrating arm is used to mark a paper tape every fiftieth of a second as it passes through the timer. By measuring the separation of the dots on the paper tape, the distance it has travelled every fiftieth of a second can be found. It is often convenient to find the distance gone in 10 fiftieths of a second (a 'tentick') and to express the speed in centimetres per tentick. o

2

3

4

5

6

7

8

9

10

11

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~

{tape A

{

{ tape B

We measure the length of 10 gaps between, say, ticks 10 or 1 and 11 and not the distance between ticks 1 and 10. What is the time interval between ticks 1 and 10110 The distance between the dots depends on the speed at which the tape is moving. Which tape in the diagrams on the left was moving at the greater speed?'!

o and

MULTIFLASH PHOTOGRAPHY The stroboscopic or multiflash photograph provides one of the most flexible methods of studying motion. A lamp which flashes at regular time intervals (left) is used to illuminate a moving object, and a time-exposure photograph is taken. Alternatively, a camera (below) with a

rotating disc in front of its open shutter can be used to photograph a fully lit moving object. In each case a series of photographs is taken at regular time inter\rals on the same negative. When the images are close together the object is moving slowly, and when they are far apart it is moving quickly. Study the next photograph, which was taken in this way, and see if you can tell when the tennis racket is moving slowly, when it is speeding up and when it is slowing down. What can you say about the movement of the ball?" If we know the time interval between the images, and from the scale of the photograph can find the distance travelled, we can calculate the average speed during each time interval. 18

This toy car was photographed every tenth of a second as it moved past a half-metre stick. What, very roughly, was its average speed?" Was it going at a steady speed all the time, or was it speeding up or slowing downi!"

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ACCELERATION Imagine a car starting from rest on a level road. A camera is set to take a photograph of the speedometer every two seconds. The above diagrams show the results that, in certain circumstances, might be obtained. A graph showing how the speed varies withtime is shown, left. Describe this motion.'? Why has the graph this shaper'"

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time

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2 elastic threads

26

Trolley experiments show that if one unit of force (1 elastic thread) produces a certain acceleration, two units of force (2 elastic threads) produce twice the acceleration (see the diagrams). Of course other factors such as mass must remain the same. If three elastic threads, rather than one, were used to pull the trolley, how much greater would the acceleration be?24

NEWTON'S FIRST AND SECOND LAWS

This result could be expressed by saying thatthe acceleration of a body is directly proportional to the net (or unbalanced) force acting on it. In other words, double the force: double the acceleration, and so on.

1 trolley

2 trolleys

Acceleration and mass Using the same apparatus as before, you find that two elastic threads produce two units of acceleration when they pull one trolley. If, however, the same force is exerted on two trolleys, only one unit of acceleration is produced Doubling the mass has halved the acceleration.

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NE WT ON 'S FIR ST AN D SEC ON D LAW S

pul l three trolleys, how Ifthe same two threads were use dto pare with the accelerawould the acceleration produced com tion of one trolleyr " by saying tha t the This resu lt could be expressed t net (or unb alan ced ) acc eler atio n pro duc ed by a con stan mass being acceleraforce is inversely proportional to the s: hal fthe acc eler atio n, ted. In oth er words, dou ble the mas and so on. ent s could be sumThe results of the two trolley exp erim atio n a of a bod y is mar ised by saying tha t the acc eler force app lied F, and is directly pro por tion al to the net s m. inversely pro por tion al to the mas F I . . . i.e. a is pro por tion a tomton 's sec ond law of Thi s is really a stat em ent of New mo tion .

TH E NE WT ON e. This is defined as the The uni t of mass is the kilogramm the Inte rna tion al Bur eau mass of a pla tinu m blo ck kep t at of Sta nda rds at Sevres nea r Paris. is defined in term s of The metre is the uni t of len gth and ctru m of kry pto n. a par ticu lar wavelength in the spe ura l frequency of the The second is bas ed on the nat caesium ato m. we can now define a Fro m the se thre e basic SI units, riat ely afte r Sir Isaa c uni t of force. It is called app rop ulta nt or net) force of New ton . Wh en an unb alan ced (res mme, the acc eler atio n 1 new ton acts on a mass of 1 kilogra second. pro duc ed is I me tre per sec ond per al to .!. to complete Use the relationship a is proportion m

the following tab le," ation of I 1 N acting on 1 kg produces acceler " " " " 1 kg " 3N " " " "1 kg " FN " " F N " "5 kg " "" " F N " "m kg

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m/ S2 m/ s2 rn/ s2 m/ S2 m/ S2

NEWTON'S FIRST AND SECOND LAWS

From the last line of this table you can see that

,(newtons)

.>

(metres/ second- )---or

F =ma

,.1... ; 1 \/ \\~\ yl •

~(kilOgrammes)

~l ~

Other systems of units may be used with this relationship, but we shall use only SI units in this book. One of the best known of all the stories about Newton tells of the time when he was forced to leave Cambridge because of the plague and return to his home in Lincolnshire. As he watched an apple fall from a tree one day he wondered if the same force might not keep the Moon circling the Earth. Whether or not such an incident really marked the beginning of Newton's theory of gravitation is not very important, but the story might help you to remember that 1 newton is approximately the force exerted by gravity on an average-sized apple. That is, the weight of an apple is about I newton.

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\ IN ER TI A

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ed how an unb alan ced In the previous section we discuss accelerate. The am oun t force applied to a body cau sed it to mass of the body: a of acc eler atio n dep end ed on the elerate; a bod y ofla rge bod y of small mass was easy to acc s pro per ty of a body to mass was mu ch mo re difficult. Thi dec eler atio n - is often resist acc eler atio n - or, of course, es refer to the inertial called its inertia. Scientists sometim mass of a body. ious masses on a tray You may have trie d pushing var ring s to red uce friction. which was covered with bal l bea ese masses ena ble d you Applying a small force to eac h ofth tha t is, how relu cta nt or to feel how the ir ine rtia s differed, ir mo tion . The larg er oth erw ise they wer e to cha nge the to accelerate. We might masses wer e mu ch mo re difficult ll mass) is easy to acsay: a body of low ine rtia (i.e. sma (i.e. large mass) is difcelerate: a body of high ine rtia ficult to accelerate.

cks is given a sha rp Wh en one block in a pile of blo e with it alth oug h the knock, tho se above it do not mov the blocks are mu ch frictional forces acting bet wee n top four blocks are gre ate r tha n the air resistance. The inal position. Tha t is, 'rel uct ant ' to move from the ir orig they have inertia. 30

Something to do Try to remove the card shown here so that the coin falls straight down into the tumbler.

Mass is sometimes defined as 'the amount of matter in a body'. This is not a very useful definition as it does not tell us how to measure the mass. If we could easily count the number of nucleons (that is, the protons and neutrons) in a substance, we might use this as a measure of mass. However, we cannot do this. Fortunately the 'amount of matter' in a body affects the ease with which it can be accelerated or decelerated, and it is this property of matter - its inertia - which we use to measure mass. For this reason the terms 'inertia' and 'mass' are often interchangeable.

INERTIAL BALANCE If you have a device for measuring force F, such as a spring balance, and a method of calculating acceleration a, perhaps a ticker timer or strobe photogr~h, then you can use the relationship F = ma or m = a to measure the mass or inertia of the body. Something to do '-- Here is a simple experiment you might like to try. Clamp a hacksaw blade to a table leg and fix a lump of plasticine to the end of it. Find how many to and fro swings there are each second (i.e. the frequency). If you put a larger lump of plasticine on the end, would you expect there to be more or less resistance to the change of speed during each to and fro movement? Would you expect this to increase or decrease the frequency? Try it and see if your prediction is correct.

In the project above you have built a simple inertial balance. Can you think how it might be calibrated to enable you to measure unknown masses?" Would it work just as well on the maori" or in a space ship in outer space?" Does the operation of this balance depend at all on the pull due to gravity, or is it independent of it ?30

31

I INERTIA I

Something to do This diagram shows an alternative form of inertial balance. Long elastic threads or springs are attached to a trolley or toy car which is then loaded so that the total mass is increased. How does the loading affect the to and fro frequency? How could this be refined to measure mass?

GRAVITATIONAL FIELD When a mass is in a gravitational field, there is a force acting on it. This gravitational force is something we are very familiar with and we call it the weight of the body. The mass of a body is the same everywhere, but the weight of the body will depend on the strength of the gravitational field. As the Moon's gravitational field has a different strength from that of the Earth, the weight of a body on the Moon will be different from its weight on the Earth. A multiflash photograph of two spheres, one bigger and heavier than the other, is shown opposite. The acceleration is clearly the same for the small mass m and the large mass M. What does this tell you about the size oftheforce acting on force f each sphere?" But acceleration = - - = - for one body mass m F or M for the other body. As they have the same acceleration, then force/mass has the same value for different bodies in a gravitational field. We use this as a way of measuring the strength of a gravitational field. In future we will measure field strength in newtons per kilogramme. 32

Using g for the gravitational field strength, we have (newtons~)

_ _ (newtons)

~(kilogrammes) Thus, at the surface of the Earth where the field strength is about 9.8 newtons/kilogramme, there will be a gravita tional force of 9.8 newtons on 1 kilogramme, 2 X 9.8 newtons on 2 kilogrammes, 3 X 9.8 newtons on 3 kilogrammes and so on. Since on the Moon the gravitational field is about one sixth of its value on the Earth, the force on each mass will be only one sixth as great.

Numerical equivalence of gravitational field strength and acceleration of a falling body Suppose a body of mass m kilogrammes is put in a gravitational fieldwhose strengthisg newtons/kilogramme. The force caused by gravity will be m X g newtons. This is called the weight of the body. If a body is released and allowed to fall freely under the influence of this force, the body will accelerate. This acceleration is given by

a But as

!is m

F m

=-

also the gravitational field strength g, the

acceleration of gravity must be g metres/second". That is, the acceleration in m/ S2 has the same numerical value: as the gravitational field strength in N/kg. Thus at the surface of the earth, where the field strength is 9.8 newtons/kilogramme, the acceleration of a body falling freely under gravity is 9.8 metres/second".

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\ PRO JEC TILE SI

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You have seen that when a ball is droppe d it accelerates toward s the ground. The multiflash photog raph (left) shows a series of images of such a ball beside a metre stick. Study the photograph carefully, and then decide which ofthefollowing statements about the ball are correct. 32 1. Its speed is increasing steadily. 2. Its acceleration is constant. 3. The force acting on it is constant. 4. The distance it drops each time interval increases by a constant amount. 5. Its average speed during each time interval increases by a constant amount. 6. Its speed is proportional to the time it has been falling. 7. Its speed is proportional to the distance it has fallen. 8. Its speed is proportional to the square root ofthe distance it has fallen. You might also like to try to calculate the acceleration of gravity from the photograph. To do this you will need to know the time interval between images. It is 0.033 seconds." In the last chapte r we saw that the mass of a body does not alter its acceleration in a gravitational field. The gravitational force is double d on twice the mass, trebled on thrice the mass and so on. If we ignore air resistance, as we will do throug hout this section, we can say that all bodies, regardless of their size or mass, accele rate toward the Earth at the same rate.

[ PROJECTILES

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Now let us consider what happens when a ball is projected horizontally and then allowed to fall. For example, we might roll a ball along a table and then allow it to run off the end. The picture at the foot of p. 34 shows a multiflash photograph of such a ball. It is taken from the side of the table.

By drawing equally spaced vertical lines on the photograph, you can see that the horizontal speed of the ball remains constant. That is, it continues to move at the same speed in its original direction. This is what you might have expected from Newton's first law of motion, since there is no horizontal unbalanced force acting on the ball. So it would appear that the speed of a body in one direction is not affected by a force acting at right angles to that direction. This is confirmed by the photograph of the same event shown below. To take this picture the camera was held above the table, so that the horizontal velocity of the ball is shown before and after leaving the table. You can see that it is constant throughout. To investigate the vertical motion of the ball a third

35

photograph was taken. This time the camera was placed in front of the table so that the ball rolled toward the camera. The result is shown left. You will see that this picture looks very similar to the left-hand photograph on p. 34. Does this mean that the vertical motion, in this case the acceleration caused by gravity, is not affected by the horizontal motion? Are the vertical and horizontal movements quite independent? To check this, a piece of apparatus was used which projected a ball-bearing horizontally and, at the same time, released a second stationary ball-bearing. A multi-

36

IPROJECTILES I

flash photograph of the event is shown here. You can see by comparing the heights of the ball-bearings at different times that they fall with the same vertical acceleration. So the vertical and horizontal motions are independent. The grid superimposed on the same photograph makes this point clear. The horizontal speed is constant, as no unbalanced force is acting in that direction (Newton's first law), and the vertical acceleration is constant because a constant vertical force (gravity) is acting on the ballbearing (Newton's second law).

37

@OJECTILES I

If these statements are correct, the speed at which the ball-bearing is projected horizontally should not affect the vertical acceleration. This is confirmed above, where a number of different horizontal velocities was used.

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I PROJECTILES

A multiflash photograph of a ball-bearing fired into the air can be seen here. Which ofthefollowing statements about the ball-bearing are correct?" 1. Its deceleration as it rises is numerically the same as its acceleration as it falls. 2. Its vertical motion is independent ofits horizontal motion. 3. It is speeding up as it falls. 4. Its horizontal speed is constant. 5. Its vertical speed is constant. 6. At any particular height above the point of projection, its speed is the same when it is rising as it is when it isfalling. Suppose that in the experiment illustrated on p. 37 the stationary ball-bearing is released from another point which is in the direct line of fire of the projected ball. Then, if there were no gravitational field, the projected ballbearing would strike the stationary one.

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19

I PROJECTILES

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Gravity, however, as we have seen, is no respecter of persons. It acts equally on both ball-bearings, so that after a given time both will have fallen through the same distance. The result is indicated in the photograph. In this experiment an electromagnet was used to release the stationary ball-bearing at the instant the other left the muzzle.

40

I PROJECTilES

I

Exactly the same result is observed if the gun fires the 'bullet' at an angle. The photographs on pages 40-1 show that a direct hit is obtained regardless of the speed at which the bullet is fired. This is sometimes called the 'monkey and hunter' experiment. Imagine that a monkey is hanging from a branch of a tree and sees that a hunter is about to shoot him. Whenever the monkey sees the flash of the gun he drops from the branch. The monkey and the bullet will both have fallen the same distance in the same time, so that the monkey's attempt to foil the hunter will not succeed. Of course, the sights of the gun must not have been adjusted to take the effects of gravity into account!

You can probably understand this more clearly if you imagine the whole operation taking place in a giant lift. If the lift rope breaks at the very moment the gun is fired and the monkey lets go, then the lift and all it contains will accelerate downward at approximately 10 metres/ second/second. If you were in the lift, you would see the bullet move straight across the lift and strike the monkey, which would be still 'hanging' beside the branch of the tree. If a stationary observer outside the lift could watch what was going on he would, of course, see the monkey and bullet falling as before, the only difference being that the hunter and tree would be falling too! 42

[PROJECTILES

I

I

I

43

EXPLOSIONS

When two trolleys spring apart, the product of the mass (m) and the speed (v) of each is found to be the same.

AND COLLISIONS

Because in all similar 'explosions' this product mv is found to be conserved, it is given a special name: momentum. If we take direction into account, we have m v or m at

- m'v'

= - m' a' t

Assuming that the trolleys start from rest and that a is the acceleration as they react together for a short time t

ma

=

-m' a'

which from Newton's second law (F F

=

=

m a) shows us that

-F'

The forces acting on the trolleys at any instant are therefore equal in size but opposite in direction. This is really a statement of Newton's third law of motion: 'to every action there is an equal and opposite reaction'. A trolley is propelled by a small carbon dioxide cylinder. As the gas is forced out of the cylinder in one direction, the cylinder, and hence the trolley, are propelled in the other direction. As the trolley accelerates, momentum m v is conserved, and the force m a acting on the trolley at any instant is exactly equal in size and opposite in direction to the force acting on the carbon dioxide. 44

EXPLOSIONS AND COLLISIONS

The photograph below shows this principle being used to launch a spacecraft, and in the diagram left, retrorockets are being fired to slow down a capsule before it enters the Earth's atmosphere.

45

A boat is propelled through the water as the propeller pushes the water in the opposite direction. Something to do I. Stand on roller skates on a smooth level surface and throw some heavy object away from you. Explain what happens. Why do you not normally observe this result when you throw a ball or even a much heavier object? 2. If a bicycle valve is fitted into the stopper of a plastic bottle, a simple 'rocket' can be made. Half fill the bottle with water and pump in air until the cork is forced out. Warning: conduct this experiment out of doors! 3. Blow up a toy balloon and throw it into the air with its mouth open. Explain what happens.

46

EXPLOSIONS

AND COLLISIONS

There is, of course, no way of recharging the balloon in the last experiment and it soon comes to rest. In a rocket engine, liquid chemicals are continuously fed under pressure into the combustion chamber. There they burn and produce a steady supply of high-temperature, highpressure gas. This gas is then ejected from the nozzle of the rocket and so propels the rocket in the opposite direction. The thrust or force exerted on the gas - and so on the rocket - is given by Newton's second law. So far we have considered the acceleration of a constant mass m accelerated by a constant force F to produce a constant acceleration a. F

m a = m (L1 v)

=

(M)

where L1 v means a 'change of velocity' and L1t the time interval during which the change takes place. In general the Greek letter L1 (delta) means 'a small change of'. Newton's second law is, however, also valid for a changing mass. The force F, in appropriate units, is equal to the rate of change of momentum; that is F =

L1 (mv)

L1t momentum L1(mv)

This change of can result from a change of speed L1 v or a change of mass L1 m, so that

F or

F

L1v mL1t L1m =-v L1t =

In the former we consider a constant mass m accelerating L1m . -L1v an d·m t h e Iatter a chanzi angmg mass --, movmg at a L1t

L1t

constant speed v. The thrust produced by a rocket motor is equal to the mass of the propellent passing through the nozzle every second, ~~, multiplied by the velocity v of the gas leaving the nozzle.

47

COLLISIONS Collision damage can be caused by a large mass moving slowly or by a small mass moving very quickly. Notice the damage caused to a stainless steel plate exposed outside a spacecraft (Gemini 8) orbiting 400 kilometres above the Earth. The damage was caused by a 10- 7 g micrometeorite travelling at about 20 km/ s. To investigate the damage caused by a car crashing into a telegraph pole at 50 krn/h, engineers at Cornell Aeronautical Laboratory dropped a vehicle on to a horizontally mounted pole. From what height must the car be droppedfor it to reach that speedr"

/ I I

/~

48

/

I

EXPLOSIONS AND COLLISIONS

Conservation of momentum When two bodies collide, the product mass X velocity is always the same before and after the collision. We can however measure this product only when both bodies are free to move. The air track (p. 22) enables accurate measurements to be taken.

In this photograph a moving vehicle has collided with, and then stuck to, a stationary one. From which side was the moving vehicle comingr'" What can you say about the mass of the vehicles?" Is this an elastic or inelastic collisionr"

An elastic collision between a stationary vehicle and a moving vehicle of the same mass is shown above. The moving vehicle stops, and the stationary one moves off at the same speed. Momentum is thus conserved.

49

EXPLOSIONS

AND COLLISIONS

In this photograph two vehicles of different masses and

moving at different speeds collide. They then move apart at different speeds. A shutter mechanism enables us to take a multiflash photograph of the straws (attached to the vehicles) before and after the collision.

The results of this experiment are shown in the above photograph. You might like to measure the four speeds and then work out the total momentum before and after the collision. Remember that momentum is a vector quantity and that you must take direction into account."

50

EXPLOSIONS

AND COLLISIONS

An air gun pellet is fired into a lump of plasticine mounted on an air track vehicle. As the mass of the vehicle with plasticine can be easily found and its velocity determined from the stop clock reading, the total momentum can be calculated. If we assume that this momentum is equal to the momentum of the pellet, the speed of a pellet of known mass can be calculated.

The dry ice puck apparatus (p. 24) can be used to study elastic collisions in two dimensions. The photograph here shows a multiflash picture in which '1 moving puck collided with a stationary one of equal mass. Use the photograph to compare the momentum before and after the collision. 51

EXPLOSIONS AND COLLISIONS

Remember that momentum is a vector quantity. You must resolve the velocities of the pucks after the collision into components in the originaldirection andat right angles to it. 40

This is a cloud chamber photograph of an atomic collision between a moving and a stationary particle. What can you say about the masses of these particles?" By considering conservation of momentum and mass/ energy, nuclear scientists can interpret bubble chamber photographs such as that illustrated below. Many new particles have been discovered in this way.

52

.-

t

EXPLOSIONS AND COLLISIONS

at rest

speeding up

THE PRODUCT Ft You have probably stood in a lift and experienced feeling heavier when the lift starts upward or that sinking feeling when it starts to go down. Imagine you are standing on a weighing machine in a lift. The pointer will indicate your weight when the lift is at rest. There are two forces acting on you, the downward pull of the earth and the upward force exerted on your feet by the platform of the weighing machine. As these two forces are equal in magnitude no unbalanced force acts on you, and therefore you do not move. When the lift starts to move upward, the platform exerts a greater force on you than it did before. This increased' force is registered by the pointer. As the earth's pull remains the same, there is now an unbalanced force acting on you and thus you accelerate upward. If the acceleration (a) produced by this unbalanced force (F) changes your speed from u to v in t seconds, we can calculate the product Ft from Newton's second law. 53

EXPLOSIONS

AND

~

COLLISIONS

-

F = ma = m (v - u) mv - mu t t Ft = mv - mu = the change in momentum

In other words, the product Ft is numerically equal to the

change in momentum. Of course, when the lift stops accelerating and moves with a steady speed, there will be no unbalanced force on you. What will the weighing machine show?" What will it show as the lift decelerates and comes to rest ?43 Let us consider a typical journey in the lift. Suppose it is at rest for 2 seconds, it accelerates upward for 4 seconds, then moves with a steady speed for 6 seconds, then decelerates for 3 seconds and finally comes to rest. Suppose your weight is 40 kgf or 400 newtons. A graph ofthe pointer readings on the weighing machine, measured in newtons, might be as shown here.

"' c

o ~

!

600 raccelerating (up)

Cl '" c

ii

steady speed

~ 400~

C

i: o