I is very useful o be able o make predicions abou he way moving objecs behave. In his chaper you will learn abou some equaions of moion ha can be used o calculae he speed and acceleraion of objecs, and he disances hey ravel in a cerain ime. Secion A: Forces and Moion Figure 1.1 The world is full of speeding objecs. Speed is a erm ha is used a grea deal in everyday life. Acion films ofen feaure high-speed chases. Speed is a cause of faal accidens on he road. Spriners srive for greaer speed in compeiion wih oher ahlees. Rockes mus reach a highenough speed o pu communicaions saellies in orbi around he Earh. This chaper will explain how speed is defined and measured and how disance ime graphs are used o show he movemen of an objec as ime passes. We shall hen look a changing speed acceleraion and deceleraion. We shall use velociy ime graphs o find he acceleraion of an objec. We shall also find how far an objec has ravelled using is velociy ime graph. You will find ou abou he difference beween speed and velociy on page 4. Speed If you were old ha a car ravelled 1 kilomeres in hours you would probably have no difficuly in working ou ha he speed (or sricly speaking he average speed see page ) of he car was 5 km/h. You would have done a simple calculaion using he following definiion of speed: disance ravelled speed = ime aken This is usually wrien using he symbol v for speed or velociy, d for disance ravelled and for ime: v = d 1
Unis of speed Typically he disance ravelled migh be measured in meres and ime aken in seconds, so he speed would be in meres per second (m/s). Oher unis can be used for speed, such as kilomeres per hour (km/h), or cenimeres per second (cm/s). In physics he unis we use are meric, bu you can measure speed in miles per hour (mph). Many cars show speed in boh mph and kph (km/h). Exam quesions should be in meric unis, so remember ha m is he abbreviaion for meres (and no miles). 1 v d Figure 1. You can use he riangle mehod for rearranging equaions like d = v. Reminder: To use he riangle mehod o rearrange an equaion, cover up he hing you wan o find. For example, in Figure 1., if you waned o work ou how long () i ook o ravel a disance (d) a a given speed (v), covering in Figure 1. leaves d/v, or disance divided by speed. If an examinaion quesion asks you o wrie ou he formula for calculaing speed, disance or ime, always give he acual equaion (such as d = v ). You may no ge he mark if you jus draw he riangle. Rearranging he speed equaion The speed equaion can be rearranged o give wo oher useful equaions: and disance ravelled, d = speed, v ime aken, ime aken, = Average speed disance ravelled, d speed, v The equaion you used o work ou he speed of he car, on page 1, gives you he average speed of he car during he journey. I is he oal disance ravelled, divided by he ime aken for he journey. If you look a he speedomeer in a car you will see ha he speed of he car changes from insan o insan as he acceleraor or brake is used. The speedomeer herefore shows he insananeous speed of he car. Speed rap! Suppose you wan o find he speed of cars driving down your road. You may have seen he police using speed guns o check ha drivers are keeping o he speed limi. Speed guns use microprocessors (compuers on a chip ) o produce an insan reading of he speed of a moving vehicle, bu you can conduc a very simple experimen o measure car speed. Measure he disance beween wo poins along a sraigh secion of road wih a ape measure or click wheel. Use a sopwach o measure he ime aken for a car o ravel he measured disance. Figure 1.3 shows you how o operae your speed rap. 1 Measure 5 m from a sar poin along he side of he road. Sar a sop clock when your parner signals ha he car is passing he sar poin. click click 3 Sop he clock when he car passes you a he finish poin. sar sop 3 Figure 1.3 Measuring he speed of a car.
Using he measuremens made wih your speed rap, you can work ou he speed of he car. Use he equaion: speed = =.5s disance ravelled ime aken So, if he ime measured is 3.9 s, he speed of he car in his experimen is: speed = 5 m = 1.8 m/s 3.9 s Disance ime graphs =.s = 1.5s Figure 1.4 A car ravelling a consan speed. = 1.s =.5s =.s Figure 1.4 shows a car ravelling along a road. I shows he car a.5 second inervals. The disances ha he car has ravelled from he sar posiion afer each.5 s ime inerval are marked on he picure. The picure provides a record of how far he car has ravelled as ime has passed. We can use he informaion in his sequence of picures o plo a graph showing he disance ravelled agains ime (Figure 1.5). You can conver a speed in m/s ino a speed in km/h. If he car ravels 1.8 meres in one second i will ravel 1.8 6 meres in 6 seconds (ha is, one minue) and 1.8 6 6 meres in 6 minues (ha is, 1 hour), which is 46 8 meres in an hour or 46.1 km/h (o one decimal place). We have muliplied by 36 (6 6) o conver from m/s o m/h, hen divided by 1 o conver from m/h o km/h (as here are 1 m in 1 km). Rule: o conver m/s o km/h simply muliply by 3.6. Disance (m) 3 4 18 1 6 from sar (s)..5 1. 1.5..5 Disance ravelled. 6. 1. 18. 4. 3. from sar (m) The disance ime graph ells us abou how he car is ravelling in a much more convenien form han he sequence of drawings in Figure 1.4. We can see ha he car is ravelling equal disances in equal ime inervals i is moving a a seady or consan speed. This fac is shown immediaely by he fac ha he graph is a sraigh line. The slope or gradien of he line ells us he speed of he car he seeper he line he greaer he speed of he car. So, in his example: speed = gradien = disance ime Speed and velociy = 3 m =1 m/s.5 s Some disance ime graphs look like he one shown in Figure 1.6. I is a sraigh line, showing ha he objec is moving wih consan speed, bu he line is sloping down o he righ raher han up o he righ. The gradien of such a line is negaive Displacemen from sarimg poin (m)..5 1. 1.5..5 (s) Figure 1.5 Disance ime graph for he ravelling car in Figure 1.4. Noe ha his graph slopes down o he righ. We call his a NEGATIVE SLOPE or negaive gradien. (s) Figure 1.6 In his graph disance is decreasing wih ime. 3
A vecor is a quaniy ha has boh size and direcion. Displacemen is disance ravelled in a paricular direcion. Force is anoher example of a vecor. The size of a force and he direcion in which i acs are boh imporan. because he disance ha he objec is from he saring poin is now decreasing he objec is reracing is pah back owards he sar. Displacemen means disance ravelled in a paricular direcion from a specified poin. So if he objec was originally ravelling in a norherly direcion, he negaive gradien of he graph means ha i is now ravelling souh. Displacemen is an example of a vecor. Velociy is also a vecor. Velociy is speed in a paricular direcion. If a car ravels a 5 km/h around a bend is speed is consan bu is velociy will be changing for as long as he direcion ha he car is ravelling in is changing. velociy = increase in displacemen ime aken Worked example Example 1 B A Figure 1.7 The screen of a global posiioning sysem (GPS). A GPS is an aid o navigaion ha uses orbiing saellies o locae is posiion on he Earh s surface. The GPS in Figure 1.7 shows wo poins on a journey. The second poin is 3 km norh wes of he firs. If a walker akes 45 minues o ravel from he firs poin o he second, wha is he average velociy of he walker? Wrie down wha you know: increase in displacemen is 3 km norh wes ime aken is 45 min (45 min=.75 h). Use: velociy = increase in displacemen ime aken average velociy = 3 km.75 h = 4. km/h norh wes Acceleraion Figure 1.8 shows some objecs whose speed is changing. The plane mus accelerae o reach ake-off speed. In ice hockey, he puck deceleraes only very slowly when i Figure 1.8 Acceleraion consan speed and deceleraion. 4
slides across he ice. When he egg his he ground i is forced o decelerae (decrease is speed) very rapidly. Rapid deceleraion can have desrucive resuls. Acceleraion is he rae a which objecs change heir velociy. I is defined as follows: acceleraion = change in velociy ime aken This is wrien as an equaion: a = (v u) or final velociy iniial velociy ime aken where a = acceleraion, v = final velociy, u = iniial velociy and = ime. (Why u? Simply because i comes before v!) Acceleraion, like velociy, is a vecor because he direcion in which he acceleraion occurs is imporan as well as he size of he acceleraion. Unis of acceleraion Velociy is measured in m/s, so increase in velociy is also measured in m/s. Acceleraion, he rae of increase in velociy wih ime, is herefore measured in m/s/s (read as meres per second per second ). We normally wrie his as m/s (read as meres per second squared ). Oher unis may be used for example, cm/s. Example A car is ravelling a m/s. I acceleraes seadily for 5 s, afer which ime i is ravelling a 3 m/s. Wha is is acceleraion? Wrie down wha you know: iniial or saring velociy, u = m/s final velociy, v = 3 m/s ime aken, = 5 s Use: a = v u a = a = 3 m/s m/s 5 s 1 m/s 5 s = m/s The car is acceleraing a m/s. Worked example I is good pracice o include unis in equaions his will help you o supply he answer wih he correc uni. Deceleraion Deceleraion means slowing down. This means ha a deceleraing objec will have a smaller final velociy han is saring velociy. If you use he equaion for finding he acceleraion of an objec ha is slowing down, he answer will have a negaive sign. A negaive acceleraion simply means deceleraion. 5
Worked example Galileo was an Ialian scienis who was born in 1564. He developed a elescope, which he used o sudy he moion of he planes and oher celesial bodies. He also carried ou many experimens on moion. Example 3 An objec srikes he ground ravelling a 4 m/s. I is brough o res in. s. Wha is is acceleraion? Wrie down wha you know: iniial velociy, u = 4 m/s final velociy, v = m/s ime aken, =. s As before, use: a = v u a = a = m/s 4 m/s. s 4 m/s. s = m/s So he acceleraion is m/s. In Example 3, we would say ha he objec is deceleraing a m/s. This is a very large deceleraion. Laer, in Chaper 3, we shall discuss he consequences of such a rapid deceleraion! Measuring acceleraion When a ball is rolled down a slope i is clear ha is speed increases as i rolls ha is, i acceleraes. Galileo was ineresed in how and why objecs like he ball rolling down a slope speeded up, and he devised an ineresing experimen o learn more abou acceleraion. A version of his experimen is shown in Figure 1.9. 5 4 3 1 ball rolling down a slope, sriking small bells as i rolls Figure 1.9 Galileo s experimen. 6 Though Galileo did no have a clockwork imepiece (le alone an elecronic imer), he used his pulse and a ype of waer clock o achieve imings ha were accurae enough for his experimens. Galileo waned o discover how he disance ravelled by a ball depends on he ime i has been rolling. In his version of he experimen, a ball rolling down a slope srikes a series of small bells as i rolls. By adjusing he posiions of he bells carefully i is possible o make he bells ring a equal inervals of ime as he ball passes. Galileo noiced ha he disances ravelled in equal ime inervals increased, showing ha he ball was ravelling faser as ime passed. Galileo did no possess an accurae way of measuring ime (here were no digial sopwaches in seveneenh-cenury Ialy!) bu i was possible o judge equal ime inervals accuraely simply by lisening.
Galileo also noiced ha he disance ravelled by he ball increased in a predicable way. He showed ha he rae of increase of speed was seady or uniform. We call his uniform acceleraion. Mos acceleraion is non-uniform ha is, i changes from insan o insan bu we shall only deal wih uniformly acceleraed objecs in his chaper. Velociy ime graphs The able below shows he disances beween he bells in an experimen such as Galileo s. Bell 1 3 4 5 (s).5 1. 1.5..5 Disance of bell from sar (cm) 3 1 7 48 75 We can calculae he average speed of he ball beween each bell by working ou he disance ravelled beween each bell, and he ime i ook o ravel his disance. For he firs bell: disance ravelled velociy = ime aken 3 cm =.5 seconds = 6 cm/s This is he average velociy over he.5 second ime inerval, so if we plo i on a graph we should plo i in he middle of he inerval, a.5 seconds. Repeaing he above calculaion for all he resuls gives us he following able of resuls. We can use hese resuls o draw a graph showing how he velociy of he ball is changing wih ime. The graph, shown in Figure 1.1, is called a velociyime graph. (s).5.75 1.5 1.75.5 Velociy (cm/s) 6 18 3 4 54 The graph in Figure 1.1 is a sraigh line. This ells us ha he velociy of he rolling ball is increasing by equal amouns in equal ime periods. We say ha he acceleraion is uniform in his case. Velociy (cm/s) 6 5 4 3 1..5 1 1.5.5 (s) Figure 1.1 Velociy ime graph for an experimen in which a ball is rolled down a slope. (Noe ha as we are ploing average velociy, he poins are ploed in he middle of each successive.5 s ime inerval.) A modern version of Galileo s experimen ligh gaes posiion posiion 1 sar inerruper posiion 4 posiion 3 air pumped in here sloping air rack elecronic imer or daa logger Figure 1.11 Measuring acceleraion. Today we can use daa loggers o make accurae direc measuremens ha are colleced and manipulaed by a compuer. A spreadshee programme can be used o produce a velociy ime graph. Figure 1.11 shows a glider on a slighly sloping air- 7
1 Velociy (cm/s) Airrack a 1.5 Airrack a 3. (s) Av Vel. (cm/s) (s) Av Vel. (cm/s).....45 11.1.3 15.9 1.35 33.3.95 47.6.5 55.6 1.56 79.4 3.15 77.8.1 111.1 5 Tips 1 When finding he gradien of a graph, draw a big riangle. Choose a convenien number of unis for he lengh of he base of he riangle o make he division easier. gradien = AB BC A (s) 3.5 Figure 1.1 Resuls of wo air-rack experimens. (Noe, once again, ha because we are ploing average velociy in he velociy ime graphs, he poins are ploed in he middle of each successive ime inerval.) rack. The air-rack reduces fricion because he glider rides on a cushion of air ha is pumped coninuously hrough holes along he air-rack. As he glider acceleraes down he sloping rack he whie card mouned on i breaks a ligh beam, and he ime ha he glider akes o pass is measured elecronically. If he lengh of he card is measured, and his is enered ino he spreadshee, he velociy of he glider can be calculaed by he spreadshee programme using v = d. Figure 1.1 shows some velociy ime graphs for wo experimens done using he air-rack apparaus. In each experimen he rack was given a differen slope. The seeper he slope of he air-rack he greaer he glider s acceleraion. This is clear from he graphs: he greaer he acceleraion he seeper he gradien of he graph. The gradien of a velociy ime graph gives he acceleraion. More abou velociy ime graphs Gradien The resuls of he air-rack experimens in Figure 1.1 show ha he slope of he velociy ime graph depends on he acceleraion of he glider. The slope or gradien of a velociy ime graph is found by dividing he increase in he velociy by he ime aken for he increase, as shown in Figure 1.13. Increase in velociy divided by ime is, you will recall, he definiion of acceleraion (see page 5), so we can measure he acceleraion of an objec by finding he slope of is velociy ime graph. The meaning of he slope or gradien of a velociy ime graph is summarised in Figure 1.14. Area under a velociy ime graph Velociy (m/s) 15 1 v v v v 5 8 B 1 3 4 5 C (s) Figure 1.13 Finding he gradien of a velociy ime graph. a) shallow gradien low acceleraion b) seep gradien high acceleraion c) horizonal (zero gradien) no acceleraion d) negaive gradien negaive acceleraion (deceleraion) Figure 1.14 The gradien of a velociy ime graph gives you informaion abou he moion of an objec a a glance.
Figure 1.15a shows a velociy ime graph for an objec ha ravels wih a consan velociy of 5 m/s for 1 s. A simple calculaion shows ha in his ime he objec has ravelled 5 m. This is equal o he shaded area under he graph. Figure 1.15b shows a velociy ime graph for an objec ha has acceleraed a a consan rae. Is average velociy during his ime is given by: average velociy = iniial velociy + final velociy In his example he average velociy is, herefore: average velociy = m/s + 1 m/s or u + v which works ou o be 5 m/s. If he objec ravels, on average, 5 meres in each second i will have ravelled meres in 4 seconds. Noice ha his, oo, is equal o he shaded area under he graph (given by he area formula for a riangle: area = 1 base heigh). The area under a velociy ime graph is equal o he disance ravelled by (displacemen of) he objec in a paricular ime inerval. Speed invesigaions using icker ape A icker imer is a machine ha makes a series of dos on a paper ape moving hrough he machine. Mos icker imers used in school physics laboraories make 5 dos each second. If he ape is pulled slowly hrough he machine, he dos are close ogeher. If he ape is pulled hrough quickly, he dos are furher apar (Figure 1.16). Ticker ape can be used o invesigae speed or acceleraion. One end of he icker ape is fasened o a rolley or air rack glider, which pulls he ape hrough he machine as i moves. The ape can hen be cu up ino lenghs represening equal ime, and used o make speed-ime graphs. As each lengh of ape represens.1 seconds, you can work ou he velociy from he lengh of he piece of ape using he equaion velociy = disance (lengh of ape)/ime (.1 seconds). a) b) Velociy (m/s) Velociy (m/s) 5 1 s area = 5m/s 1s = 5 m = disance ravelled 4 6 8 1 (s) 1 a) power inpu area of a riangle = 1/ base heigh 5 m/s 1 m/s area = disance ravelled 1 4 s 3 4 (s) Figure 1.15 a) An objec ravelling a consan velociy, b) An objec acceleraing a a consan rae. coil icker ape magne vibraing bar carbon paper disc a) 6 b) 6 c) 6 5 5 5 b) speed (cm/s) 4 3 1 speed (cm/s) 4 3 1 speed (cm/s) 4 3 1.1 sec c) d).1 sec.1..3 ime (s).4.5.1..3 ime (s) Figure 1.17 Disance-ime graphs made from icker ape. a) Consan speed, b) Acceleraing, c) Deceleraing..4.5.1..3 ime (s).4.5 Figure 1.16 a) A icker imer, b) A ape pulled hrough a a seady, slow speed. The icker imer makes 5 dos each second, so every 5 dos show he disance moved in.1 second. c) A ape pulled hrough a a seady, faser speed. d) A ape being acceleraed hrough he imer. 9
End of Chaper Checklis You will need o be able o do he following: undersand and use he equaion average speed, v = disance ravelled, d ime aken, recall ha he unis of speed are meres per second, m/s recall ha disance ime graphs for objecs moving a consan speed are sraigh lines undersand ha he gradien of a disance ime graph gives he speed recall ha disance ravelled in a specified direcion is called displacemen; displacemen is a vecor quaniy undersand ha velociy is speed in a specified direcion. I is also a vecor quaniy undersand and use he equaion acceleraion = change in velociy ime aken, or a = (v u) recall ha he unis of acceleraion are meres per second squared, m/s undersand ha acceleraion is a vecor undersand ha velociy ime graphs of objecs moving wih consan velociy are horizonal sraigh lines undersand ha he gradien of a velociy ime graph gives acceleraion; a negaive gradien (graph line sloping down o he righ) indicaes deceleraion work ou he disance ravelled from he area under a velociy ime graph undersand and use he equaion average velociy = explain how o use icker ape o measure speed. iniial velociy + final velociy or u + v Quesions More quesions on speed and acceleraion can be found a he end of Secion A on page 57. 1 A spriner runs 1 meres in 1.5 seconds. Work ou her speed in m/s. A je can ravel a 35 m/s. How far will i ravel a his speed in: a) 3 seconds b) 5 minues c) half an hour? 3 A snail crawls a a speed of.4 m/s. How long will i ake o climb a garden cane 1.6 m high? 4 Look a he following skeches of disance ime graphs of moving objecs. Disance A Disance B In which graph is he objec: a) moving backwards b) moving slowly c) moving quickly d) no moving a all? Disance C D 1
5 Skech a disance ime graph o show he moion of a person walking quickly, sopping for a momen, hen coninuing o walk slowly in he same direcion. 6 Plo a disance ime graph using he daa in he following able. Draw a line of bes fi and use your graph o find he speed of he objec concerned. Disance (m). 1.6 3.5 4.8 6.35 8. 9.6 (s)..5.1.15..5.3 7 The diagram below shows a rail of oil drips made by a car as i ravels along a road. The oil is dripping from he car a a seady rae of one drip every.5 seconds. a) Wha can you ell abou he he way he car is moving? b) The disance beween he firs and he sevenh drip is 135 meres. Wha is he average speed of he car? 8 A car is ravelling a m/s. I acceleraes uniformly a 3 m/s for 5 s. a) Draw a velociy ime graph for he car during he period ha i is acceleraing. Include numerical deail on he axes of your graph. b) Calculae he disance he car ravels while i is acceleraing. 9 Explain he difference beween he following erms: a) average speed and insananeous speed b) speed and velociy. 1 A spors car acceleraes uniformly from res o 4 m/s in 6 s. Wha is he acceleraion of he car? 11 Skech velociy ime graphs for: a) an objec moving wih a consan velociy of 6 m/s b) an objec acceleraing uniformly a m/s for 1 s c) an objec deceleraing a 4 m/s for 5 s. 1 A plane saring from res acceleraes a 3 m/s for 5 s. By how much has he velociy increased afer: a) 1 s b) 5 s c) 5 s? 13 Look a he following skeches of velociy ime graphs of moving objecs. Velociy A Velociy B In which graph is he objec: a) no acceleraing b) acceleraing from res c) deceleraing Velociy d) acceleraing a he greaes rae? C Velociy D 14 Skech a velociy ime graph o show how he velociy of a car ravelling along a sraigh road changes if i acceleraes uniformly from res for 5 s, ravels a a consan velociy for 1 s, hen brakes hard o come o res in s. 15 Plo a velociy ime graph using he daa in he following able. Velociy (m/s)..5 5. 7.5 1. 1. 1. 1. 1. 1. (s). 1.. 3. 4. 5. 6. 7. 8. 9. Draw a line of bes fi and use your graph o find: a) he acceleraion during he firs 4 s b) he disance ravelled in i) he firs 4 s of he moion shown ii) he las 5 s of he moion shown c) he average speed during he 9 seconds of moion shown. 16 The leaky car from quesion 7 is sill on he road! I is sill dripping oil bu now a a rae of one drop per second. The rail of drips is shown on he diagram below as he car ravels from lef o righ. The disance beween he firs and second oil drip is.5 m. Does he spacing of he oil drips show ha he car is acceleraing a a seady rae? Explain how you would make and use measuremens from he oil drip rail o deermine his. Work ou he rae of acceleraion of he car. 11