19Kinematics UNCORRECTED PAGE PROOFS

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1 19Kinemaics 9.1 Kick off wih CAS 9. Inroducion o kinemaics 9.3 Velociy ime graphs and acceleraion ime graphs 9.4 Consan acceleraion formulas 9.5 Insananeous raes of change 9.6 Review

2 9.1 Kick off wih CAS U N C O R R EC TE D PA G E PR O O FS To come Please refer o he Resources ab in he Prelims secion of your ebookplus for a comprehensive sep-by-sep guide on how o use your CAS echnology. c9kinemaics.indd 317 6/4/15 1:9 PM

3 9. Inroducion o kinemaics Our lives are perpeually involved in movemen. Walking around he house, being ranspored o school, hrowing a ball, riding a bicycle, picking up a pen, climbing sairs and going on a holiday are jus a few eamples. Mos of our movemens are rouine, and we don give hem a second hough. However, someimes we do need o hink abou wha we are doing; for eample, undersanding moion can be a maer of life and deah in siuaions such as crossing a road safely, deciding when i is pruden o overake when driving or calculaing where a cyclone is heading. Even in less dramaic siuaions like keeping an appoinmen on ime, or judging how and when o hrow a ball while playing spor, we give more hough o moion. Then we sar o employ quesions of judgemen: How far is i? How long will i ake? How will I ge here? Our ineres in analysing moion eends far beyond hese eamples aken from our daily lives. People have long been fascinaed by movemen in he world abou hem: by he moion of he planes and sars, by he fligh of birds, by he oscillaions of pendulums and by he growh of plans, o name a few eamples. The sudy of moion is fundamenal in all branches of science. The name kinemaics is given o he sudy of he moion of bodies, objecs or paricles. In his chaper, we consider moion ha is only one-dimensional; ha is, sraigh-line moion. This is called recilinear moion (o disinguish i from curvilinear moion, which deals wih curves). Eamples of recilinear moion include a ball ravelling along a pool able in a single direcion and an ice-hockey puck ha has been hi along he ice. For mahemaical convenience, all moving objecs ha we consider in his chaper will be reaed as poins; ha is, he objecs do no roae or change shape. To look a how we migh analyse moion, le s consider he laes jump by Bill he Bungee jumper. Bill jumps from a bridge ha is 1 meres above he ground and is aached o an 8-mere elasic rubber rope. He falls verically owards he ground. In he firs seconds he falls meres and in he ne seconds he falls a furher 6 meres. Afer 8 meres 1 meres he bungee rope sars o srech, and herefore slows he fall so ha Bill ravels a furher meres in seconds. The sreched bungee rope hen pulls him up a disance of 15 meres in seconds, passing wha is called he equilibrium posiion. (This is he posiion ha Bill would evenually 5 meres remain in, once he sopped bouncing on he rope.) He coninues ravelling up a furher 1 meres in seconds. Bill coninues bouncing unil he is lowered safely o ground level. 318 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

4 If we ake he saring poin, S, o be meres, hen he firs 1 seconds of Bill s jump can be displayed as follows. S Sage 1 Sage Sage 3 meres a = s A meres a = s B Posiion The posiion of a paricle moving in a sraigh line is esablished by is disance from a fied reference poin on Q 3 1 O 1 3 P 4 5 he line. This is usually he origin, O, wih posiions o he righ of O normally being aken as posiive. Consider he paricles P and Q, which boh sar from he origin, O. The posiion of paricle P is 4 unis o he righ, herefore = 4. Paricle Q is 3 unis o he lef of he origin and herefore has a posiion of = 3. We could describe Bill s moion by noing his posiion a various imes. S A B We show his on a sraigh line (verical or O 8 horizonal) by indicaing his locaion relaive Posiive o a reference poin, usually he origin, O. direcion Poin S, a he origin (acually 1 meres above he ground), shows Bill s saring posiion. Taking downwards as posiive, poin A is a and poin B is a 8. Displacemen The displacemen of a moving paricle is is change in posiion relaive o a fied poin. Displacemen gives boh he disance and direcion ha a paricle is from a poin. This can be represened on a posiion ime line (or displacemen ime line), as shown a righ, for he firs 1 seconds of Bill he bungee jumper s pah. 8 meres a = 4 s 1 meres a = 6 s 75 meres a = 1 s 85 meres a = 8 s E D = 8 S A = 1 C = = B = 4 = 6 O Noe: The direcion of he moion is indicaed by he arrows. C E D (m) (m) Topic 9 Kinemaics 319

5 Bill ravels from C (1 meres) o E (75 meres). The displacemen from C o E is he change in posiion from C o E. Displacemen = final posiion iniial posiion = 75 1 = 5 meres The disance from C o E is 5 meres bu he displacemen is 5 meres. Displacemen is a vecor quaniy and has boh magniude and direcion. (In his case he magniude is 5 meres and he direcion is negaive.) Disance is a scalar quaniy and has magniude only. For he firs 1 seconds of Bill s jump, his displacemen is 75 meres (75 ). However, he disance Bill has moved is 15 meres. Noe: A poin C, Bill is momenarily a a sop (his velociy is ) and his moion changes direcion from down o up. Velociy Velociy is also a vecor quaniy. The average velociy of a paricle is he rae of change of is posiion wih respec o ime. This can be shown on a posiion ime graph. The red line shows he posiion of he paricle,, a ime,. change in posiion Average velociy = change in ime final posiion iniial posiion = change in ime = 1 1 = δ δ Bill s average velociy over he firs 1 seconds of his jump can be calculaed as follows: Average velociy = 1 1 = 75 1 = 75 1 = 7.5 m/s 1 Time The commonly used unis of velociy are cm/s, m/s and km/h. Noe: 1 m/s = 3.6 km/h. The insananeous velociy is he velociy a a given poin of ime. Tha is, i is he gradien of he displacemen ime graph a a given poin. Posiion 1 δ Change in posiion Change in ime δ 3 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

6 WORKED EXAMPLE 1 HiNK Speed Speed is he magniude of velociy, and herefore is a scalar quaniy. Average speed = disance ravelled ime aken Insananeous speed is he magniude of insananeous velociy and is always posiive. Bill s average speed over he firs 1 seconds of his jump can be calculaed as follows: Average speed = 15 1 = 1.5 m/s (compared o he average velociy of 7.5 m/s) The following posiion ime line shows a paricle ha moves from S o A in seconds, hen from A o F in 3 seconds. Find: a he saring posiion, S b he final posiion, F c he displacemen of F from S d he disance ravelled from S o F e he average velociy from S o F g he average speed from S o F. WRie a Read he posiion of poin S. a The posiion of poin S is. b Read he posiion of poin F. b The posiion of poin F is. c Displacemen = final posiion iniial posiion d Add he disance from S o A o he disance from A o F. change in posiion e Average velociy = change in ime c Displacemen = = 4 unis o he righ of S d Disance = = unis e Average velociy = S = 5 F A f Average speed = disance ravelled ime aken = 4 5 =.8 unis/second in he posiive direcion f Average speed = 5 = 4 unis/second Topic 9 KInEMATICS 31

7 Consan velociy Velociy can be deermined by he gradien of a posiion ime graph. If he posiion ime graph is a series of conneced sraigh-line secions, hen he velociy is consan over he duraion of each sraigh-line secion. The velociy is consan from = o = 4. 4 The velociy is consan from = 4 o = 1. 1 WORKED EXAMPLE Tuorial eles 1535 Worked eample A Luna Park here is a new game called Hi he duck. To win, you mus knock down a mobile duck ha moves back and forh in a sraigh line on a 5-mere rack. You have hree shos wih small sandbags. The posiion ime graph shows he posiion of he duck, cenimeres o he righ of is saring poin, along he rack a various imes, seconds. a Wha is he iniial posiion of he duck? b How long did he game las? c Wha is he final displacemen of he duck from is saring posiion? d Wrie he imes for which he velociy is: i posiive ii negaive iii zero. e Hence, find he velociy for each of he hree ime inervals in par d. f Wha was he average speed of he duck during his game? HiNK a The iniial posiion of he duck is when =. b The graph finishes when = 1. c Displacemen = final posiion iniial posiion d i Look for where he gradien slopes upwards o he righ. ii Look for where he gradien slopes downwards o he righ. iii Look for where he gradien is horizonal. change in posiion e Velociy = change in ime WRie a When =, he iniial posiion of he duck is cm o he righ of is saring poin. b The game lased for 1 seconds. c Displacemen = 1 = 1 cm d i The gradien is posiive from = o = 5. ii The gradien is negaive from = 6 o = 1. iii The gradien is zero from = 5 o = 6. e i Velociy = 1 1 = 4 5 = 5 = 4 cm/s Posiion (cm) MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

8 disance ravelled f Average speed = ime aken WORKED EXAMPLE 3 Posiion epressed as a funcion of ime When he posiion is epressed as a funcion of ime, he posiion ime graph can be skeched and he moion hen analysed. If he posiion ime graph is curved, hen he velociy (or gradien) is always changing and never consan. A paricle moves in a sraigh line so ha is posiion, cm, from a fied poin, O, on he line a ime seconds is given by he rule: = 1 ( 1), [, 5] The posiion ime graph is shown a righ. a Copy and complee he able below b Wha is he iniial posiion of he paricle? c Wha is he significance of he posiion a = 1? d Show he movemen of he paricle on a posiion ime line. e i Wha is he displacemen of he paricle? f ii Velociy = 1 1 ii Hence, deermine he average velociy of he paricle. Posiion (cm) i Wha is he disance ravelled by he paricle? ii Hence, deermine he paricle s average speed. = = 3 4 = 75 cm/s iii Velociy = 1 1 = = = cm/s f Average speed = 5 1 = 5 cm/s Topic 9 KInEMATICS 33

9 THINK a 1 Subsiue each value of ino he rule = 1 ( 1) and evaluae for. Complee he able. WRie a When =, = 1 ( 1) = 1 ( 1) =.5 When = 1, = 1 (1 1) = 1 () = When =, = 1 ( 1) = 1 (1) =.5 When = 3, = 1 (3 1) = 1 () = When = 4, = 1 (4 1) = 1 (3) = 4.5 When = 5, = 1 (5 1) = 1 (4) = b Sae he posiion of he paricle when =. b The iniial posiion is.5 cm from O. c A = 1 he paricle is a he posiion =, and he posiion ime graph shows ha he paricle is changing direcion. d The paricle sars a =.5, moves o =, hen urns and finishes a = 8. e i Displacemen = final posiion iniial posiion change in posiion ii Average velociy = change in ime c A = 1 he paricle is changing direcion. d = 1 = = = cm e i Displacemen = 8.5 = 7.5 cm ii Average velociy = 1 1 = = = 1.5 cm/s 34 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

10 f i Add he disance ravelled from = o = 1 o he disance ravelled from = 1 o = 5. ii Average speed = disance ravelled ime aken f i The disance from = o = 1 is.5 cm and he disance from = 1 o = 5 is 8 cm. The oal disance is 8.5 cm. ii Average speed = = 1.7 cm/s EXERCISE 9. PRacise Inroducion o kinemaics 1 WE1 Each of he following posiion ime lines shows a paricle ha moves from S o A in seconds, hen from A o F in 3 seconds. In each case, find: a c i he saring posiion, S ii he final posiion, F iii he displacemen of F from S iv he disance ravelled from S o F v he average velociy from S o F vi he average speed from S o F. S 1 S F A F A b F S A Each of he following posiion ime lines shows a paricle ha moves from S o A in seconds, hen from A o F in 3 seconds. In each case, find: i he saring posiion, S ii he final posiion, F iii he displacemen of F from S iv he disance ravelled from S o F v he average velociy from S o F vi he average speed from S o F. a F b A A S S F WE The posiion ime graph a righ shows he posiion of a moving paricle, cenimeres o he righ of he origin, O, a various imes, seconds. a Wha is he iniial posiion of he paricle? b Wha is he final displacemen of he paricle from is saring posiion? c Wrie he imes for which he velociy is: i posiive ii negaive iii zero. d Hence, find he velociy for each of he hree ime inervals in par c. e Wha was he average speed of he paricle? Posiion (cm) Topic 9 Kinemaics 35

11 CONSOLIDATE 4 The posiion ime graph a righ shows 6 he posiion of a moving paricle, cenimeres o he righ of he origin, O, a various imes, seconds. a Wha is he iniial posiion of he paricle? b Wha is he final displacemen of he paricle from is saring posiion? c Wrie he imes for which he 1 velociy is: i posiive ii negaive iii zero d Hence, find he velociy for each of he hree ime inervals in par c. e Wha was he average speed of he paricle? 5 WE3 A paricle moves in a sraigh line so ha is posiion, cm, from a fied poin, O, on he line a 18 ime seconds is given by he rule: 16 = 1 ( ), [, 8] 14 1 The posiion ime graph is shown a righ. 1 8 a Copy and complee he able below b Wha is he significance of he posiion a =? c Show he movemen of he paricle on a posiion ime line. d Deermine he average velociy of he paricle. e Wha is he paricle s average speed? 6 A paricle moves in a sraigh line so ha is posiion, cm, from a fied poin, O, on he line a ime seconds is given by he rule: = 8 + 1, [, 8] a Copy and complee he able below b Skech he posiion ime graph for he paricle. Check your answer using a CAS calculaor. c Wha is he significance of he posiion a = 4? d Show he movemen of he paricle on a posiion ime line. e Deermine he average velociy of he paricle. f Wha is he paricle s average speed? 7 Represen each of he following siuaions on a posiion ime line. a A paricle sars a S, unis o he lef of he origin. I is hen displaced 1 unis o A and undergoes a final displacemen of 5 unis o F. b A paricle sars a S, 3 unis o he lef of he origin. I is hen displaced 1 unis o A and undergoes a final displacemen of 8 unis o F. Posiion (cm) Posiion (cm) 36 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

12 c A paricle sars a S, 6 unis o he righ of he origin. I is hen displaced 8 unis o A and undergoes a final displacemen of 7 unis o F. d A paricle sars a S, 4 unis o he lef of he origin. I is hen displaced 11 unis o A and undergoes a final displacemen of 6 unis o F. e A paricle sars a S, 3 unis o he lef of he origin. I is hen displaced 8 unis o A, followed by a displacemen of 7 unis o B and undergoes a final displacemen of 5 unis o F. f A paricle sars a S, 8 unis o he righ of he origin. I is hen displaced 3 unis o A, followed by a displacemen of 4 unis o B and undergoes a final displacemen of unis o F. 8 Each movemen from S o F C = 5 F = 8 described in quesion 7 akes 6 S = B = 4 seconds and he measuremens A = 3 are in cenimeres. In each case deermine: i he displacemen of F from S ii he oal disance ravelled by he paricle iii he average velociy iv he average speed. Use he posiion ime line a righ o answer quesions 1 o Draw a Posiion ime graph for each of he following: a An objec moving wih a consan posiive velociy b An objec moving in a posiive direcion wih a consan slow speed a firs and hen a faser consan speed. 1 The displacemen of F from S, in cm, is: A 4 B 4 C 3 D 14 E The disance ravelled in moving from S o F, in cm, is: A 4 B 34 C 44 D 34 E 56 1 The average speed in moving from S o F, in cm/s, is: A 4.5 B 7 C 5.5 D 6.8 E 3 13 The average velociy in moving from A o C, in cm/s, is: A B 1 C 1 D E.5 14 A rain moves in a sraigh line so ha is disance, meres, from a fied poin, O, a seconds is given by he rule: = 6 + 9, [, 8] a Draw a posiion ime graph for he movemen of he rain. b Deermine he average velociy of he rain in he firs 4 seconds. c Wha is he rain s average speed in he firs 4 seconds? 15 A paricle moves in a sraigh line so ha is posiion, cm, from a fied poin, O, on he line a ime seconds is given by he rule: = 4 5, [, 6] a Skech he posiion ime graph for he paricle. Check your answer using a CAS calculaor. b Show he movemen of he paricle on a posiion ime line. Topic 9 Kinemaics 37

13 Ineraciviy in-67 Moion graphs (kinemaics) MASTER 9.3 c Deermine he average velociy of he paricle. d Wha is he paricle s average speed? 16 A paricle moves in a sraigh line so ha is posiion, cm, from a fied poin, O, on he line a ime seconds is given by he rule: = + + 8, [, 6] a Skech he posiion ime graph for he paricle. Check your answer using a CAS calculaor. b Show he movemen of he paricle on a posiion ime line. c Deermine he average velociy of he paricle. d Wha is he paricle s average speed? 17 A paricle moves in a sraigh line so ha is posiion, cm, from a fied poin, O, on he line a ime seconds is given by he rule: = 7 + 1, [, 8] a Skech he posiion ime graph for he paricle. b Show he movemen of he paricle on a posiion ime line. c Wha is he displacemen of he paricle? d Deermine he average velociy of he paricle. e Wha is he disance ravelled by he paricle? f Deermine he paricle s average speed. 18 A paricle moves in a sraigh line so ha is posiion, cm, from a fied poin, O, on he line a ime seconds is given by he rule: = 5 + 6, [, 6] a Skech he posiion ime graph for he paricle. b Show he movemen of he paricle on a posiion ime line. c Wha is he displacemen of he paricle? d Deermine he average velociy of he paricle. e Wha is he disance ravelled by he paricle? f Deermine he paricle s average speed, correc o decimal places. Velociy ime graphs and acceleraion ime graphs Velociy ime graphs Le us ake anoher look a he posiion ime line for he bungee jump performed by Bill ha was described a he sar of he chaper. S = A = E D = 8 = 1 C = 6 B = Meres This siuaion can be represened on a posiion ime graph as shown below. The curve reflecs he fac ha he change of posiion over ime (velociy) is no consan. 38 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

14 Posiion (m) (6, 1) 1 9 C (8, 85) 8 B (4, 8) D (1, 75) 7 E A (, ) 1 S We can calculae he average velociy in each of he sages as follows: From S o A: Average velociy = 1 1 = = = 1 m/s From B o C: Average velociy = = 6 4 = = 1 m/s From D o E: Average velociy = = = 1 = 5 m/s From A o B: Average velociy = = 4 = 6 = 3 m/s From C o D: Average velociy = 1 1 = = 15 = 7.5 m/s Noe: The negaive velociies occur when he moion is upwards, because we decided o define downwards as posiive. We can now represen he moion of Bill s bungee jump during each sage on a velociy ime graph (or more precisely, an average velociy ime graph). v av 3 Average velociy (m/s) Topic 9 Kinemaics 39

15 Noice ha he graph shows ha he velociy is consan during each of he sages (shown as he sep formaion of he graph). This is because we have calculaed he average velociy of each sage. If we were o analyse he average velociy over smaller ime inervals, we would ge more seps wih smaller widhs, as is displayed in he second graph. If we allowed hese ime inervals (sep widhs) o ge closer and closer o zero, hen he associaed average velociies would effecively become a series of conneced poins ha would collecively produce a velociy ime graph somehing like he one displayed a righ. This is a velociy ime graph as i shows Bill s velociy a every insance of he firs 1 seconds of moion during his bungee jump. There are no horizonal lines (seps) because he velociy is changing every insan over he course of he moion. This change in velociy over ime is called acceleraion. Acceleraion is also a vecor quaniy. For he firs 4 seconds of moion, he graph is a sraigh line because Bill is subjeced only o acceleraion due o graviy, which is consan a 9.8 m/s. This means ha every second, Bill s velociy increases by 9.8 m/s while he is moving downwards. For he period of ime where he bungee rope is sreched (longer han 8 m), from = 4 seconds o abou = 9 seconds, he elasiciy of he rope causes he acceleraion o coninually change according o he ension in he bungee rope. Tha is why he velociy ime graph is curved during his ime. From abou = 9 seconds o = 1 seconds (where he bungee rope is shorer han 8 m), he rope is again slack and Bill is subjec o acceleraion due only o graviy again. A his sage he moion is upwards, bu because acceleraion due o graviy acs downwards, Bill is slowing down or deceleraing. change in velociy Average acceleraion = change in ime = v v 1 1 = δv δ The mos common unis of acceleraion are cm/s and m/s. For he momen we will consider only eamples ha involve consan acceleraion. Average velociy (m/s) Average velociy (m/s) v av v MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

16 WORKED EXAMPLE 4 Draw a velociy ime graph o mach he following descripion. An objec ha is moving in a sraigh line has an iniial velociy of 5 m/s. I acceleraes a a consan rae unil i reaches a velociy of 1 m/s afer 6 seconds. I mainains his velociy for 8 seconds and hen deceleraes a a consan rae for a furher 4 seconds unil i comes o res. HiNK 1 The velociy ranges from m/s o 1 m/s. The oal ime is = 18 seconds. 3 Draw a se of aes wih velociy on he verical ais and ime on he horizonal ais. Label each ais appropriaely. 4 Skech a sraigh line from (, 5) o (6, 1) o show he acceleraion in he firs sage. 5 Draw a horizonal line from (6, 1) o (14, 1) o show he consan velociy during he second sage. 6 Draw a sraigh line from (14, 1) o (18, ) o show he final sage of deceleraion. WRie/DRaW v 1 Velociy (m/s) Noice ha he gradien of each sraigh-line secion of he velociy ime graph gives he acceleraion of he objec. Analysing he velociy ime graph The gradien of a velociy ime graph allows us o calculae he acceleraion of an objec moving in a sraigh line. In addiion o his, he area beween he velociy ime graph and he ime ais provides useful informaion relaing o displacemen and disance. Earlier, i was shown ha: change in posiion Average velociy = change in ime or v av = δ δ where v av represens average velociy. Rearranging his resuls in: δ = v av δ In oher words, he signed area beween a velociy ime graph and he ime ais is equal o he change in posiion or displacemen. When we calculae he signed area, we ake he area above he ime ais as posiive displacemen and he area below he ime ais as negaive displacemen. If he disance (raher han he displacemen ha he paricle has ravelled) is required, hen here is no need o sign he areas. Tha is, he disance ravelled is he oal area Topic 9 KInEMATICS 331

17 beween he velociy ime graph and he ime ais. Using he average velociy ime graph describing Bill s bungee jump from earlier, he informaion described above can be highlighed as follows. The displacemen is equal o he sum of he signed areas of he recangles. Average velociy (m/s) v av Displacemen = = = 75 meres The disance is equal o he sum of all he unsigned areas of he recangles. Disance = = = 15 meres The following can be obained from he figure shown below. Velociy (m/s) v 5 5 Area Area 1. The objec is ravelling a a consan velociy of 5 m/s unil = 5 s. I slows down unil i sops a = 7 s, hen i changes direcion and increases is speed o 5 m/s a = 9 s. The objec hen slows down and sops when = 1 s.. The gradien of he line beween = s and = 5 s is zero, so he acceleraion is m/s. 1 Beween = 5 s and = 9 s he gradien is, so he acceleraion is m/s. Beween = 9 s and = 1 s he gradien is 5, so he acceleraion is 5 m/s. 3. Toal displacemen = Area 1 Area. 4. Toal disance = Area 1 + Area. Noe: When appropriae, break he area beween he velociy ime graph and he ime ais ino simple shapes, for eample recangles, riangles or rapeziums. Area of a recangle = L W Area of a riangle = 1 bh Area of a rapezium = 1 (a + b)h 33 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

18 WORKED EXAMPLE 5 TUTORIAL eles-1536 Worked eample 5 Consider he velociy ime graph obained in Worked eample 4 o find: a he acceleraion in he firs 6 seconds b he acceleraion in he las 4 seconds c he oal displacemen d he oal disance ravelled. Velociy (m/s) v 1 5 HiNK change in velociy a Average acceleraion = change in ime change in velociy b Average acceleraion = change in ime c 1 The displacemen is equal o he oal signed area under he velociy ime graph. Divide he given graph ino wo rapeziums, one from = o = 6 and he oher from = 6 o = 18. WRie/DRaW a Average acceleraion = v v 1 1 = = 5 6 m/s b Average acceleraion = v v 1 1 c Velociy (m/s) v = = 1 4 =.5 m/s 1 1 Area 1 Area Calculae he area of each rapezium. Area 1 = 1 (5 + 1) 6 = = 45 unis Area = 1 (8 + 1) 1 1 = 1 = 1 unis Topic 9 KInEMATICS 333

19 4 Find he displacemen. Displacemen = Area 1 + Area = = 145 m d The disance is equal o he oal unsigned area under he velociy ime graph. Acceleraion ime graphs Jus as he gradien of a posiion ime graph gives he rae of change of posiion or velociy, he gradien of a velociy ime graph gives he rae of change of velociy or acceleraion. Where he velociy is increasing, he acceleraion is posiive. Where he velociy is decreasing, he acceleraion is negaive. Where he velociy is no changing, he acceleraion is zero. Consider a modified velociy ime graph of he v B firs 1 seconds of moion of Bill s bungee jump. 4 We will assume he acceleraion is consan, bu 3 differen hrough each of he sages of he jump. change in velociy As average velociy = change in ime, he acceleraion for each sage is: From S o B: Average acceleraion = v v 1 1 = 4 4 = 4 4 = 1 m/s From B o C: Average acceleraion = v v 1 1 = = = m/s From C o D: Average acceleraion = v v = = = 7.5 m/s d The disance is equal o 145 m. Noe: Because he velociy is always posiive in his eample, he disance is equal o he displacemen. Velociy (m/s) 1 1 S C E D 334 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

20 WORKED EXAMPLE 6 Tuorial eles-1537 Worked eample 6 From D o E: Average acceleraion = v v 1 1 = = 15 = 7.5 m/s Therefore, he acceleraion ime graph would look like he graph below. Acceleraion (m/s ) a Noe: The signed area under he acceleraion ime graph gives he change in velociy. In he graph above, he area beween he graph and he ime ais from = s o = 4 s is 4, which is verified on he previous acceleraion ime graph. Consider he moion of an elevaor ha has he velociy ime graph shown. Take posiive values o represen upward moion. a In wha secions OA, AB, BC ec. is he lif: i acceleraing posiively ii acceleraing negaively iii ravelling a a consan velociy? b Deermine he acceleraion for each secion of he lif s journey. c Skech he acceleraion ime graph. d If he lif sared a ground level, meres, deermine is posiion a: i C ii G. e Deermine he average velociy of he lif. f How far did he lif ravel? Velociy (m/s) g Wha was he lif s average speed? v 8 4 O C D G A B E F HiNK a i Acceleraion is posiive where he velociy is increasing. WRie a i The acceleraion is posiive from O o A and from F o G. Topic 9 KInEMATICS 335

21 ii Acceleraion is negaive where he velociy is decreasing. iii Acceleraion is zero where he velociy is no changing. change in velociy b Average acceleraion = change in ime = v v 1 1 c The acceleraion is consan in each secion, so he acceleraion ime graph is a series of horizonal lines (seps). ii The acceleraion is negaive from B o C and from D o E. iii The acceleraion is zero from A o B, from C o D and from E o F. b From O o A, average acceleraion = v v 1 1 c = 8 5 = 8 5 = 1.6 m/s From A o B, average acceleraion = = 13 = m/s From B o C, average acceleraion = = = 4 m/s From C o D, average acceleraion = = 5 5 = m/s From D o E, average acceleraion = 1 7 = 1 = 6 m/s 1 1 From E o F, average acceleraion = = = m/s From F o G, average acceleraion = 1 a = 1 5 =.4 m/s Acceleraion (m/s ) MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

22 d i Because he lif sared a posiion meres, he posiion a poin C is he signed area under he rapezium OABC. ii The posiion a poin G is he signed area under he rapezium DEFG plus posiion a poin C. change in posiion e Average velociy = change in ime f The oal disance ravelled by he lif is he oal area beween he velociy ime graph and he ime ais. disance ravelled g Average speed = ime aken Eercise 9.3 PRacise d i The posiion a C is he area of rapezium OABC = 1 (13 + ) 8 = = 13 meers ii The posiion a G is he signed area under he rapezium DEFG plus posiion a poin C 1 = (8 + 15) = Velociy ime graphs and acceleraion ime graphs = = 6 meres (ha is, he lif ends up 6 meres below ground level). e Average velociy = 1 1 = 6 = 6 4 =.15 m/s f The oal disance ravelled by he lif is = 7 meres. g Average speed = 7 1 WE4 Draw a velociy ime graph o mach each of he following descripions. a An objec, which is moving in a sraigh line, has an iniial velociy of 4 m/s. I acceleraes a a consan rae unil, afer 5 seconds, i reaches a velociy of 9 m/s. I mainains his velociy for 1 seconds and hen deceleraes a a consan rae for a furher 5 seconds, when i comes o res. b An objec, which is moving in a sraigh line, has an iniial velociy of 6 m/s. I acceleraes a a consan rae unil, afer 8 seconds, i reaches a velociy of 1 m/s. I mainains his velociy for 15 seconds and hen deceleraes a a consan rae for a furher 5 seconds unil i reaches a velociy of 8 m/s. c An objec, which is moving in a sraigh line, has an iniial velociy of 5 m/s. I acceleraes a a consan rae unil, afer 1 seconds, i reaches a velociy of 4 m/s. I mainains his velociy for 1 seconds and hen deceleraes a a consan rae for a furher 9 seconds, when i comes o res. Draw a velociy ime graph o mach each of he following descripions. a An objec, which is moving in a sraigh line, has an iniial velociy of 5 m/s. I deceleraes a a consan rae unil, afer 6 seconds, i reaches a velociy of 5 m/s. I mainains his velociy for 4 seconds and hen acceleraes a a consan rae for a furher 6 seconds, when i comes o res. b An objec, which is moving in a sraigh line, has an iniial velociy of 8 m/s. I mainains his velociy for 1 seconds and hen acceleraes a a consan rae 4 4 = 6.75 m/s Topic 9 Kinemaics 337

23 unil, afer 8 seconds, i reaches a velociy of 4 m/s. I mainains his velociy for 1 seconds and hen deceleraes a a consan rae for a furher 4 seconds, when i reaches a velociy of m/s, which i mainains. 3 WE5 Consider he velociy ime v 1 graph shown o find: a he acceleraion in he firs 8 5 seconds b he acceleraion in he las 4 5 seconds c he oal displacemen d he oal disance ravelled. v 4 Consider he velociy ime 16 graph shown o find: a he acceleraion in he firs 6 seconds b he acceleraion in he las 6 seconds c he oal displacemen d he oal disance ravelled WE6 Consider he moion of an elevaor whose velociy ime graph is as shown. Take v A B posiive values o represen upward moion. 6 a In wha secions, OA, AB, BC ec., is he lif: i acceleraing posiively O C D G ii acceleraing negaively iii ravelling a a consan velociy? b Deermine he acceleraion for each secion 8 of he lif s journey. E F c Skech he acceleraion ime graph. d If he lif sared a ground level, meres, deermine is posiion a: i C ii G. e Deermine he average velociy of he lif. f How far did he lif ravel? g Wha was he lif s average speed? 6 Consider he moion of a lif in a high-rise building. The lif s velociy ime graph is as shown. The lif sars from he 5h floor, which is 1 meres above ground level. Take posiive values o represen upward moion. v 6 E F Velociy (m/s) Velociy (m/s) Velociy (m/s) Velociy (m/s) O C D G A B 338 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

24 CONSOLIDATE a In wha secions, OA, AB, BC ec., is he lif: i acceleraing posiively ii acceleraing negaively iii ravelling a a consan velociy? b Deermine he acceleraion for each secion of he lif s journey. c Skech he acceleraion ime graph. d Deermine he lif s posiion a: i C ii G. e Deermine he average velociy of he lif. f How far did he lif ravel? g Wha was he lif s average speed? 7 Consider he velociy ime graph shown o find: a he acceleraion in he firs 6 seconds b he acceleraion in he las 1 seconds c he oal displacemen d he oal disance ravelled. Use he velociy ime graph below o answer quesions 8 o 1. Velociy (m/s) a A B C D E The magniude of he acceleraion is greaes beween he poins: A A and B B B and C C A and B and D and E D D and E E E and F 9 The average velociy from A o F is equal o: A 3.3 m/s B.3 m/s C 4 m/s D.8 m/s E 4 m/s 1 The average speed from A o F is equal o: A 3.3 m/s B.3 m/s C 4 m/s D.8 m/s E 4 m/s 11 A learner driver sars her car from res and drives for 8 seconds unil she reaches a speed of 16 m/s. Wha is he average acceleraion? Velociy (m/s) F v Topic 9 Kinemaics 339

25 MASTER 1 A racing car acceleraes uniformly from 15 m/s o 45 m/s in.6 seconds. Deermine he average acceleraion of he car, correc o 1 decimal place. 13 A cyclis increases his speed from 4. m/s o 6.3 m/s over 5.3 seconds. Wha is he average acceleraion, correc o 1 decimal place? 14 Sarah ook a new car for a es drive. From res, he car ravelled a disance of 4 meres in 16 seconds. a Wha is he average velociy? b Assuming a consan acceleraion: i Wha is Sarah s final velociy? ii Wha is he average acceleraion of he car, correc o 1 decimal place? 15 A driver brakes suddenly o avoid hiing a kangaroo on he road. The car is ravelling a 3 m/s and sops in.58 seconds. a Wha is he average velociy from he ime he driver brakes? b Wha is he sopping disance of he car? 16 A car is ravelling a a consan speed of 18 km/h when i passes a saionary police moorcycle. Four seconds laer he moorcycle ses off in pursui wih a consan acceleraion of 5 m/s unil i reaches a speed of 16 km/h, which i hen mainains. (1 m/s = 3.6 km/h) a For how long does he moorcycle accelerae? b Skech a velociy ime graph ha represens he moion of boh he car and he moorcycle. c How long afer he car firs passes he moorcycle does i ake for he moorcycle o cach up o he car? d How far have hey ravelled? 17 Polly is leading a 5-kilomere bicycle race when her bicycle ges a puncure 36 meres from he finish line. She changes her yre, and he insan she akes off again, Molly passes her, ravelling a a consan speed of 14 m/s. Polly acceleraes a a consan rae for 5 seconds unil she reaches a speed of 16 m/s, which she mainains unil he finish. a Skech a velociy ime graph ha represens he moion of boh Polly and Molly. b Verify ha Polly sill wins he race. c How far from he finish line are hey when Polly caches up o Molly? d If Molly sared o accelerae a a consan rae from he momen ha Polly caugh up o her, wha would her acceleraion be if hey were o dead hea? 18 Ma he monkey is climbing a coconu ree in a sraigh line o find a coconu for lunch. His moion is described as follows. Ma sars from res a ground level wih consan acceleraion unil he reaches a speed of 1.5 m/s afer 4 seconds. He mainains his speed for 8 seconds, hen he deceleraes o a sop afer anoher seconds. Afer a furher 9 seconds, Ma heads back down he ree wih consan acceleraion, reaching a speed of.5 m/s 34 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

26 9.4 in seconds. He mainains his speed for 5 seconds, hen he jumps from he ree. (Take posiive as up.) a Draw a velociy ime graph represening he moion of he monkey unil he leaves he ree. b A wha heigh did Ma leap off he ree? c Wha was he oal disance ravelled by Ma on he ree? d Wha was Ma he monkey s average speed: i while on he ree ii while in moion on he ree? Challenge: When Ma begins his descen, a palm leaf falls from he ree a a heigh of 5 meres. I falls wih a consan acceleraion of m/s. e Verify ha Ma he monkey is sill on he ree when he palm leaf his he ground and deermine where Ma is a his ime. Consan acceleraion formulas Acceleraion due o graviy is usually 9.8 m/s. I varies slighly depending on he disance from he cenre of he Earh. This means ha a falling objec or an objec hrown ino he air is subjec o a consan (or uniform) downward acceleraion of 9.8 m/s. Since acceleraion is a vecor quaniy, when he objec is moving upwards, i is subjec o an acceleraion of 9.8 m/s ; ha is, a deceleraion or reardaion. Consider an objec moving in a sraigh line ha has an iniial v v velociy of u. I acceleraes consanly unil i reaches a velociy of v afer seconds. Is velociy ime graph is shown a righ. u We can use his graph o derive various formulas ha can be applied o problems involving consan acceleraion. Since acceleraion, a, is he change in velociy over ime: Muliply boh sides by : Make v he subjec, so: a = δv δ = v u a = v u v = u + a [1] Furhermore, since average velociy is he change in posiion, s, over ime: So, average velociy = δs δ or u + v s = u + v Therefore, s = 1 (u + v) [] Velociy (m/s) Topic 9 Kinemaics 341

27 WORKED EXAMPLE 7 Subsiuing v = u + a (equaion [1]) ino equaion []: s = 1 1u + u + a = 1 1u + a = 1 1u + a Therefore, s = u + 1 a [3] From [1], = v u a Subsiuing = v u ino equaion []: a s = 1 u (u + v)qv a = 1 u Qv a as = v u Therefore, v = u + as [4] In summary, if u is he iniial velociy, v is he final velociy, s is he displacemen, a is he consan acceleraion and is he ime inerval, hen he following formulas apply for sraigh-line moion: v = u + a [1] s = 1 (u + v) [] s = u + 1 a [3] v = u + as [4] Noes 1. A res means he velociy is zero.. 1 m/s = 3.6 km/h. (Verify his.) 3. When an objec is ravelling in one direcion, u can be reaed as he iniial speed, v as he final speed and s as he disance ravelled. A sone is dropped from a bridge ha is 15 meres above a river. Find: a he ime aken for he sone o reach he river b he sone s speed on impac. Give answers o he neares enh. R R HiNK a 1 Lis he given informaion and wha has o be found. Find using s = u + 1 a by subsiuing s = 15, a = 9.8 and u =. WRie a Given: s = 15, a = 9.8 and u = Require: =? s = u + 1 a 15 = MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

28 3 Solve he equaion for. 15 = = =!3.61 = Sae he soluion. The sone reaches he river afer approimaely 5.5 seconds. b 1 Lis he given informaion and wha has o be found. Find v using v = u + as by subsiuing u =, a = 9.8 and s = 15. b Given: s = 15, a = 9.8 and u = Require: v =? v = u + as = Solve he equaion for v. v = 94 v =!94 = Sae he soluion. The sone reaches he river a a speed of 54. m/s. WORKED EXAMPLE 8 A driver is forced o suddenly apply he brakes when a dog appears in fron of his car i. The car skids in a sraigh line, sopping cenimeres shor of he sarled dog. The car skidded a disance of 1 meres for seconds. a A wha speed was he car ravelling as i began o skid? b Wha was he acceleraion of he car during he skid? HiNK a 1 Lis he given informaion and wha has o be found. Find u using s = 1 (u + v) by subsiuing s = 1, = and v =. WRie a Given: s = 1, = and v = Require: u =? s = 1 (u + v) 1 = 1 (u + ) 3 Solve he equaion for u. 1 = 1 u u = 1 4 Sae he soluion. The iniial speed of he car was 1 m/s. b 1 Lis he given informaion and wha has o be found. b Given: v =, u = 1 and = Require: a =? Find a using v = u + a by subsiuing v =, u = 1 and =. v = u + a = 1 + a 3 Solve he equaion for a. 1 = a a = 6 4 Sae he soluion. The acceleraion of he car was 6 m/s. Topic 9 KInEMATICS 343

29 WORKED EXAMPLE 9 A ball is hrown upwards a 14.7 m/s from a ower ha is 5 meres above he ground. a Deermine he oal ime ha he ball is in he air before i reaches he ground. b Find he maimum heigh reached by he ball. c Find he ball s speed when i firs srikes he ground. d Give answers o he neares enh and le up be he posiive direcion. HiNK a 1 Lis he given informaion and wha has o be found. Find using s = u + 1 a by subsiuing u = 14.7, a = 9.8 and s = 5. 3 Solve he quadraic equaion by using he quadraic formula. b 1 Lis he given informaion and wha has o be found. Find using v = u + a by subsiuing u = 14.7, a = 9.8 and v =. 3 Find s using s = u + 1 a by subsiuing u = 14.7, a = 9.8 and = 1.5. WRie a Given: u = 14.7, a = 9.8, s = 5 Require: =? s = u + 1 a 5 = ( 9.8) 5 = = = b ± "b 4ac a = 14.7 ± "(14.7) 4(4.9)( 5) (4.9) =. and 5. = 5. seconds, as ime canno be negaive. b Given: u = 14.7, a = 9.8 and v = Require: =? s =? v = u + a = = = 1.5 s = u + 1 a s = = Add s o he heigh of he ower. Maimum heigh = = Sae he soluion. The maimum heigh reached by he ball is 61. m. c 1 Lis he given informaion and wha has o be found. Find v using v = u + as by subsiuing u =, a = 9.8 and s = c Given: u =, a = 9.8 and s = 61.5 Require: v =? v = u + as = = MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

30 3 Solve he equaion for v. v =! = Sae he soluion. The ball firs srikes he ground a a speed of 34.6 m/s. Eercise 9.4 PRacise CONSOLIDATE Consan acceleraion formulas 1 WE7 A sone is dropped from a bridge ha is 98 meres above a river. Giving answers o he neares enh, find: a he ime aken for he sone o reach he river b he sone s speed on impac. An objec ravelling a 8 m/s acceleraes uniformly over a disance of meres unil i reaches a speed of 18 m/s. Find: a he acceleraion b he ime aken. 3 WE8 A driver is forced o suddenly apply he brakes when a ca appears in fron of her car. The car skids in a sraigh line, sopping 8 cm shor of he sarled ca. The car skidded a disance of 15 meres for 3 seconds. a A wha speed was he car ravelling as i began o skid? b Wha was he acceleraion of he car during he skid? 4 A falcon is hovering in he air when i suddenly dives verically down o swoop on is prey, which is 15 meres direcly below i. If he acceleraion is uniform and i akes he falcon 5 seconds o reach is prey, find: a he final speed of he falcon in m/s and km/h b he acceleraion of he falcon. 5 WE9 A ball is hrown upwards a 9.8 m/s from a ower ha is 3 meres above he ground. a Deermine he oal ime ha he ball is in he air before i reaches he ground. b Find he ball s speed when i firs srikes he ground. Give answers o he neares enh. 6 A ball is hrown upwards a m/s from a ower ha is 8 meres above he ground. a Deermine he oal ime ha he ball is in he air before i reaches he ground. b Find he ball s speed when i firs srikes he ground. Give answers o he neares enh. 7 A paricle moving from res wih consan acceleraion reaches a speed of 16 m/s in 4 seconds. Find: a he acceleraion b he disance ravelled. Topic 9 Kinemaics 345

31 8 A parachuis free-falls from an aircraf for 6 seconds. Find: a he speed of he parachuis afer 6 seconds b he disance ravelled afer 6 seconds. 9 A ball is dropped from a ower and reaches he ground in 4 seconds. Find: a he heigh of he ower b he speed of he ball when i his he ground. 1 How long does i ake for: a a car o accelerae on a sraigh road a a consan 6 m/s from an iniial speed of 17 m/s o a final speed of 8 m/s b a downhill skier o accelerae from res a a consan m/s o a speed of 1 m/s? 11 A skaeboarder is ravelling down a genly sloping pah a a speed of 1 m/s when he sops skaing. He rolls a furher 6 meres before coming o a sop. Assuming he acceleraion is uniform, find: a he acceleraion b he ime i akes o come o a sop. 1 A ram is ravelling a 16 m/s when he brakes are applied, reducing he speed o 6 m/s in seconds. Assuming he reardaion is consan, find: a he acceleraion b he disance ravelled during he seconds afer he brakes are applied c he braking disance of he ram. 13 A rain ravels a disance of 18 meres in 9 seconds while acceleraing uniformly from res. a The speed of he rain afer 9 seconds can be deermined using he formula: A v = u + a B s = 1 (u + v) C C = πr D s = u + 1 a E v = u + as b The speed in km/h afer 9 seconds is: A b 36 c 144 d 16 e 4 c The speed in km/h afer 45 seconds is: A 7 b 36 c 144 d 16 e d The disance ravelled afer 45 seconds is: A 5 m b 9 m c 675 m d 135 m e 45 m 14 An objec is projeced verically upwards from he op of a building ha is 5 meres above he ground. Is iniial speed is 8 m/s. If he objec hen falls o he ground, find: a is maimum heigh above he ground b he oal ime aken o reach he ground c he speed of he objec when i reaches he ground. 346 MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

32 MASTER A car moving from res wih uniform acceleraion akes 1 seconds o ravel 144 meres. Wha is is speed afer 6 seconds? 16 A bird s egg falls from a nes in a ree. If i is iniially 39. meres above he ground, calculae: a is speed when i is halfway o he ground b is speed on sriking he ground c he ime aken o reach he ground. 17 A cage is descending ino a well a a consan speed of m/s when a sone falls hrough he wire in he cage. If he sone reaches he waer a he boom of he well 1 seconds before he cage, find he heigh above he waer a which he sone fell ou of he cage. 18 A ho-air balloon is rising wih a speed of 19.6 m/s when a gas cylinder falls off he balloon. If he balloon is 8 meres above he ground when he cylinder falls off, how long will i ake he cylinder o reach he ground and wha will is speed be hen? Insananeous raes of change Insananeous velociy As we have discussed previously, he insananeous velociy a a given ime is in fac he gradien of he posiion ime graph a ha ime. We have also seen ha when he velociy is variable he posiion ime graph will be curved. Consider a paricle moving in a sraigh line such ha is posiion, cm, a any ime, seconds, is described by he rule: () = 3, [, 3] Compleing a able of values will give: The posiion ime graph is shown a righ. The velociy a any given ime (say a = seconds) is equal o he gradien of he curve a ha given ime ( = ). The gradien of a curve a any given poin is he gradien of he angen o he curve a ha poin. So, he velociy a = is equal o he gradien of he angen o he curve a =. Physically deermining he gradien of he angen ofen leads o inaccurae resuls. Care needs o be aken, firsly o draw an accurae and smooh curve, hen o place he angen a eacly he righ posiion. There is oo much room for error wih his process. Posiion (cm) Posiion (cm) () = Tangen a = 1 3 Topic 9 Kinemaics 347

33 WORKED EXAMPLE 1 Insead, we can apply he rule: Average velociy = δ δ o esimae he gradien (velociy). This involves aking wo poins on he curve on eiher side of =. To ensure ha he poin a = is in he middle of he wo poins chosen, each poin mus be he same disance, h, eiher side of =. The gradien of he line ha joins he wo poins on he curve a = h and = + h esimaes he gradien a =. Finding he rise and he run beween he wo poins allows us o calculae he gradien as: v() = ( + h)3 ( h) 3. h The smaller he value of h, he closer his gradien will be o he rue gradien of he angen. For eample, using a calculaor o find v() when h = 1,.1 and.1 produces he resuls shown in he able below. h v() I is quie clear from his able ha as h ges smaller and smaller, he value of v() is approaching 1. If i is no already obvious, i becomes even more so if h =.1 or.1 and so on. In summary, he insananeous velociy a =, v( ), of a paricle moving in a sraigh line wih is posiion described as () is found by evaluaing: v( ) = ( + h) ( h) h for very small values of h (h > ). This echnique uses he same process as ha of differeniaing from firs principles, which was covered in Mahemaical Mehods (CAS) Unis 1 and, and hus we can say: v() = d he derivaive of wih respec o d or v() = () Posiion (cm) [( + h), ( + h) 3 ] (, 8) [( h), ( h) 3 ] h h 1 3 A paricle is ravelling in a sraigh line wih is posiion, cm, a any ime, seconds, given as () = 3, ([, 3]. Find he velociy of he paricle afer 1.5 seconds. HiNK 1 Given he epression () = 3, we wan v(1.5). WRie () = MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and

34 Find he velociy equaion by differeniaing posiion,, wih respec o ime, (v() = ()). v() = () v() = Subsiue = 1.5 seconds. v(1.5) = 3(1.5) 1 v(1.5) = Sae he soluion. The velociy of he paricle a = 1.5 seconds is 5.75 cm/s. WORKED EXAMPLE 11 Insananeous acceleraion When he acceleraion is variable, he velociy ime graph is curved. The insananeous acceleraion a a given ime is he gradien of he velociy ime graph a ha ime. So, like he insananeous velociy: The insananeous acceleraion a =, a( ), of a paricle moving in a sraigh line wih is velociy described as v() is found by evaluaing, he cenral difference formula: a( ) = v( + h) v( h) h for very small values of h (h > ). Again he echnique uses he same process o ha of differeniaing from firs principles, and we can say: or a() = dv d a() = v () he derivaive of v wih respec o A paricle is ravelling in a sraigh line wih is velociy, v cm/s, a any ime, seconds, given as v() = 8 + 1, Find he acceleraion of he paricle afer 1 second. HiNK WRie 1 Given he epression v() = 8 + 1, we wan a(1). v() = Find he acceleraion equaion by differeniaing velociy v () = 8 wih respec o ime (a() = v ()) using a calculaor. ( + 1) 3 Subsiue = 1 second ino he formula for a(). 8 a(1) = (1 + 1) = 8 4 = 4 Sae he soluion. The acceleraion of he paricle a = 1 seconds is cm/s. Topic 9 KInEMATICS 349

35 WORKED EXAMPLE 1 Tuorial eles-1538 Worked eample 1 Approimaing velociy ime graphs We have already seen ha he disance ravelled by a v paricle ravelling in a sraigh line is he unsigned area beween he velociy ime graph and he ime ais. When he acceleraion is consan, we calculae he areas of recangles, riangles or rapeziums. If he acceleraion is variable, he velociy ime graph is curved and so i needs o be approimaed by sraighline funcions. This will resul in he area under he graph comprising eiher recangles, riangles or rapeziums. Then he disance ravelled can be esimaed. One way o approimae he velociy ime curve is o use a series of horizonal seps over he required domain or v ime values. This can be achieved by firs dividing v 4 he domain inerval ino n equally sized ime inervals, each h unis long. Ne, evaluae he v 3 velociy a he midpoin of each of hese inervals. v v 1 Each of hese velociies can be reaed as he average velociy over is corresponding inerval. The resul will be a sep funcion graph somehing like he figure on he righ The area of he rapezium gives he disance ravelled. The unsigned area under his velociy ime graph can be found by deermining he sum of each recangular area (h v n ). This gives an esimae for he disance ravelled over a given period of ime. As he recangle widh (or inerval widh), h, ges smaller and smaller, he number of recangles, n, increases and herefore he esimae ges closer and closer o he eac disance. The following worked eample oulines he seps involved, wih he aid of graphs. v Noe: 4 3 = 3 = 1 = 1 = h unis A paricle is ravelling in a sraigh line wih is velociy, v m/s, a any ime, seconds, given as: v() = +, Esimae he disance ravelled during he firs 4 seconds of is moion by approimaing he velociy wih sep funcions each 1 uni wide. HiNK 1 Skech he graph of v() = + over he domain [, 4]. DRaW/WRie v Velociy (m/s) v() = MATHS QUEST 11 SPECIALIST MATHEMATICS VCE Unis 1 and