Kinematics. introduction to kinematics 15A

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1 15 15A Inroducion o kinemaics 15B Velociy ime graphs and acceleraion ime graphs 15C Consan acceleraion formulas 15D Insananeous raes of change Kinemaics AreAS of STuDy Diagrammaic and graphical represenaion of empirical posiion ime daa for a single paricle in recilinear moion, including examples wih ariable elociy (daa may be obained by a suden moing along a 1 mere ape according o a gien se of insrucions, daa logging or preious experimenal daa) Graphical modelling and numerical analysis of posiion ime and elociy ime relaionships based on coninuous hybrid funcions formed by sraigh line segmens, including consideraion of aerage elociy and disance raelled oer an ineral Modelling and analysis of recilinear moion under consan acceleraion, including use of consan acceleraion formulas: = u + a, = u + as, s = 1 (u + ) and s = u + 1 a Qualiaie graphical analysis of he relaionship beween posiion ime, elociy ime and acceleraion ime graphs for simple cases of recilinear moion inoling ariable acceleraion Numerical approximaion o insananeous rae of change of a funcion f a ime = a by ealuaion of he cenral difference f(a a+h)- f( a- h) for small alues of h using h echnology and is applicaion o approximae ealuaion of insananeous elociy and insananeous acceleraion in simple cases of recilinear moion inoling ariable elociy and ariable acceleraion Approximaion of elociy ime relaionships by sep funcions and is applicaion o approximae ealuaion of disance raelled in simple cases of recilinear moion inoling ariable elociy and ariable acceleraion, as a sum of areas of recangles, using echnology ebookplus 15A inroducion o kinemaics Digial doc 1 Quick Quesions Our lies are perpeually inoled in moemen. Walking around he house, being ranspored o school, hrowing a ball, riding a bicycle, picking up a pen, climbing sairs, going on a holiday are jus a few examples. Mos of our moemens are rouine, and we don gie hem a second hough. Howeer, someimes we do need o hink abou wha we are doing; for example, undersanding moion can be a maer of life and deah in siuaions such as crossing a road safely, deciding when i is pruden o oerake when driing, or calculaing where a cyclone is heading. Een in less-dramaic siuaions like keeping an appoinmen on ime, or judging how and when o hrow a ball while playing spor, we gie 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? Chaper 15 Kinemaics 511

2 Our ineres in analysing moion exends far beyond hese examples aken from our daily lies. People hae long been fascinaed by moemen 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. The sudy of moion is fundamenal in all branches of science. The name kinemaics is gien 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 curilinear moion, which deals wih cures). Examples of recilinear moion include a ball raelling along a pool able in a single direcion, or an ice-hockey puck ha has been hi along he ice. For mahemaical conenience, all moing 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 Bungy jumper. Bill jumps from a bridge ha is 1 meres aboe he ground and is aached o an 8-mere elasic rubber rope. He falls erically owards he ground. In he firs seconds he falls meres and in he nex seconds he falls a furher 6 meres. Afer 8 meres he bungy rope sars o srech, and herefore slows he fall so ha Bill raels a furher meres in seconds. The sreched bungy 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 eenually remain in, once he sopped bouncing on he rope.) He coninues raelling up a furher 1 meres in seconds. Bill coninues bouncing unil he is lowered safely o ground leel. 1 meres 5 meres 51 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

3 If we ake he saring poin, S, o be meres, hen he firs 1 seconds of Bill s jump can be displayed as follows. Sage 1 Sage Sage 3 S meres a = s A meres a = s B 8 meres a = 4 s E D 75 meres a = 1 s 85 meres a = 8 s C 1 meres a = 6 s Posiion The posiion of a paricle moing in a sraigh line is esablished by is disance from a fixed reference poin on he line. This is usually he origin, O, wih posiions o he righ of O normally being aken as posiie. 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 x = 4. Paricle Q is 3 unis o he lef of he origin and herefore has a posiion of x = - 3. We could describe Bill s moion by noing his posiion a arious imes. We show his on a sraigh line (erical or horizonal) by indicaing his locaion relaie o a reference poin, usually he origin, O. Posiions o he righ of O are normally aken as posiie O Poin S, a he origin, (acually 1 meres aboe he ground), shows Bill s saring posiion. Taking downwards as posiie, poin A is a and poin B is a 8. Q S A B P Posiie direcion 5 x O 8 x (m) Displacemen The displacemen of a moing paricle is is change in posiion relaie o a fixed poin. Displacemen gies boh he disance and direcion ha a paricle is from E D = 8 a poin. = 1 S A This can be represened on a C = 6 posiion ime line (or displacemen = = B = 4 ime line), as shown a righ, for he O x (m) firs 1 seconds of Bill he bungy jumper s pah. Noe: The direcion of he moion is indicaed by he arrows. Chaper 15 Kinemaics 513

4 Bill raels 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 ecor quaniy and has boh magniude and direcion. (In his case he magniude is 5 meres and he direcion is negaie.) Disance is a scalar quaniy and has magniude only. For he firs 1 seconds of Bill s jump, his displacemen is 75 meres (75 ). Howeer, he disance Bill has moed is 15 meres. Noe: A poin C, Bill is momenarily a a sop (his elociy is ) and his moion changes direcion from down o up. Velociy Velociy is also a ecor quaniy. The aerage elociy 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, x, a ime,. Aerage elociy = change in posiion change in ime = final posiion - iniial posiion change in ime x - x1 = - 1 = δ x δ Bill s aerage elociy oer he firs 1 seconds of his jump can be calculaed as follows: x-x1 Aerage elociy = = 1 - = 75 1 = 7.5 m/s The commonly used unis of elociy are cm/s, m/s or km/h. Noe: 1 m/s = 3.6 km/h. The insananeous elociy is he elociy a a gien poin of ime. Tha is, i is he gradien of he displacemen ime graph a a gien poin. Speed Speed is he magniude of elociy and so i is a scalar quaniy. Aerage speed = disance raelled ime aken Insananeous speed is he magniude of insananeous elociy and is always posiie. Bill s aerage speed oer he firs 1 seconds of his jump can be calculaed as follows: Aerage speed = 15 1 = 1.5 m/s (compared o he aerage elociy of 7.5 m/s). Posiion x x x 1 1 Time Change in posiion δ x Change in ime δ 514 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

5 WorKeD example 1 The following posiion ime line shows a paricle ha moes from S o A in seconds hen from A o F in 3 seconds. Find: a he saring posiion, S F b he final posiion, F S A c he displacemen of F from S d he disance raelled from S o F e he aerage elociy from S o F f he aerage speed from S o F x ThinK WriTe 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 c Displacemen = - - = 4 unis o he righ of S d Add he disance from S o A o he disance from A o F. d Disance = = unis e Aerage elociy = change in posiion x - x1 e Aerage elociy = change in ime - 1 = f Aerage speed = disance raelled ime aken = 4 5 =.8 unis/second in he posiie direcion f Aerage speed = 5 = 4 unis/second Consan elociy 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 elociy is consan oer he duraion of each sraigh-line secion. x The elociy is consan from = o = 4. 4 The elociy is consan from = 4 o = 1. 1 WorKeD example A Luna Park here is a new game called Hi he duck. To win, you mus knock down a mobile duck ha moes back and forh in a sraigh line on a 5-mere rack. You hae hree shos wih small sandbags. The posiion ime graph shows he posiion of he duck, x cenimeres o he righ of is saring poin, along he rack a x arious imes, seconds. 5 a Wha is he iniial posiion of he duck? 4 b How long did he game las? c Wha is he final displacemen of he duck from is saring 3 posiion? d Wrie he imes for which he elociy is: 1 i posiie ii negaie iii zero. e Hence, find he elociy for each of he hree ime inerals in par d. f Wha was he aerage speed of he duck during his game? Posiion (cm) ebookplus Tuorial in-1178 Worked example Chaper 15 Kinemaics 515

6 Think Wrie a The iniial posiion of he duck is when =. a When =, he iniial posiion of he duck is cm o he righ of is saring poin. b The graph finishes when = 1. b The game lased for 1 seconds. c Displacemen = final posiion iniial posiion. c Displacemen = 1 = 1 cm d i Look for where he gradien slopes upwards o he righ. ii Look for where he gradien slopes downwards o he righ. d i The gradien is posiie from = o = 5. ii The gradien is negaie from = 6 o = 1. iii Look for where he gradien is horizonal. iii The gradien is zero from = 5 o = 6. e Velociy = change in posiion x e i - x1 Velociy = change in ime = 5- = 5 = 4 cm/s x - x1 ii Velociy = = 1-6 = = 75 cm/s x - x1 iii Velociy = = 6-5 f Aerage speed = disance raelled ime aken = 1 = cm/s f Aerage speed = 5 1 = 5 cm/s Posiion expressed as a funcion of ime When he posiion is expressed as a funcion of ime, he posiion ime graph can be skeched and he moion hen analysed. If he posiion ime graph is cured, hen he elociy (or gradien) is always changing and neer consan. Worked Example 3 A paricle moes in a sraigh line so ha is posiion, x cm, from a fixed poin, O, on he line, a ime,, seconds, is gien by he rule: x = 1 ( 1), [, 5] 516 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

7 The posiion ime graph is shown: a Copy and complee he able below x b Wha is he iniial posiion of he paricle? c Wha is he significance of he posiion a = 1? d Show he moemen of he paricle on a posiion ime line. e i Wha is he displacemen of he paricle? ii Hence, deermine he aerage elociy of he paricle. f i Wha is he disance raelled by he paricle? ii Hence, deermine he paricle s aerage speed. Posiion (cm) x Think a 1 Subsiue each alue of ino he rule x = 1 ( 1) and ealuae for x. Wrie a When =, x = 1 ( 1) = 1 ( 1) =.5 When = 1, x = 1 (1 1) = 1 () = When =, x = 1 ( 1) = 1 (1) =.5 When = 3, x = 1 (3 1) = 1 () = When = 4, x = 1 (4 1) = 1 (3) = 4.5 When = 5, x = 1 (5 1) = 1 (4) Complee he able = 8 x 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 x = and he posiion ime graph shows ha he paricle is changing direcion. d The paricle sars a x =.5, moes o x = hen urns and finishes a x = 8. c A = 1 he paricle is changing direcion. d = 1 = = cm = x Chaper 15 Kinemaics 517

8 e i Displacemen = final posiion iniial posiion ii Aerage elociy = change in posiion change in ime f i Add he disance raelled from = o = 1, o he disance raelled from = 1 o = 5. ii Aerage speed = disance raelled ime aken e i Displacemen = 8.5 = 7.5 cm ii Aerage elociy = x - x = = = 1.5 cm/s 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 Aerage speed = = 1.7 cm/s REMEMBER 1. A paricle s posiion gies is locaion relaie o a reference poin, usually he origin, O.. A paricle s displacemen is he change in is posiion relaie o a fixed poin. Displacemen gies boh he disance and direcion ha he paricle is from a poin. Displacemen = final posiion iniial posiion 3. The aerage elociy of a paricle is he rae of change of is posiion wih respec o ime. change in posiion Aerage elociy = change in ime final posiion - iniial posiion = change in ime disance raelled 4. Aerage speed = ime aken Exercise 15A Inroducion o kinemaics 1 WE 1 Each of he following posiion ime lines shows a paricle which moes 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 i he disance raelled from S o F he aerage elociy from S o F i he aerage speed from S o F. a F b F S A S A x x c S e F A x A S F x d A F S x 518 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

9 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. 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. 3 Each moemen from S o F described in quesion akes 6 seconds and he measuremens are in cenimeres. In each case deermine: i he displacemen of F from S ii he oal disance raelled by he paricle iii he aerage elociy i he aerage speed. Use he posiion ime line a righ o C = 5 answer quesions 4 o 7. S = B = 4 4 MC The displacemen of F from S, in cm, is: A - 4 B 4 C 3 D 14 E 56 5 MC The disance raelled in moing from S o F, in cm is: A 4 B 34 C 44 D - 34 E 56 6 MC The aerage speed in moing from S o F, in cm/s is: A = 3 A 4.5 B 7 C 5.5 D E 3 F = 8 x 7 MC The aerage elociy in moing from A o C, in cm/s is: A B 1 C - 1 D - E WE The posiion ime graph shows he posiion of a moing paricle, x cenimeres o he righ of he origin, O, a arious 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 elociy is: i posiie ii negaie iii zero. d Hence, find he elociy for each of he hree ime inerals in par c. e Wha was he aerage speed of he paricle? Posiion (cm) x Chaper 15 Kinemaics 519

10 9 The posiion ime graph shows he posiion of a moing paricle, x cenimeres o he righ of he origin, O, a arious 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 elociy is: i posiie ii negaie iii zero. d Hence, find he elociy for each of he hree ime inerals in par c. e Wha was he aerage speed of he paricle? 1 WE 3 A paricle moes in a sraigh line so ha is posiion, x cm, from a fixed poin, O, on he line a ime, seconds, is gien by he rule: x = 1 ( ), [, 8] The posiion ime graph is shown below: x Posiion (cm) a Copy and complee he able below x b Wha is he significance of he posiion a =? c Show he moemen of he paricle on a posiion ime line. d Deermine he aerage elociy of he paricle. e Wha is he paricle s aerage speed? 11 A paricle moes in a sraigh line so ha is posiion, x cm, from a fixed poin, O, on he line a ime seconds is gien by he rule: x = 8 + 1, [, 8] a Copy and complee he able below x 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 moemen of he paricle on a posiion ime line. e Deermine he aerage elociy of he paricle. f Wha is he paricle s aerage speed? 1 A paricle moes in a sraigh line so ha is posiion, x cm, from a fixed poin, O, on he line a ime seconds is gien by he rule: x = 4 5, [, 6] a Skech he posiion ime graph for he paricle. Check your answer using a CAS calculaor. b Show he moemen of he paricle on a posiion ime line. Posiion (cm) x Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

11 15B c Deermine he aerage elociy of he paricle. d Wha is he paricle s aerage speed? 13 A paricle moes in a sraigh line so ha is posiion, x cm, from a fixed poin, O, on he line a ime, seconds, is gien by he rule: x = , [, 6] a Skech he posiion ime graph for he paricle. Check your answer using a CAS calculaor. b Show he moemen of he paricle on a posiion ime line. c Deermine he aerage elociy of he paricle. d Wha is he paricle s aerage speed? Velociy ime graphs and acceleraion ime graphs Velociy ime graphs Le us ake anoher look a he posiion ime line for he bungy jump performed by Bill ha was described a he sar of he chaper. S = A = E = 1 D = 8 B = 4 C = x Meres This siuaion can be represened on a posiion ime graph shown a righ. The cure reflecs he fac ha he change of posiion oer ime (elociy) is no consan. We can calculae he aerage elociy in each of he sages as follows: From S o A: Aerage elociy = x - x = - - Posiion (m) From A o B: Aerage elociy = x ebookplus Ineraciiy in-67 Moion graphs (kinemaics) x (6, 1) 1 9 C (8, 85) 8 B 7 (4, 8) D (1, 75) E A 1 (, ) S x = 4- = = 1 m/s = 3 m/s From B o C: Aerage elociy = x - x 1 From C o D: Aerage elociy = x - x = 6-4 = = = 8-6 = - 15 = 1 m/s = m/s Chaper 15 Kinemaics 51

12 From D o E: Aerage elociy = x - x = = - 1 = 5 m/s Noe: The negaie elociies occur when he moion is upwards, since we decided o define downwards as posiie. We can now represen he moion of Bill s bungy jump during each sage on a elociy ime graph (or more paricularly, an aerage elociy ime graph). Noice ha he graph shows ha he elociy is consan during each of he sages (shown as he sep formaion of he graph). This is because we hae calculaed he aerage elociy of each sage. If we were o analyse he aerage elociy oer smaller ime inerals, we would ge more seps wih smaller widhs, as is displayed in he second graph. If we allowed hese ime inerals (sep widhs) o ge closer and closer o zero, hen he associaed aerage elociies would effeciely become a series of conneced poins ha would colleciely produce a elociy ime graph somehing like he one displayed a righ. This is a elociy ime graph as i shows Bill s elociy a eery insance of he firs 1 seconds of moion during his bungy jump. There are no horizonal lines (seps) because he elociy is changing eery insan oer he course of he moion. This change in elociy oer ime is called acceleraion. Acceleraion is also a ecor quaniy. For he firs 4 seconds of moion, he graph is a sraigh line because Bill is subjeced only o acceleraion due o graiy, which is consan a 9.8 m/s. This means ha eery second, Bill s elociy increases by 9.8 m/s while he is moing downwards. For he period of ime where he bungy rope is sreched, (greaer 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 bungy rope. Tha is why he elociy ime graph is cured during his ime. From abou = 9 seconds o = 1 seconds, (where he bungy rope is less han 8 m) he rope is again slack and Bill is subjec o acceleraion due only o graiy again. A his sage he moion is upwards, bu since acceleraion due o graiy acs downwards, Bill is slowing down or deceleraing. Aerage acceleraion = change in elociy change in ime = = δ δ The mos common unis of acceleraion are cm/s or m/s. For he momen we will consider only examples ha inole consan acceleraion. Aerage elociy (m/s) Aerage elociy (m/s) Aerage elociy (m/s) a a Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

13 Worked Example 4 Draw a elociy ime graph o mach he following descripion. An objec ha is moing in a sraigh line has an iniial elociy of 5 m/s. I acceleraes a a consan rae unil i reaches a elociy of 1 m/s afer 6 seconds. I mainains his elociy for 8 seconds and hen deceleraes a a consan rae for a furher 4 seconds when i comes o res. Think 1 The elociy ranges from m/s o 1 m/s. The oal ime is = 18 seconds. 3 Draw a se of axes wih elociy on he erical axis and ime on he horizonal axis. Label each axis 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 elociy during he second sage. 6 Draw a sraigh line from (14, 1) o (18, ) o show he final sage of deceleraion. Velociy (m/s) Wrie/Draw Noice ha he gradien of each sraigh-line secion of he elociy ime graph gies he acceleraion of he objec. Analysing he elociy ime graph The gradien of a elociy ime graph allows us o calculae he acceleraion of an objec moing in a sraigh line. In addiion o his, he area beween he elociy ime graph and he ime axis also proides useful informaion relaing o displacemen and disance. Earlier, i was shown ha: Aerage elociy = change in posiion change in ime or a = δ x where δ a represens aerage elociy. Rearranging his resuls in: δx = a δ In oher words, he signed area beween a elociy ime graph and he ime axis is equal o he change in posiion or displacemen. When we calculae he signed area, we ake he area aboe he ime axis as posiie displacemen and he area below he ime axis as negaie displacemen. If he disance (raher han he displacemen ha he paricle has raelled) is required, hen here is no need o sign he areas. Tha is, he disance raelled is he oal area beween he elociy ime graph and he ime axis. a Using he aerage elociy ime graph describing Bill s 4 bungy jump from earlier, he informaion described aboe 3 can be highlighed as follows. The displacemen is equal o he sum of he signed areas of he recangles. 1 Displacemen = = = 75 meres 1 Aerage elociy (m/s) Chaper 15 Kinemaics 53

14 The disance is equal o he sum of all he unsigned area of he recangles. Disance = = Area 1 = 15 meres 9 1 The following can be obained from he figure shown a righ. 5 7 Area 1. The objec is raelling a a consan elociy of 5 m/s unil = 5 s. I slows down unil i sops a = 7 s, before i changes direcion and increases is speed o 5 m/s a = 9 s. 5 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. Beween = 5 s and = 9 s, he gradien is - 1, 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 elociy ime graph and he ime axis ino simple shapes; for example recangles, riangles or rapeziums. Area of a recangle = L W Area of a riangle = 1 bh Area of a rapezium = 1 (a + b)h Velociy (m/s) WorKeD example 5 Consider he elociy ime graph obained in worked example 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 raelled. Velociy (m/s) 1 5 ebookplus Tuorial in-1179 Worked example ThinK a Aerage acceleraion = b Aerage acceleraion = change in elociy change in ime change in elociy change in ime WriTe/DrAW a Aerage acceleraion = b Aerage acceleraion = = = 5 6 m/s = = = -.5 m/s 54 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

15 c 1 The displacemen is equal o he oal signed area under he elociy ime graph. c Diide he gien graph ino wo rapeziums, one from = o = 6 and he oher from = 6 o = 18. Velociy (m/s) Area 1 Area d 3 Calculae he area of each rapezium. Area 1 = 1 (5 + 1) 6 = = 45 unis Area = 1 (8 + 1) 1 = 1 1 = 1 unis 4 Find he displacemen. Displacemen = Area 1 + Area = = 145 m The disance is equal o he oal unsigned area under he elociy ime graph. d The disance is equal o 145 m. Noe: Since he elociy is always posiie in his example, he disance is equal o he displacemen. Acceleraion ime graphs Jus as he gradien of a posiion ime graph gies he rae of change of posiion or elociy, he gradien of a elociy ime graph gies he rae of change of elociy or acceleraion. Where he elociy is increasing he acceleraion is posiie. Where he elociy is decreasing he acceleraion is negaie. Where he elociy is no changing he acceleraion is zero. Consider a modified elociy ime graph of he firs 1 seconds of moion of Bill s bungy jump. We will assume he acceleraion is consan, bu differen hrough each of he sages of he jump. Since aerage acceleraion = change in elociy, he change in ime acceleraion for each sage is: From S o B: Aerage acceleraion = = 4-4- = 4 4 = 1 m/s Velociy (m/s) S 1 B C E D Chaper 15 Kinemaics 55

16 From B o C: Aerage acceleraion = = = - 4 = - m/s From C o D: Aerage acceleraion = = = - 15 = m/s From D o E: Aerage acceleraion = = = 15 = 7.5 m/s Therefore, he acceleraion ime graph would look like he graph aboe. Noe: The signed area under he acceleraion ime graph gies he change in elociy. In he graph on he preious page, he area beween he graph and he ime axis from = s o = 4 s is 4, which is erified on he preious elociy ime graph. Acceleraion (m/s ) a 1 1 WorKeD example 6 Consider he moion of an eleaor, which has is elociy ime graph as shown. Take posiie alues o represen upward moion. a In wha secions OA, AB, BC, ec. is he lif: i acceleraing posiiely? ii acceleraing negaiely? iii raelling a a consan elociy? A B b Deermine he acceleraion for each secion of 8 he lif s journey. 4 c Skech he acceleraion ime graph. C O d If he lif sared a ground leel, meres, 18 4 deermine is posiion a: i C ii G. 8 e Deermine he aerage elociy of he lif. 1 f How far did he lif rael? g Wha was he lif s aerage speed? Velociy (m/s) ebookplus Tuorial in-118 Worked example D G E F ThinK a i Acceleraion is posiie where he elociy is increasing. ii Acceleraion is negaie where he elociy is decreasing. iii Acceleraion is zero where he elociy is no changing. WriTe a i The acceleraion is posiie from O o A and from F o G. ii The acceleraion is negaie 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. 56 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

17 change in elociy b Aerage acceleraion = change in ime = b From O o A, aerage acceleraion = = 8-5- = 8 5 = 1.6 m/s From A o B, aerage acceleraion = = 13 = m/s - 8 From B o C, aerage acceleraion = -18 = - 8 = 4 m/s - From C o D, aerage acceleraion = 5 - = 5 = m/s From D o E, aerage acceleraion = From E o F, aerage acceleraion = = - 1 = 6 m/s = 8 = m/s From F o G, aerage acceleraion = = 1 5 =.4 m/s c The acceleraion is consan in each secion, so he acceleraion ime graph is a series of horizonal lines (seps). c Acceleraion (m/s ) a Chaper 15 Kinemaics 57

18 d i Since he lif sared a posiion meres, he posiion a poin C is he signed area under he rapezium OABC. d i The posiion a C is he area of rapezium OABC = 1 (13 + ) 8 = = 13 meres ii The posiion a poin G is he signed area under he rapezium DEFG plus posiion a poin C. ii The posiion a G is he signed area under he rapezium DEFG plus posiion a poin C = 1 (8 + 15) = = = 6 meres (ha is, he lif ends up 6 meres below ground leel). e Aerage elociy = change in posiion change in ime f The oal disance raelled by he lif is he oal area beween he elociy ime graph and he ime axis. g Aerage speed = disance raelled ime aken e Aerage elociy = x - x - f 1 1 = = =.15 m/s The oal disance raelled by he lif is = 7 meres. g Aerage speed = 7 4 = 6.75 m/s REMEMBER change in posiion 1. Aerage elociy = change in ime disance raelled. Aerage speed = ime aken change in elociy 3. Aerage acceleraion = change in ime 4. The signed area beween a elociy ime graph and he ime axis is equal o he change in posiion or displacemen. The area aboe he ime axis is posiie displacemen and he area below he ime axis is negaie displacemen. 5. The unsigned area beween a elociy ime graph and he ime axis is equal o he disance raelled. 6. Final posiion = displacemen + iniial posiion 58 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

19 Exercise 15B Velociy ime graphs and acceleraion ime graphs 1 WE4 Draw a elociy ime graph o mach each of he following descripions. a An objec, which is moing in a sraigh line, has an iniial elociy of 4 m/s. I acceleraes a a consan rae unil, afer 5 seconds, i reaches a elociy of 9 m/s. I mainains his elociy 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 moing in a sraigh line, has an iniial elociy of 6 m/s. I acceleraes a a consan rae unil, afer 8 seconds, i reaches a elociy of 1 m/s. I mainains his elociy for 15 seconds and hen deceleraes a a consan rae for a furher 5 seconds unil i reaches a elociy of 8 m/s. c An objec, which is moing in a sraigh line, has an iniial elociy of 5 m/s. I acceleraes a a consan rae unil, afer 1 seconds, i reaches a elociy of 4 m/s. I mainains his elociy for 1 seconds and hen deceleraes a a consan rae for a furher 9 seconds, when i comes o res. d An objec, which is moing in a sraigh line, has an iniial elociy of 5 m/s. I deceleraes a a consan rae unil, afer 6 seconds, i reaches a elociy of 5 m/s. I mainains his elociy for 4 seconds and hen acceleraes a a consan rae for a furher 6 seconds, when i comes o res. e An objec, which is moing in a sraigh line, has an iniial elociy of 8 m/s. I mainains his elociy for 1 seconds and hen acceleraes a a consan rae unil, afer 8 seconds, i reaches a elociy of 4 m/s. I mainains his elociy for 1 seconds and hen deceleraes a a consan rae for a furher 4 seconds, when i reaches a elociy of m/s, which i mainains. WE5 Consider he elociy ime graph shown o find: a he acceleraion in he firs 5 seconds b he acceleraion in he las 5 seconds c he oal displacemen d he oal disance raelled. Velociy (m/s) Consider he elociy ime 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 raelled. Velociy (m/s) Chaper 15 Kinemaics 59

20 4 Consider he elociy 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 raelled. Velociy (m/s) Use he elociy ime graph a righ o answer quesions 5 o 7. 5 MC 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 6 MC The aerage elociy from A o F is equal o: A 3.3 m/s C 4 m/s E 4 m/s B.3 m/s D.8 m/s Velociy (m/s) 8 4 A B C D E F 7 MC The aerage 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 8 WE6 Consider he moion of an eleaor, whose elociy ime graph is as shown. Take posiie alues o represen upward moion. a In wha secions, OA, AB, BC, ec. is he lif: i acceleraing posiiely? ii acceleraing negaiely? iii raelling a a consan elociy? 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 leel, meres, deermine is posiion a: i C ii G. e Deermine he aerage elociy of he lif. f How far did he lif rael? g Wha was he lif s aerage speed? 9 Consider he moion of a lif in a high-rise building. The lif s elociy ime graph is as shown. The lif sars from he weny-fifh floor, which is 1 meres aboe ground leel. Take posiie alues o represen upward moion. Velociy (m/s) Velociy (m/s) 6 O 8 6 O 9 A B C A B D G C D G E E F F 53 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

21 a In wha secions, OA, AB, BC, ec. is he lif: i acceleraing posiiely? ii acceleraing negaiely? iii raelling a a consan elociy? 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 aerage elociy of he lif. f How far did he lif rael? g Wha was he lif s aerage speed? 1 A car is raelling 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 elociy ime graph which 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 hae hey raelled? 11 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, raelling a a consan speed of 14 m/s. Polly acceleraes a a consan rae for 5 seconds, when she reaches a speed of 16 m/s, which she mainains unil he finish. a Skech a elociy 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? Chaper 15 Kinemaics 531

22 ebookplus Digial doc WorkSHEET C 1 Max he monkey is climbing a coconu ree in a sraigh line, o find a coconu for lunch. His moion is described as follows. Max sars from res a ground leel wih consan acceleraion unil he reaches a speed of 1.5 m/s afer 4 seconds. He mainains his speed for 8 seconds, when he deceleraes o a sop afer anoher seconds. Afer a furher 9 seconds, Max heads back down he ree wih consan acceleraion, reaching a speed of.5 m/s in seconds. He mainains his speed for 5 seconds, when he jumps from he ree. (Take posiie as up.) a Draw a elociy ime graph represening he moion of he monkey unil he leaes he ree. b A wha heigh did Max leap off he ree? c Wha was he oal disance raelled by Max on he ree? d Wha was Max he monkey s aerage speed: i while on he ree? ii while in moion on he ree? Challenge: When Max 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 Max he monkey is sill on he ree when he palm leaf his he ground and deermine where Max is a his ime. Consan acceleraion formulas Acceleraion due o graiy is usually 9.8 m/s. I can ary 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 ecor quaniy, when he objec is moing upwards, i is subjec o an acceleraion of m/s ; ha is, a deceleraion or reardaion. Consider an objec moing in a sraigh line, which has an iniial elociy of u. I acceleraes consanly unil i reaches a elociy of afer seconds. Is elociy ime graph is shown a righ. We can use his graph o derie arious formulas, which can be applied o problems inoling consan acceleraion. Since acceleraion, a, is he change in elociy oer ime, u δ a = δ u = - Muliply boh sides by : a = u Make he subjec, so: = u + a [1] Furhermore, since aerage elociy is he change in posiion, s, oer ime, δs u+ aerage elociy = or δ So, s u = + Therefore, s = 1 (u + ) [] Subsiuing = u + a (equaion [1]) ino equaion [] s = 1 (u + u + a) Velociy (m/s) = 1 (u + a) = 1 (u + a ) 53 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

23 Therefore, s = u + 1 a [3] From [1], u Subsiuing = - ino equaion []: a u = - a s = 1 (u + ) - u a 1 - u = a as = u Therefore, = u + as [4] In summary, if u is he iniial elociy, is he final elociy, s is he displacemen, a is he consan acceleraion and is he ime ineral, hen he following formulas apply for sraigh line moion: = u + a [1] s = 1 (u + ) [] s = u + 1 a [3] = u + as [4] Noes 1. A res means he elociy is zero.. 1 m/s = 3.6 km/h. (Verify his.) 3. When an objec is raelling in one direcion, u can be reaed as he iniial speed, as he final speed and s as he disance raelled. Worked Example 7 A sone is dropped from a bridge ha is 15 meres aboe a rier. Find: a he ime aken for he sone o reach he rier b he sone s speed on impac. Gie answers o he neares enh. Think a 1 Lis he gien informaion and wha has o be found. Find using s = u + 1 a by subsiuing in s = 15, a = 9.8 and u =. Wrie a Gien: s = 15, a = 9.8 and u = Require: =? s = u + 1 a 15 = Sole he equaion for. 15 = = = = Sae he soluion. The sone reaches he rier afer approximaely 5.5 seconds. b 1 Lis he gien informaion and wha has o be found. b Gien: s = 15, a = 9.8 and u = Require: =? Find using = u + as by subsiuing u =, a = 9.8 and s = 15. = u + as = Chaper 15 Kinemaics 533

24 3 Sole he equaion for. = 94 = 94 = Sae he soluion. The sone reaches he rier a a speed of 54. m/s. Worked Example 8 A drier is forced o suddenly apply he brakes of his car when a dog appears in fron of 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 raelling as i began o skid? b Wha was he acceleraion of he car during he skid? Think a 1 Lis he gien informaion and wha has o be found. Find u using s = 1 (u + ) by subsiuing s = 1, = and =. Wrie a Gien: s = 1, = and = Require: u =? s = 1 (u + ) 1 = 1 (u + ) 3 Sole 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 gien informaion and wha has o be found. Find a using = u + a by subsiuing =, u = 1 and =. 3 Sole he equaion for a. - 1 = a a = 6 b Gien: =, u = 1 and = Require: a =? = u + a = 1 + a 4 Sae he soluion. The acceleraion of he car was - 6 m/s. Worked Example 9 A ball is hrown upwards a 14.7 m/s from a ower ha is 5 meres aboe 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. (Gie answers o he neares enh.) Think Wrie a 1 Le u be up as he posiie direcion. a Lis he gien informaion and wha has o be found. Gien: u = 14.7, a = - 9.8, s = - 5 Require: =? 3 Find using s = u + 1 a by subsiuing u = 14.7, a = - 9.8, s = - 5. s = u + 1 a - 5 = (- 9.8) - 5 = = 534 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

25 4 Sole he quadraic equaion by using he quadraic formula. b 1 Lis he gien informaion and wha has o be found. Find using = u + as by subsiuing in u =, a = 9.8 and s = Sole he equaion for. = - b± b -4ac a 14. 7± ( 147.) -449 (. )(- 5) = 4 (.9) = -. and 5. = 5. seconds, since ime can no be negaie b Gien: u =, a = 9.8 and s = 61.5 Require: =? = u + as = = = = Sae he soluion. The ball firs srikes he ground a a speed of 34.6 m/s. REMEMBER 1. If u is he iniial elociy, is he final elociy, s is he displacemen, a is he consan acceleraion and is he ime ineral, hen he following formulas apply for sraigh line moion: (a) = u + a (b) s = 1 (u + ) (c) s = u + 1 a (d) = u + as. When an objec is raelling in one direcion, u can be reaed as he iniial speed, as he final speed and s as he disance raelled. 3. A res means ha he elociy is zero m/s = 3.6 km/h 5. Acceleraion due o graiy is 9.8 m/s for falling objecs and m/s for objecs raelling upwards. Exercise 15c Consan acceleraion formulas 1 WE 7 A sone is dropped from a bridge ha is 98 meres aboe a rier. Giing answers o he neares enh, find: a he ime aken for he sone o reach he rier b he sone s speed on impac. A paricle moing from res wih consan acceleraion reaches a speed of 16 m/s in 4 seconds. Find: a he acceleraion b he disance raelled. Chaper 15 Kinemaics 535

26 3 An objec raelling a 8 m/s acceleraes uniformly oer a disance of meres unil i reaches a speed of 18 m/s. Find: a he acceleraion b he ime aken. 4 A parachuis free-falls from an aircraf for 6 seconds. Find: a he speed of he parachuis afer 6 seconds b he disance raelled afer 6 seconds. 5 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. 6 We 8 A drier is forced o suddenly apply he brakes of his car when a ca appears in fron of i. 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 raelling as i began o skid? b Wha was he acceleraion of he car during he skid? 7 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? 8 A skaeboarder is raelling 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. 9 A falcon is hoering in he air when i suddenly dies erically 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. 1 A ram is raelling 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 raelled seconds afer he brakes are applied c he braking disance of he ram. 11 MC A rain raels a disance of 18 meres in 9 seconds while acceleraing uniformly from res. 536 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

27 a The speed of he rain afer 9 seconds can be deermined using he formula: A = u + a B s = 1 (u + ) C C = πr D s = u + 1 a E = 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 raelled afer 45 seconds is: A 5 m B 9 m C 675 m D 135 m E 45 m 1 We 9 A ball is hrown upwards a 9.8 m/s from a ower ha is 3 meres aboe 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. (Gie answers o he neares enh.) 13 A ball is hrown upwards a m/s from a ower ha is 8 meres aboe 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. (Gie answers o he neares enh.) 14 An objec is projeced erically upwards from he op of a building ha is 5 meres aboe he ground. Is iniial speed is 8 m/s. If he objec hen falls o he ground, find: a is maximum heigh aboe he ground b he oal ime aken o reach he ground c he speed of he objec when i reaches he ground. 15 A car moing from res wih uniform acceleraion akes 1 seconds o rael 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 aboe 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. Chaper 15 Kinemaics 537

28 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 aboe he waer a which he sone fell ou of he cage. 18 A 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 aboe 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? 15D Insananeous raes of change Insananeous elociy As we hae discussed preiously, he insananeous elociy a a gien ime is in fac he gradien of he posiion ime graph a ha ime. We hae also seen ha when he elociy is ariable he posiion ime graph will be cured. Consider a paricle moing in a sraigh line such ha is posiion, x cm, a any ime, seconds, is described by he rule: x() = 3, [, 3] Compleing a able of alues will gie: 1 3 x The posiion ime graph is shown a righ. Posiion (cm) x x() = Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

29 The elociy a any gien ime, say a = seconds, is equal o he gradien of he cure a =. The gradien of a cure a any gien poin is he gradien of he angen o he cure a ha poin. So, he elociy a = is equal o he gradien of he angen o he cure a =. To physically deermine he gradien of he angen ofen leads o inaccurae resuls. Care needs o be aken, firsly o draw an accurae and smooh cure, hen o place he angen a exacly he righ posiion. There is oo much room for error wih his process. Insead, we can apply he rule: Aerage elociy = δ x δ o esimae he gradien (elociy). This inoles aking wo poins on he cure 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 cure 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 ( + h) 3-( - h) 3 ( ) =. h The smaller he alue of h, he closer his gradien will be o he rue gradien of he angen. For example, using a calculaor o find () when h = 1,.1 and.1 produces he resuls shown in he able. I is quie clear from his able ha as h ges smaller h () and smaller he alue of () is approaching 1. If i is no already obious i becomes een more so if h =.1 or.1 and so on. In summary, he insananeous elociy a =, ( ), (of a paricle moing in a sraigh line) wih is posiion described as x() is found by ealuaing: x ( + h) - x( - h) ( ) = h for ery small alues of h (h > ). This echnique uses he same process o ha of differeniaing from firs principles which was coered in Mahemaical Mehods (CAS) Unis 1 &, and hus we can say: or Posiion (cm) Posiion (cm) dx ()= he deriaie of x wih respec o. d () = x () x x Tangen a = 1 3 (, 8) [( + h), ( + h) 3 ] [( h), h h ( h) 3 ] 1 3 Worked Example 1 A paricle is raelling in a sraigh line wih is posiion, x cm, a any ime, seconds, gien as x() = 3 -, [, 3]. Find he elociy of he paricle afer 1.5 seconds. Chaper 15 Kinemaics 539

30 Think Wrie 1 Gien he expression x() = 3 -, we wan (1.5). x() = 3 - Find he elociy equaion by differeniaing posiion, x, () = x () wih respec o ime, (() = x ()). () = Subsiue = 1.5 seconds. (1.5) = 3(1.5) - 1 (1.5) = Sae he soluion. The elociy of he paricle a = 1.5 seconds is 5.75 cm/s. Insananeous acceleraion When he acceleraion is ariable, hen he elociy ime graph is cured. The insananeous acceleraion a a gien ime is he gradien of he elociy ime graph a ha ime. So, like he insananeous elociy: The insananeous acceleraion a =, a( ), (of a paricle moing in a sraigh line) wih is elociy described as () is found by ealuaing: ( + h) -( - h) a ( ) = h for ery small alues of h (h > ). Again he echnique uses he same process o ha of differeniaing from firs principles, and we can say: d a ()= he deriaie of wih respec o. d or a() = () Worked Example 11 A paricle is raelling in a sraigh line wih is elociy, cm/s, a any ime, seconds, gien as 8 () =,. + 1 Find he acceleraion of he paricle afer 1 second. Think 1 Gien he expression, ()= we wan a(1) Find he acceleraion equaion by differeniaing elociy wih respec o ime (a() = ()). To do his, on he Main screen, complee he enry lines as: 8 Define () = +1 d (()) d -8 Define a() = ( + 1) Press E afer each enry. Wrie/display 8 ()= Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

31 3 Subsiue = 1 second ino he formula for a(). 4 Sae he soluion. The acceleraion of he paricle a = 1 seconds is - cm/s. Approximaing elociy ime graphs We hae already seen ha he disance raelled by a paricle raelling in a sraigh line is he unsigned area beween he elociy ime graph and he ime axis. When he acceleraion is consan, we calculae he areas of recangles, riangles or rapeziums. If he acceleraion is ariable, he elociy ime graph is cured and so i needs o be approximaed by sraigh-line funcions. This will resul in he area under he graph comprising eiher recangles, riangles or rapeziums. Then he disance raelled can be esimaed. One way o approximae he elociy ime cure is o use a series of horizonal seps oer he required domain or ime alues. This can be achieed by firs diiding he domain ineral ino n equally sized ime inerals, each h unis long. Nex, ealuae he elociy a he midpoin of each of hese inerals. Each of hese elociies can be reaed as he aerage elociy oer is corresponding ineral. The resul will be a sep funcion graph somehing like he figure following. The area of he rapezium gies he disance raelled Noe: 4 3 = 3 = 1 = 1 = h unis 1 The unsigned area under his elociy ime graph can be found by deermining he sum of each recangular area (h n ). This gies an esimae for he disance raelled oer a gien period of ime. As he recangle widh (or ineral widh), h, ges smaller and smaller, he number of recangles, n, increases and herefore he esimae ges closer and closer o he exac disance. 3 4 Chaper 15 Kinemaics 541

32 The following worked example will ouline he seps inoled, wih he aid of graphs. WorKeD example 1 A paricle is raelling in a sraigh line wih is elociy, (in m/s), a any ime seconds, gien as: () = +, Esimae he disance raelled during he firs 4 seconds of is moion by approximaing he elociy wih sep funcions each 1 uni wide. ebookplus Tuorial in-1181 Worked example 1 ThinK WriTe/DrAW 1 Skech he graph of () = + oer he domain [, 4]. Since h is 1 and he domain is [, 4], hen he inerals are from o 1, 1 o, o 3 and 3 o 4. 3 The midpoins of each ineral are.5, 1.5,.5 and 3.5. Velociy (m/s) () = Ealuae (.5), (1.5), (.5) and (3.5). These represen he heigh of each recangle. (.5) = =.75 (1.5) = = 3.75 (.5) = = 8.75 (3.5) = = Skech he sep funcion graph oer he domain [, 4] as an approximaion of he elociy ime relaionship. Velociy (m/s) Deermine he sum of each recangular area under he sep funcion. 7 Sae he unsigned area as he approximae disance raelled. Area of each recangle = lengh widh Toal area = (.75 1) + (3.75 1) + (8.75 1) + ( ) = 1( ) = 9 The paricle raels approximaely 9 meres during he firs 4 seconds. (Compared o he exac disance of 9 1 meres.) 3 54 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

33 In summary, if he acceleraion is ariable, hen he disance, d, raelled by a paricle can be esimaed from a elociy ime funcion by ealuaing: d = h[( 1 ) + ( ) + ( 3 ) + + ( n )] where h = sep funcion widh (ime ineral widh) n = he number of inerals n = midpoin of ime ineral, n ( n ) = elociy a ime, n. The mehod shown aboe is an approximaion of he displacemen (area under he cure), ha can be improed by reducing he sep funcion widh. Howeer, o calculae he exac displacemen (area under he cure), calculus is used. Using your knowledge from Mahemaical Mehods (CAS) Unis 1 &, his can be achieed as shown in worked example 13. Tha is, a formula for he disance raelled by an objec, d(), can be found by finding he anideriaie of he formula for is elociy, (), wih respec o ime. A formula for is elociy, () can be found by finding he anideriaie of is acceleraion, a(), wih respec o ime. d () = d () () = ad () Displacemen x() Differeniae Anidiffereniae Velociy () Differeniae Anidiffereniae Acceleraion a() Noes 1. The signed area beween a elociy ime cure and he -axis gies he displacemen.. If he elociy is posiie oer he gien ime ineral hen he displacemen is equal o he disance. WorKeD example 13 A paricle is raelling in a sraigh line wih is elociy, (in m/s), a any ime, seconds, gien as: () = +, Calculae he exac disance raelled during he firs 4 seconds of is moion. ebookplus Tuorial in-118 Worked example 13 ThinK WriTe/DiSPlAy Mehod 1: Using he rule 1 In order o calculae he disance, d(), raelled during he firs 4 seconds of moion, he anideriaie of he expression, () = +, from = o = 4 needs o be found. d () = d () 4 d () = ( + ) d Chaper 15 Kinemaics 543

34 3 Anidiffereniae he expression. d ()= Subsiue he limis and sole. d ()= d ()= + 3 d ()= Sae he exac disance raelled. The exac disance raelled during he firs 4 seconds of is moion, is 9 1 meres. 3 Mehod : Using a CAS calculaor 1 On he Main screen, using he sof keyboard, ap: ) - P Complee he enry line as: 4 ( + ) d Then press E. Sae he exac disance raelled. The exac disance raelled during he firs 4 seconds of is moion, is 9 1 meres. 3 WorKeD example 14 A car acceleraes from res a m/s for 5 seconds. a Wrie an equaion for he acceleraion. b Wrie he equaion for he elociy. c Calculae he disance coered in he firs 5 seconds. ebookplus Tuorial in-1183 Worked example 14 ThinK Mehod 1: Using he rule WriTe a 1 The acceleraion is m/s. a a() = b 1 The elociy equaion is produced by anidiffereniaing he formula for acceleraion. I is gien ha = when =. Calculae he consan, c, using his informaion. b a() = () = d () = + c = () + c c = () =, Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

35 c 1 To calculae he disance, d(), coered in he firs 5 seconds, anidiffereniae he elociy equaion wih limis and 5. Sae he disance coered in he firs 5 seconds. 5 c d() = ()d 4 d() = [ ] 5 d() = [5 - ] d() = 5 The disance raelled in he firs 5 seconds is 5 meres. Mehod : Using a CAS calculaor a 1 The acceleraion is m/s. a a() = b 1 To deermine he equaion for b elociy, gien a() =, on he Main screen, using he sof keyboard, ap: ) - P Complee he enry line as: ( ) d + c Then press E. Wrie he equaion for elociy. () = + c 3 I is gien ha = when =. Calculae he consan, c, using his informaion. = () + c c = () =, 5 c 1 To calculae he disance, d(), c coered in he firs 5 seconds, on he Main screen, complee he enry line as: 5 ( d ) Then press E. Sae he disance coered in he firs 5 seconds. The disance raelled in he firs 5 seconds is 5 meres. Chaper 15 Kinemaics 545

36 REMEMBER 1. The insananeous elociy a =, ( ), (of a paricle moing in a sraigh line) wih is posiion described as x() is found by ealuaing: x ( + h) -x( - h) ( ) =, for ery small alues of h (h > ). h. The insananeous acceleraion a =, a( ), (of a paricle moing in a sraigh line) wih is elociy described as () is found by ealuaing: ( + h) -( - h) a ( ) =, for ery small alues of h (h > ). h 3. If he acceleraion is ariable, hen he disance, d, raelled by a paricle can be esimaed from a elociy ime funcion by ealuaing: d = h[( 1 ) + ( ) + ( 3 ) ( n )] where h = sep funcion widh (ime ineral widh) n = he number of inerals n = midpoin of ime ineral n ( n ) = elociy a ime n. 4. Displacemen x() Differeniae Anidiffereniae Velociy () Differeniae Anidiffereniae Acceleraion a() Exercise 15D Insananeous raes of change 1 WE 1 A paricle is raelling in a sraigh line wih is posiion, x cm, a any ime, seconds, gien as x() = 3 +, [, 5]. Find he elociy of he paricle afer seconds. A paricle is acceleraing in a sraigh line wih is posiion, x cm, a any ime, seconds, gien as x() = 4, [, 4]. Find he elociy of he paricle afer 3.5 seconds. 3 A missile raelling in a sraigh line has is posiion, x m, a any ime, seconds, gien by x() = 3 4, [, 6]. Find he elociy of he missile afer 4 seconds. 4 A paricle is raelling in a sraigh line wih is posiion, x cm, a any ime, seconds, gien as x ()=,. Find he elociy of he paricle afer 3 seconds The posiion of a lif, x m, a any ime, seconds, is gien as x ()= 1 -, [, 8]. Find he elociy of he lif afer 1.5 seconds Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

37 6 WE 11 A paricle is raelling in a sraigh line wih is elociy, (in m/s), a any ime, seconds, gien as ()=,. Find he acceleraion of he paricle afer seconds An an is raelling in a sraigh line wih is elociy, (in cm/s), a any ime, seconds, gien 8 as () =,. Find he acceleraion of he an afer 3.5 seconds. ( + 1) 8 A lif moes wih is elociy, (in m/s), a any ime seconds, gien as ()= - [, 4] Find he acceleraion of he lif afer 1 second. 9 An objec is acceleraing in a sraigh line such ha is elociy, (in cm/s), a any ime, seconds, is gien as () = , [, 6]. Find he acceleraion of he objec afer seconds. Quesions 1 and 11 refer o he following informaion: The posiion of an objec raelling in a sraigh line is gien by x m. A any ime, seconds, is posiion is x() = log e ( + 1),. 1 MC The elociy a = 3 is neares o: A.9 m/s B.77 m/s C.51 m/s D 1.37 m/s E.5 m/s 11 MC The elociy a = 6 is neares o: A 1 m/s B.9 m/s C 3.74 m/s D 1.84 m/s E 5. m/s Quesions 1 and 13 refer o he following informaion: An objec is raelling in a sraigh line such ha is elociy, (in m/s), a any ime, seconds, is gien as: () = 3e,. 1 MC The acceleraion, in m/s, afer seconds is equal o: A 1.5 B 3 C 1.5 D 3 E 6 13 MC Using a sep funcion. seconds wide o approximae he elociy, he disance raelled afer seconds is neares o: A 1.57 m B 9 m C 18 m D m E 16.8 m 14 WE1 A paricle is raelling in a sraigh line wih is elociy, (in m/s), a any ime, seconds, gien as: () = + 3,. Esimae he disance raelled during he firs 6 seconds of is moion by approximaing he elociy wih sep funcions each 1 uni wide. 15 An objec is raelling in a sraigh line wih is elociy, (in m/s), a any ime, seconds, gien as () = 3 +,. Calculae he exac disance raelled during he firs 3 seconds of is moion. 16 WE13 A paricle is raelling in a sraigh line wih is elociy, (in m/s), a any ime, seconds, gien as () = + 3,. Calculae he exac disance raelled during he firs 6 seconds of is moion. 17 A paricle sars a res and raels in a sraigh line wih is elociy, (in m/s), a any ime seconds, gien as () = 3 +,. a Find he equaion for he posiion of he paricle wih respec o ime. b Calculae he disance coered in he firs 3 seconds. Chaper 15 Kinemaics 547

38 ebookplus Digial doc WorkSHEET We14 An objec iniially sars from res and acceleraes in a sraigh line, a (in m/s ), a any ime, seconds, gien as a() = + 1,. a Find he equaion for he elociy of he objec wih respec o ime. b Find he equaion for he posiion of he objec wih respec o ime. c Calculae he disance raelled in he firs 4 seconds. 19 An objec iniially raelling a 15 m/s acceleraes in a sraigh line, a (in m/s ), a any ime, seconds, gien as a() = ,. a Find he equaion for he elociy of he objec wih respec o ime. b Find he equaion for he posiion of he objec wih respec o ime. c Calculae he disance raelled in he firs seconds. 548 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

39 Summary Posiion and elociy A paricle s posiion gies is locaion relaie o a reference poin, usually he origin, O. A paricle s displacemen is he change in is posiion relaie o a fixed poin. Displacemen gies boh he disance and direcion ha a paricle is from a poin. Displacemen = final posiion iniial posiion The aerage elociy of a paricle is he rae of change of is posiion wih respec o ime. disance raelled Aerage speed = ime aken Aerage elociy = Velociy ime graphs and acceleraion ime graphs change in posiion change in ime final posiion - iniial posiion = change in ime change in elociy Aerage acceleraion = change in ime The signed area beween a elociy ime graph and he ime axis is equal o he change in posiion or displacemen. The area aboe he ime axis is posiie displacemen and he area below he ime axis is negaie displacemen. The unsigned area beween a elociy ime graph and he ime axis is equal o he disance raelled. Final posiion = displacemen + iniial posiion Consan acceleraion formulas If u is he iniial elociy, is he final elociy, s is he displacemen, a is he consan acceleraion and is he ime ineral, hen he following formulas apply for sraigh line moion: = u + a s = u + 1 a s = 1 (u + ) = u + as When an objec is raelling in one direcion, u can be reaed as he iniial speed, as he final speed and s as he disance raelled. A res means ha he elociy is zero 1 m/s = 3.6 km/h. Acceleraion due o graiy is 9.8 m/s for falling objecs and m/s for objecs raelling up. Insananeous raes of change The insananeous elociy a =, ( ), (of a paricle moing in a sraigh line) wih is posiion described as x() is found by ealuaing: x ( + h) -x( - h) ( ) = h for ery small alues of h (h > ). The insananeous acceleraion a =, a( ), (of a paricle moing in a sraigh line) wih is elociy described as () is found by ealuaing: for ery small alues of h (h > ). ( + h) -( - h) a ( ) = h Chaper 15 Kinemaics 549

40 If he acceleraion is ariable, hen he disance, d, raelled by a paricle can be esimaed from a elociy ime funcion by ealuaing: d = h[( 1 ) + ( ) + ( 3 ) + + ( n )] where h = sep funcion widh (ime ineral widh) n = he number of inerals n = midpoin of ime ineral, n ( n ) = elociy a ime, n. Displacemen x() Differeniae Anidiffereniae Velociy () Differeniae Anidiffereniae Acceleraion a() 55 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

41 chaper reiew Shor answer 1 a Represen he following siuaion on a posiion ime line. A paricle sars a S, 5 unis o he lef of he origin. I is hen displaced 6 unis o A, followed by a displacemen of - 4 unis o B. I undergoes a final displacemen of 1 unis o F. b If he moemen described aboe akes 4 seconds and he measuremens are in cenimeres, deermine: i he displacemen of F from S ii he oal disance raelled by he paricle iii he aerage elociy i he aerage speed. The posiion ime graph shows he posiion of a moing paricle, x cenimeres o he righ of he origin, O, a arious imes, seconds. Posiion (cm) x 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 elociy is: i posiie ii negaie iii zero. d Hence, find he elociy for each of he hree ime inerals in par c. 3 Draw a elociy ime graph o mach he following descripion. An objec, which is moing in a sraigh line, has an iniial elociy of - 6 m/s. I acceleraes a a consan rae unil i reaches a elociy of 6 m/s afer 1 seconds. I mainains his elociy for 6 seconds and hen deceleraes a a consan rae for a furher 3 seconds, when i comes o res. 4 Consider he elociy ime graph shown o find: Velociy (m/s) a he acceleraion in he firs 4 seconds b he acceleraion in he las 3 seconds c he oal displacemen d he oal disance raelled. 5 Consider he moion of an eleaor, which has is elociy ime graph as shown. Take posiie alues o represen upward moion. Velociy (m/s) O A B C D G a Deermine he acceleraion for each secion of he lif s journey. b Skech he acceleraion ime graph. c If he lif sared a ground leel, meres, deermine is posiion a: i C ii G. d Deermine he aerage elociy of he lif. e How far did he lif rael? E F Chaper 15 Kinemaics 551

42 6 An objec raelling a 5 m/s acceleraes uniformly oer a disance of 6 meres unil i reaches a speed of m/s. Find: a he acceleraion b he ime aken. 7 A ball is dropped from a ower and reaches he ground in 3 seconds, using a = 1 m/s. Find: a he heigh of he ower b he speed of he ball when i his he ground. 8 A ram is raelling a 1 m/s when he brakes are applied, reducing he speed o 4 m/s in 3 seconds. Assuming he reardaion is consan, find: a he acceleraion b he disance raelled seconds afer he brakes are applied c how long i akes he ram o sop afer he brakes are applied d he braking disance of he ram. 9 The posiion of a lif, x m, a any ime, seconds, is gien as: x() = ( + 3) + 5, [, 8] Find he elociy of he lif afer 5 seconds. 1 A paricle is raelling in a sraigh line wih is elociy, (in m/s), a any ime, seconds, gien as () = ( + 3),. a Find he acceleraion of he paricle afer seconds. b Find he disance raelled during he firs 4 seconds of is moion. 11 A passenger je of mass 48 kg moes from res wih consan acceleraion along a runway due o a oal hrus of 15 6 newons supplied by is engines. Assume ha air resisance and oher fricional forces are negligible. The magniude of he acceleraion of he je is. m/s. a How many seconds, correc o one decimal place, does i ake he je o reach is lif-off speed of 7 m/s? b Wha disance is needed, correc o he neares mere, for he je o ake off? [ VCAA 6] Muliple choice Use he posiion ime line below o answer quesions 1 o 4. C = 6 F = 8 S = The displacemen of F from S, in cm, is: A B 4 C 6 D E 1 The disance raelled in moing from S o F, in cm, is: A B 6 C 48 D E 8 3 The aerage speed in moing from S o F, in cm/s, is: A.5 B c -.5 D - 4 E The aerage elociy in moing from B o C, in cm/s, is: A 1 B 5 C - 1 D - 5 E -.1 Quesions 5 o 7 refer o he following elociy ime graph. 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 C and D D D and E E E and F 6 The aerage elociy from A o F is equal o: A 14 m/s B - 4 m/s C 4.5 m/s D m/s E 6 m/s 7 The aerage speed is equal o: A 5.5 m/s B m/s C 4.5 m/s D m/s E 6 m/s F B = 4 x 55 Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

43 Quesions 8 o 11 refer o he following informaion: A drag car raels a disance of 6 meres in 1 seconds while acceleraing uniformly from res. 8 The acceleraion of he drag car can be deermined using he formula: A = u + a B s = 1 (u + ) C C = πr D s = u + 1 a E = u + as 9 The speed, in km/h, afer 1 seconds is: A 7 B 36 C 18 D 48 E 3 1 The speed, in km/h, afer 6 seconds is: A 36 B 9 C 4 D 18 E The disance raelled afer 6 seconds is: A 3 m B m C 15 m D 5 m E 45 m Quesions 1 and 13 refer o he following: A paricle is raelling in a sraigh line wih is posiion, x m, a any ime, seconds, gien as x() = 3 4,. 1 The elociy a = 3 is: A 3 m/s B 4.5 m/s C 3 m/s D 1.5 m/s E 5 m/s 13 The acceleraion a = 3 is: A 9 m/s B 1 m/s C 3 m/s D 9 m/s E 1 m/s Quesions 14 and 15 relae o he following informaion: An objec is raelling in a sraigh line such ha is elociy, (in m/s), a any ime, seconds, is gien as () = ( + 1),. 14 The acceleraion, in m/s, afer seconds is equal o: A 9 B - 3 C 6 D 3 E The disance raelled from = o = 4: A 8 m 3 B 3 m 3 C 411 m 3 D m E m Exended response 1 An objec moes in a sraigh line so ha is posiion, x cm, from a fixed poin, O, on he line a ime, seconds, is gien by he rule: x() = , [, 6] a Skech he posiion ime graph for he paricle. b Deermine he aerage elociy of he paricle. c Show he moemen of he paricle on a posiion ime line. d Wha is he paricle s aerage speed? e Find he elociy of he paricle a: i = 1 and ii = 3 seconds. f During wha imes is he paricle raelling faser han i is a = 1 and = 3 seconds? During he filming of a moie, a sunman has o chase a moing bus and jump ino i. The sunman is required o sand sill unil he bus 1 passes him. He mus hen sar chasing i. The elociy ime graph a 8 righ describes he moion of he sunman and he bus from he insan he bus door passes he sunman. 6 a A wha insan did he sunman reach he same speed as he bus? 4 b Wha is he acceleraion of he sunman during he firs 4 seconds? c A wha insan did he sunman cach up o he bus? d How far did he sunman run o reach he door of he bus? Suppose he bus acceleraes afer 8 seconds a 1 m/s unil i reaches 11 m/s and he sunman mainains his speed of 1 m/s. e How far behind he bus is he sunman afer 8 seconds? f Verify ha he sunman is sill behind he bus when he bus sops acceleraing. g Explain why he sunman will neer cach he bus. Velociy (m/s) Sunman Bus Chaper 15 Kinemaics 553

44 3 A girl a he boom of a 1-m high cliff hrows a ennis ball erically upwards. A he same insan, a boy a he ery op of he cliff drops a golf ball so ha i his he ennis ball while boh balls are sill in he air. The acceleraion of boh balls can be aken as 1. m/s downwards. a If he balls collide when he ennis ball is a he op of is pah, wha is he posiion of he ennis ball when i srikes he golf ball? b Wih wha speed is he ennis ball hrown for his o occur? c Wha is he speed of he golf ball when i srikes he ennis ball? d How long has each ball been in moion when hey collide? 4 Alana can accelerae o her maximum speed of 8 m/s in 1.6 seconds. Her siser Lily can accelerae o her maximum speed of 8. m/s in seconds. Assume ha hey boh accelerae uniformly and hey can mainain heir maximum speed once hey reach i. Alana challenges Lily o a 1-mere race. a Who will win he race? b Wha is he winning margin? The girls broher Blake has a srong ineres in handicap racing. He works ou wo ariaions of a handicap ha will resul in a dead hea. c If boh girls run he full disance, how much earlier should he loser hae sared for a dead hea o resul? d If hey sar a he same ime, how much less disance should he loser hae o coer for a dead hea o resul? Blake rains he loser o accelerae fas enough for a dead hea o resul. e Find his acceleraion. ebookplus Digial doc Tes Yourself Chaper Mahs Ques 11 Adanced General Mahemaics for he Casio ClassPad

45 ebookplus ACTiViTieS Chaper opener Digial doc 1 Quick Quesions: Warm up wih en quick quesions on kinemaics. (page 511) 15A Inroducion o kinemaics Tuorial We in-1178: Wach how o use a posiion ime graph o deermine alues for ime, displacemen, elociy and aerage speed. (page 515) 15B Velociy ime graphs and acceleraion ime graphs Ineraciiy Moion graphs (kinemaics) in-67: Consolidae your undersanding of moion graphs using he ineraciiy. (page 51) Tuorials We 5 in-1179: Wach how o use a elociy ime graph o deermine alues for acceleraion, displacemen and he disance raelled of an objec. (page 54) We6 in-118: Wach how o use a elociy ime graph o deermine alues for acceleraion, displacemen and he disance raelled of an objec. (page 56) Digial doc WorkSHEET 15.1: Inerpre and creae posiion ime lines, inerpre and creae posiion-ime and elociy-ime graphs o aid in soling worded problems. (page 53) 15D Insananeous raes of change Tuorials We 1 in-1181: Wach how o esimae he disance raelled of a paricle moing in a sraigh line using a elociy ime graph. (page 54) We13 in-118: Wach how o calculae he exac disance raelled of a paricle gien is elociy as a funcion of ime. (page 543) We14 in-1183: Wach how o wrie he equaions for acceleraion and elociy of a car. (page 544) Digial doc WorkSHEET 15.: Calculae duraion, speed, elociy and acceleraion of bodies in moion. (page 548) Chaper reiew Digial doc Tes Yourself: Take he end-of-chaper es o es your progress. (page 554) To access ebookplus aciiies, log on o Chaper 15 Kinemaics 555

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