Velocity and Acceleration 4.6

Transcription

1 Velociy and Acceleraion 4.6 To undersand he world around us, we mus undersand as much abou moion as possible. The sudy of moion is fundamenal o he principles of physics, and i is applied o a wide range of opics, such as auomoive engineering (crash ess, racing, performance ess), saellie and rocke launches, and improvemen of ahleic performance (kinesiology). We have already defined and compued velociies in Secion 3.4. Now we can compue hem more easily wih he aid of he differeniaion formulas developed in his chaper. Suppose an objec moves along a sraigh line. (Think of a ball being hrown verically upward or a car being driven along a road or a sone being dropped from a cliff.) Suppose he posiion funcion of he objec is s(). Anoher way o say his is ha s() is he displacemen (direced disance) of he objec from he origin a ime. Recall ha he insananeous velociy of he objec a ime is defined as he limi of he average velociies over shorer and shorer ime inervals. s h s v = lim ( ) ( ) h h ds The velociy funcion is he derivaive of he posiion funcion, v s ( ). Example 1 TV Adverisemen for a New Fooball Helme To dramaically adverise he effeciveness of a new fooball helme, a melon is aped inside i and he helme is dropped from a ower 78.4 m high. The heigh of he helme, h, in meres, afer seconds is given by h , unil i his he ground. a) Find he velociy of he helme afer 1 s and s. b) When will he helme hi he ground? c) The manufacurer s demonsraion will be effecive if he melon is undamaged afer impac. The melon will remain inac if he impac speed is less han 3 m/s. Is he demonsraion effecive? Soluion a) The posiion funcion is h , so he velociy a ime is dh = 98. The velociy afer 1 s is given by dh 981. ( ) The velociy afer 1 s is 9.8 m/s. The velociy afer s is given by dh 98. ( ) m/s The velociy afer s is 19.6 m/s. 4.6 Velociy and Acceleraion MHR 7

2 The negaive signs for he velociies indicae ha he helme is moving in he negaive direcion (down). b) The helme will hi he ground when he heigh is. h = = = 78.4/4.9 = 16 Since >, = 4. So, he helme his he ground afer 4 s. c) Subsiue = 4 ino he velociy funcion. v(4) = h (4) = -9.8(4) = -39. Since he speed (ha is, he absolue value of he velociy) of 39. m/s is greaer han 3 m/s, he melon is damaged on impac. Thus, he demonsraion fails. Example Analysis of a Cheeah s Moion The posiion funcion of a cheeah moving across level ground in a sraigh line chasing afer prey is given by he equaion s() = where is measured in seconds and s in meres. a) Wha is he cheeah s velociy afer 1 s? 4 s? 8 s? b) When is he cheeah momenarily sopped? c) Wha are he posiions of he cheeah in par b)? d) When is he cheeah moving in he posiive direcion? When is i moving in he negaive direcion? e) Find he posiion of he cheeah afer 1 s. f) Find he oal disance ravelled by he cheeah during he firs 1 s. g) Compare he resuls of pars e) and f). Soluion a) The velociy afer seconds is v = s () = Afer 1 s, s (1) = 3(1) - 3(1) 63 = 36 The velociy afer 1 s is 36 m/s. Afer 4 s, s (4) = 3(4) - 3(4) 63 = -9 The velociy afer 4 s is -9 m/s. 8 MHR Chaper 4

3 Afer 8 s, s (8) 3(8) 3(8) The velociy afer 8 s is 15 m/s. b) The cheeah sops momenarily when v() ( 3)( 7) 3 or 7 Thus, he cheeah is momenarily sopped afer 3 s and 7 s. c) A 3, s(3) (3) 3 15(3) 63(3) 81 A 7, s(7) (7) 3 15(7) 63(7) 49 The cheeah momenarily sops when i is 81 m and 49 m from is saring posiion. d) The cheeah moves in he posiive direcion when v(), ha is, 1 1 ( 3)( 7) This inequaliy is saisfied when boh facors are posiive, ha is, (7, ), or when boh facors are negaive, ha is, [, 3). Thus, he cheeah moves in he posiive direcion in he ime inervals [, 3) and (7, ). I moves in he negaive direcion when (3, 7). We can demonsrae his using a number line wih he facors lised underneah. The,, and symbols show where he quaniies are posiive, zero, and negaive. ( 3) 3 7 ( 7) ( 3) ( 7) The graph of he displacemen, s() , is shown below. e) A 1 s, s(1) (1) 3 15(1) 63(1) 13 The cheeah is 13 m from where i sared afer 1 s. Window variables: x [, 1], y [, 1] 4.6 Velociy and Acceleraion MHR 9

4 f) The pah of he cheeah is modelled in he figure. (Remember ha he cheeah goes back and forh along a sraigh line) =, s= = 3s, s = 81 m =7s, s =49m = 1s, s = 13 m To find he oal disance ravelled, firs deermine he change in posiion over each inerval for which he cheeah does no change direcion. The oal disance ravelled is he sum of hese quaniies. From par d), he cheeah changes direcion a 3 and 7. The disance ravelled in he ime inerval [, 3] is s(3) s() The disance ravelled in he ime inerval [3, 7] is s(7) s(3) The disance ravelled in he ime inerval [7, 1] is s(1) s(7) The oal disance ravelled by he cheeah is ( ) m or 194 m. g) The difference beween he oal disance ravelled by he cheeah and is final posiion is (194 13) m or 64 m. The cheeah s displacemen from is saring poin is 64 m less han is oal disance run. Acceleraion The acceleraion of an objec is he rae of change of is velociy wih respec o ime. Therefore, he acceleraion funcion a() a ime is he derivaive of he velociy funcion: a () v () dv Since he velociy is he derivaive of he posiion funcion s(), i follows ha: The acceleraion is he second derivaive of he posiion funcion. Thus, a() v () s () or dv ds a = = If s is measured in meres and in seconds, he unis for acceleraion are meres per second squared, or m/s. 3 MHR Chaper 4

5 Invesigae & Inquire: Moion Wih Variable Acceleraion 1. Use a moion deecor (CBR) conneced o a graphing calculaor o record he moion of a lab car aced on by a non-consan force, such as a sreched spring or a long chain hanging over he edge of a able. The mass of he chain hanging over he car edge causes boh car and chain o accelerae. The mass hanging over he edge is no consan.. Compare he slopes of he angens o he velociy-ime (v-) graph o he acceleraion from he acceleraion-ime (a-) graph a he same ime. Record your observaions in a able similar o he one shown below. Time (s) Slope of v- graph Acceleraion from a- graph 3. How is he slope of he v- graph relaed o he acceleraion? 4. Given he velociy of an objec as a funcion of ime, how would you find he acceleraion? 5. Wha is he geomerical inerpreaion of acceleraion? Example 3 Moion Wih Non-Consan Acceleraion The posiion funcion of a person on a bicycle pedalling down a seep hill wih seadily increasing effor in pedalling is s (), where s is measured in meres and in 6 seconds, [, 4]. a) Find he velociy and acceleraion as funcions of ime. b) Find he acceleraion a s. Soluion a) The velociy is ds v = 1 = ( ) ( ) 1 1 = Velociy and Acceleraion MHR 31

6 and he acceleraion is dv a = = 1 b) We subsiue in he acceleraion equaion. a 1 3 Afer s, he acceleraion is 3 m/s. Example 4 Analysis of a Posiion-Time Graph The graph shows he posiion funcion of a bicycle. Deermine he sign of he velociy and acceleraion in each inerval, and relae hem o he slope of he graph and wheher he slope is increasing or decreasing (ha is, how he graph bends). s C D E F G A B Soluion Inerval Descripion of graph Velociy Acceleraion A o B Slope, horizonal segmen B o C Posiive slope, slope increasing posiive posiive C o D Posiive slope, slope decreasing posiive negaive D Slope, momenarily horizonal negaive D o E Negaive slope, slope decreasing negaive negaive E o F Negaive slope, slope increasing negaive F o G Slope, horizonal segmen posiive Noe ha he able in Example 4 reflecs our undersanding ha he velociy of he bicycle is he slope of he posiion-ime graph. Furhermore, he acceleraion is he rae of change of velociy, which corresponds o he rae of change of he slope of he posiion-ime graph. Example 5 Moion Along a Verical Line In a circus sun, a person is projeced sraigh up ino he air, and hen falls sraigh back down ino a safey ne. The posiion funcion is s() 3 5 for he upward par of he moion and s() 4.8( 3) 45 for he downward par of he moion, where s is in meres and is in seconds. a) Deermine he acceleraion for he upward par of he moion. b) Deermine he acceleraion for he downward par of he moion. c) Compare he resuls of pars a) and b) and explain he difference. 3 MHR Chaper 4

7 Soluion a) s() = 3-5 ds = 3 1 ds a= = 1 The acceleraion for he upward par of he moion is 1 m/s, direced downward. b) s() = -4.8( - 3) 45 ds = 9.6( 3) ds a= = 9. 6 The acceleraion for he downward par of he moion is 9.6 m/s, direced downward. c) The acceleraion due o graviy near Earh s surface is abou -9.8 m/s. When he person is moving upward, graviy acs o slow he person. Since air resisance opposes he moion, i also acs o slow he person down. The effec of graviy and air resisance ogeher produce he acceleraion of 1 m/s. When he person is moving downward, graviy acs o make he person go faser. In his case, air resisance sill opposes he moion, which means i acs o slow he person down. Thus, air resisance opposes graviy, so ha he absolue value of he oal acceleraion is a lile less han wha graviy by iself would produce¾only 9.6 m/s insead of 9.8 m/s. Negaive acceleraion, dv a= < indicaes ha velociy is decreasing (as a poin A in he figure). This follows from he fac ha he acceleraion is he slope of he angen o he graph of he velociy funcion. Likewise, posiive acceleraion, dv > a= means ha velociy is increasing (as a B). v A B This can be a lile confusing, since here is a disincion beween velociy and speed. Velociy has boh magniude and direcion, whereas speed is simply he magniude of he velociy, wih no direcion indicaed. If he velociy is negaive and he acceleraion is negaive, hen he velociy is decreasing bu he speed is increasing (because speed is he absolue value of he velociy). A summary of he effec of acceleraion on speed (how fas he objec is moving wihou regard for he direcion) is as follows: If a()v() >, hen he objec is speeding up a ime. If a()v() <, hen he objec is slowing down a ime. 4.6 Velociy and Acceleraion MHR 33

8 In words, we can say ha, if he acceleraion and he velociy have he same sign a a paricular ime (hey are in he same direcion), hen he objec is speeding up a ha ime. This occurs because he objec is being pushed in he direcion of moion. If he acceleraion and he velociy have opposie signs a a paricular ime (hey are in opposie direcions), hen he objec is slowing down a ha ime, since he objec is being pushed in a direcion opposie o is moion. We can use similar reasoning o deermine if an objec is moving away from he origin or oward he origin. The origin is defined as he posiion equal o. If an objec has a posiive posiion and A: slows down B: speeds up C: slows down D: speeds up posiive velociy, i is moving away from he origin. Similarly, if an objec has a negaive posiion and a negaive velociy, i is moving away from he origin. In boh cases, he produc of he posiion and he velociy is posiive. v Objec s<, v<, sv> Objec is moving away from origin. If an objec has a posiive posiion and negaive velociy, i is moving oward he origin. Similarly, if an objec has a negaive posiion and a posiive velociy, i is moving oward he origin. In boh cases, he produc of he posiion and he velociy is negaive, indicaing ha he objec is moving oward he origin. Objec v s <, v >, sv < Objec is moving oward he origin. To summarize: If s()v(), hen he objec is moving away from he origin a ime. If s()v(), hen he objec is moving oward he origin a ime. Example 6 Speeding Up or Slowing Down? Objec s>, v>, sv> Objec is moving away from origin. v Objec s>, v<, sv< Objec is moving oward he origin. In a science-ficion movie, a compuer-generaed robo appears o be moving abou he scene. The acors and he robo are no on he se a he same ime. The robo is added o he sho laer. For he movie o look realisic, he acors mus be given exac insrucions abou he robo s moion so hey can respond a exacly he righ ime. The posiion of he robo is defined by he funcion s() , where is measured in seconds, [, 1], and s in meres. a) Find he velociy and he acceleraion a ime. b) Find he velociy and he acceleraion afer 3 s and 5 s. Is he robo speeding up or slowing down a hese imes? c) Graph he posiion, velociy, and acceleraion funcions for he firs 1 s. d) When is he robo speeding up? When is i slowing down? y A B C D v a v s 34 MHR Chaper 4

9 Soluion a) s() ds v () = = dv a () = = 6 4 b) A 3, v 3(3) 4(3) 36 9 and a 6(3) 4 6 Afer 3 s, he velociy is 9 m/s and he acceleraion is 6 m/s. Since a(3)v(3) 54, he robo is speeding up a his ime. A 5, v 3(5) 4(5) 36 9 and a 6(5) 4 6 Afer 5 s, he velociy is 9 m/s and he acceleraion is 6 m/s. Since a(5)v(5) 54, he robo is slowing down a his ime. c) The hree graphs are shown below on he same se of axes. Window variables: x [, 1], y [ 5, 16] d) Facoring he velociy funcion gives v ( )( 6) and so he velociy is posiive for [, ) and (6, 1] and negaive for (, 6) (see he graph). For he acceleraion funcion, we have a 6 4 6( 4) and so he acceleraion is negaive for [, 4) and posiive for (4, 1] (see he graph). 4.6 Velociy and Acceleraion MHR 35

10 We can summarize his informaion in he following number line diagrams. ( ) Velociy ( 6) ( ) ( 6) 6 Acceleraion The able and he graph below show when he robo is slowing down and when i is speeding up. Slowing Down Speeding Up v, a v, a v, a v, a [, ) (4, 6) (, 4) (6, 1] 4 s a v 6 7 slows down speeds up slows down speeds up Key Conceps If a paricle has posiion funcion s(), hen is velociy funcion is ds v () s () and is acceleraion funcion is dv ds a () v () s () If a()v(), hen he objec is speeding up. If a()v(), hen he objec is slowing down. If s()v(), hen he objec is moving away from he origin. If s()v(), hen he objec is moving oward he origin. 36 MHR Chaper 4

11 Communicae Your Undersanding 1. Explain how o find he velociy and acceleraion of a paricle given is posiion funcion.. Given he posiion funcion of a paricle, explain how o deermine when a) he velociy is zero b) he acceleraion is zero 3. a) Under wha condiions on he acceleraion is he velociy of a paricle increasing? Under wha condiions is he velociy of a paricle decreasing? b) Under wha condiions on he acceleraion is a paricle speeding up? Under wha condiions is a paricle slowing down? c) Why are he resuls of pars a) and b) differen? 4. Discuss he validiy of each saemen. Provide examples o illusrae your answer. a) If he acceleraion is posiive, he objec is speeding up. If he acceleraion is negaive, he objec is slowing down. b) If he velociy is posiive, he objec is moving away from he origin. If he velociy is negaive, he objec is moving oward he origin. B Apply, Solve, Communicae 1. Communicaion The graph shows he posiion funcion of a car. s () a) Wha is he iniial velociy of he car? b) Is he car going faser a A or a B? c) Is he car slowing down or speeding up a A, B, and C? d) Wha happens beween C and D? e) Wha happens a F?. The graph of a velociy funcion is shown. s () B A C D F A B E C E D Sae wheher he acceleraion is posiive, zero, or negaive a) from o A b) from A o B c) from B o C d) from C o D e) from D o E 3. The graph of a posiion funcion is shown. a) For he par of he graph from o A, use slopes of angens o decide wheher he velociy is increasing or decreasing. Is he acceleraion posiive or negaive? b) Sae wheher he acceleraion is posiive, zero, or negaive i) from A o B ii) from B o C iii) from C o D iv) from D o E 4. Communicaion On he graph are shown he posiion funcion, velociy funcion, and acceleraion funcion of an objec. Idenify each curve and explain your reasoning. y s () A E B D C a c b 4.6 Velociy and Acceleraion MHR 37

12 5. The figure shows he graphs of he posiion funcion, he velociy funcion, and he acceleraion funcion of a car ha is undergoing a es o deermine is performance level. Deermine when he car is speeding up and when i is slowing down. y v s a Each posiion funcion gives s, in meres, as a funcion of, in seconds. Find he velociy and acceleraion as funcions of ime. a) s 9 4 b) s 3 5 c) s d) 3 s = 1 e) f) s = ( 6 s = 1 )( ) 7. Each posiion funcion gives s, in meres, as a funcion of, in seconds. Find he velociy and acceleraion for any and for 4s. a) s 6 3 b) s c) s d) 3 s = 1 e) s = ( ) f ) 3 s = 4 8. If a sone is hrown downward wih a speed of 1 m/s, from a cliff ha is 9 m high, is heigh, h, in meres, afer seconds is h Find he velociy afer 1 s and afer s. 9. Inquiry/Problem Solving If a ball is hrown direcly upward wih an iniial velociy of 9.4 m/s, hen he heigh, h, in meres, afer seconds, is given by h a) Find he velociy of he ball afer 1 s, s, 4 s, and 5 s. b) When does he ball reach is maximum heigh? c) Wha is is maximum heigh? d) When does i hi he ground? e) Wih wha velociy does i hi he ground? 1. The posiion of a moorcycle moving down 1 a sraigh highway is s 15, where is measured in seconds and s, in meres. a) When does he moorcycle reach i) 15 m/s? ii) 5 m/s? b) Wha is he acceleraion of he moorcycle? 11. The posiion of a person on a skaeboard is given by s 8 1, where s is measured in meres and, in seconds. a) Find he velociy afer s and 6 s. b) When is he person a res? c) When is he person moving in he posiive direcion? d) Draw a diagram o illusrae he moion of he person. 1. Inquiry/Problem Solving A new moion deecor is used o graph he posiion of a spors car. If he moion deecor is no calibraed properly, i migh give unrealisic graphs and daa. The posiion of he car is described by he funcion s 3 1 6, [, 6], where s is measured in meres and, in seconds. I is known ha he larges possible acceleraion of he car is 1 m/s. Invesigae if he deecor is working properly. a) i) Wha is he iniial velociy of he car? ii) When is he car a res? iii) When is he car moving in he forward direcion? b) Draw a diagram o illusrae he moion of he car. c) Find he oal disance ravelled by he car in he firs 6 s. d) Find he maximum acceleraion of he car. e) Lis all he evidence ha you can o show he company ha he equaion of moion is no realisic for a spors car. 13. Applicaion If a weighed and coiled rescue rope is hrown upward wih a velociy of m/s 38 MHR Chaper 4

13 from he op of a 3 m cliff, hen he heigh, in meres, of he rope above he base of he cliff, afer seconds, is s a) When does he rope reach is maximum heigh? b) Use he quadraic formula o find how long i akes for he rope o reach he ground. c) Find he approximae velociy wih which he rope srikes he ground. 14. A posiion funcion is given by 1 s s v g, where s, v, and g are consans. Find a) he iniial posiion b) he iniial velociy c) he acceleraion 15. Applicaion A sun car driver is pracising for a movie. The posiion funcion of he car is 1 s 3 5,, where s is measured in 3 meres and, in seconds. Find he acceleraion a he insan he velociy is zero. 16. An unoccupied es rocke mean for orbi is fired sraigh up from a launch pad. Somehing goes wrong and he rocke comes back down. The heigh of he rocke above he ground is modelled by h 5 1 3,, where h is 6 measured in meres and, in seconds. a) When is he acceleraion posiive and when is i negaive? b) When is he velociy zero? c) Find he maximum heigh of he rocke. d) A wha velociy does he rocke hi he ground? C e) When is he rocke speeding up? When is he rocke slowing down? 17. The posiion of a cougar chasing is prey is given by he funcion s 3 6 9, where is measured in seconds and s, in meres. a) Find he velociy and acceleraion a ime. b) When is he cougar moving oward he origin? When is i moving away? c) When is he cougar speeding up? When is i slowing down? d) Graph he posiion, velociy, and acceleraion for he firs 4 s. 18. The shaf of a well is 15 m deep. A sone is dropped ino he well and a splash occurs afer 1.6 s. How far down he well is he surface of he waer? 19. Applicaion A hawk is flying a km/h horizonally and drops is prey from a heigh of 3 m. a) Sae equaions represening he horizonal displacemen, velociy, and acceleraion of he falling prey. b) If he prey falls a an acceleraion due o graviy of 9.8 m/s, sae equaions represening he verical displacemen, velociy, and acceleraion. c) When is he verical speed greaer han he horizonal speed? d) Develop an equaion represening he overall velociy of he prey. e) Wha is he equaion represening he overall acceleraion of he prey? f) Wha is he acceleraion of he prey afer s? 4.6 Velociy and Acceleraion MHR 39