Variable acceleration, Mixed Exercise 11

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1 Variable acceleraion, Mixed Exercise 11 1 a v 1 P is a res when v b s 0 0 v d (1 ) The disance ravelled by P is 7. m. 1 a v 6+ a d v 6 + When, a 6+ 0 The acceleraion of P when is 0 m s. b x v d 1 (6 + )d + + k, where k is a consan of inegraion. 8 When 0, x k k x + 8 When, x OP 7 m Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 1

2 a v a d ( ) + c, where c is a consan of inegraion. When 0, v c c 8 v + 8 Le s m be he displacemen from A a ime seconds. s v d ( 8)d k, where k is a consan of inegraion. When 0, s k k 0 Displacemen of P from A a ime seconds + 8 b The greaes posiive displacemen of P occurs when d s v 0: ( + )( ) 0 > 0, so When, s < 0 Hence, P does no reach B. c P reurns o A when s ( ) 0 b ± b ac a ± ( ) (1)( ) (1) ± 10 > 0, so 6.6 P reurns o A when 6.6. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

3 d Disance beween wo insans when P passes hrough A maximum disance found in par b Toal disance ravelled by P beween he wo insans when i passes hrough A is 160 m. a a d v so speed has maximum value when a > 0, so v a d (8 )d 8 + c, where c is a consan of inegraion. When 0, v c c 0 v 8 When, v (8 ) The greaes posiive speed of he paricle is m s 1. b s v d 8 + k, where k is a consan of inegraion. 6 When 0, s k k 0 s 6 When, s The disance covered by he paricle during he firs wo seconds is 0 m. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

4 a s + 11 v d s + m s 1 b P is a res when v ( )( 6) 0 or 6 P is a res when and 6. c a d v 6 + d a 0 when The acceleraion is zero when 11. e The speed of P is 16 when v 16 and v 16. When v 16, ( 10)( ) 0 10 or When v 16, ± 8 ± or 6.9 From he graph in par d, he required values are 0 < 0.8, 10 < <. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

5 6 a The body is a res when v ( )( ) 0 or The body is a res when and. b a d v 6 11 When, a (6 ) When, he acceleraion is 1 m s. c From par a, he body changes direcion when and. s1 displacemen for 0 s displacemen for s displacemen for s1 0 ( ) ( ) s ( )d s ( )d Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free.

6 11 6 c s Sign o be ignored when calculaing disance: Toal disance s1 + s + s The oal disance ravelled is 16.0 m (1 d.p.). 7 a v a d ( 8 )d + c, where c is a consan of inegraion. When 0, v c c ( 0 ) + c v + 6 b s v d k, where k is a consan of inegraion. 10 When 0, s k k 0 s c Paricle is a res when v ( )( 6) 0 0, so or 6 The paricle is a res when and 6. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 6

7 x 0 dx Maxima and minima occur when d x ( 9 + 1) 0 ( )( 7) 0 So urning poins are a 0, and 7. From he skech graph, he drone is a a greaer heigh when han when 0, and 7 corresponds o he minimum heigh over he given inerval. (1 ) + (8 ) + 00 When, x (1 7 ) + (8 7 ) + 00 When 7, x The maximum heigh reached by he drone is 8.6 m, and he minimum heigh is 1.1 m. 9 When 0, v u 800, s 100 When, v 0 Using v u + a: a 800 a v a d + c, where c is a consan of inegraion. When 0, v c c 800 v 800 s v d (800 ) Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 7

8 9 s k, where k is a consan of inegraion. When 0, s k k 100 s So a 100, b 800, c a v a d (0 6 ) 0 + c, where c is a consan of inegraion. When 0, v c c 7 v b When 7, v P s maximum speed in he inerval 0 7 is when d v dv When 10, v The greaes speed of P in he inerval 0 7 is 0 m s 1. c s 7 (7 + 0 )d The disance ravelled by P in he inerval 0 7 is 196 m. 11 a a (7 ) So a k(7 ) 7k k v a d (7 )d k k k 7k + c, where c is a consan of inegraion. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 8

9 11 a When 0, v c c 0 k v 7k When, v 6 6 1k 9k 1k 6 k 1 7 v 6 s v d 7 6 d 7 + d, where d is a consan of inegraion. When 0, v d d 0 7 s 1 ( ) 1 a Time canno be negaive so 0. When 0, s 0 When, s So when, he mouse is again a a disance of zero from he hole: i has reurned. b s 10 + When mouse is a he greaes disance, d s 0 ds When d s 0, ( 1 + ) 0 ( )( ) 0 s 0 when 0 and, so maximum is when.. When., s The greaes disance of he mouse from he hole is 9.1 m. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 9

10 1 a Any wo from: As he shule rises, i burns large amouns of fuel, reducing mass and herefore allowing he same force o produce greaer acceleraion. (Hence posiive erms in he equaion.) While he shule remains in he amosphere, he air resisance forces on i will be changing in a complex way: he increasing speed will cause hem o increase, bu reduced densiy of he amosphere a greaer heighs will reduce heir effec. A greaer heighs, he graviaional pull of he Earh is less, which increases he resulan force on he shule and increases he acceleraion. (In pracice, his effec is small compared o ha of he mass reducion.) As he fuel from each ank in he booser rockes is used up, hey may become less efficien, reducing he hrus hey produce. (The fuel feed mechanisms are designed o preven his and ensure smooh ransiions beween each sage, bu any asronau can ell you ha here is no such hing as a smooh journey ino space!) b v a d 7 (( ) ( ) ) ( ) (1. 10 ) c, where c is a consan of inegraion. When 1, v ( )(1) (1. 10 )(1) + 0.0(1) (1) + c, c 66 v ( ) (1. 10 ) , c When 6, v ( )(6) (1. 10 )(6) + 0.0(6) (6) When 6s, he velociy of he space shule is 90 m s -1 (.9 km s 1 ). d For his secion of he fligh: a 8.6, u 90, v 780 m s 1 v u + a Toal ime o reach escape velociy ( s.f.) The shule cus is main engines 1 s afer launch. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 10

11 Challenge 1 v a d ( 18+ 0) c, where c is a consan of inegraion. When 0, v c c 0 v Checking for crossing poins o find if he velociy becomes negaive during firs s: ( 9 + 0) 0 ( )( ) 0 This means ha displacemen is posiive for he firs four seconds and negaive in he fifh second (see skech graph), so need o find disances separaely. s1 disance ravelled in firs s s disance ravelled in fifh second s1 ( )d s ( )d Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 11

12 1 Toal disance The paricle covers.7 m in he firs s of is moion. v a d (6+ ) + + c, where c is a consan of inegraion. Assuming ha velociy does no change direcion during his ime, disance ravelled beween and v ( + + )d + + c c ( + + c) ( + + c) c 7 9 c + c So 0 + c c 6 v When, v A s, he velociy is 91 m s 1. Pearson Educaion Ld 017. Copying permied for purchasing insiuion only. This maerial is no copyrigh free. 1

MEI Mechanics 1 General motion. Section 1: Using calculus

MEI Mechanics 1 General motion. Section 1: Using calculus Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy