MEI Mechanics 1 General motion. Section 1: Using calculus

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1 Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a When =, v 4 a (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy is ms -. (ii) v Since his is negaive, here are no imes for which he velociy is zero. (iii) When s = 0, 0 ( )( ) 0 or The paricle is a he origin when.. (i) s 5 v 4 5 When = 0, v = -5 so he iniial velociy is -5 ms -. a 6 4 When = 0, a = -4 so he iniial acceleraion is -4 ms -. (ii) When v = 0, ( 5 )( ) 0 5 or The velociy is zero afer secon. of 8 /0/ ME

2 MEI M General moion Exercise soluions (iii) When he velociy is a is minimum value, a = When v 4 5. (This mus be a minimum poin since he equaion of he velociy is a quadraic wih posiive coefficien of x².), (i) s 6 (6 ) 40 s v (4 ) v (ii) The paricle is a O when = 0 and when = 6. (iii) The greaes displacemen is when he velociy is zero. (4 ) 0 0 or 4 From he graph, he greaes displacemen is when = 4 s 4 (6 4) The greaes displacemen is m. (iv) The greaes posiive speed is when he acceleraion is zero a 6 When a = 0, = When =, v (4 ) The greaes negaive speed in he ime inerval is when = 6 (from he graph). When = 6, v 6(4 6) 6 of 8 /0/ ME

3 MEI M General moion Exercise soluions So he greaes speed in he ime inerval is 6 ms (i) s v When = 0, v = -, so he iniial velociy is - ms -. When = 0, s = 0 When = 4, s The difference beween is posiions is 6 m. (ii) When = 0, v = - When = 4, v 4 6 The velociy is iniially negaive, bu when = 4 he velociy is posiive, so he paricle has changed direcion. (iii) When v = 0, 0 4 Since mus be posiive, he paricle changes direcion when = When =, s 6 So beween = 0 and =, he paricle ravels from s = 0 o s = -6, so i ravels 6 m. Beween = and = 4, i ravels from s = -6 o s = 6, so i ravels m. So he oal disance ravelled is 48 m. 6. v 9 s 9 c 4 When = 0, s = 0 0 c 4 s 0 a 6 8 When acceleraion is zero, ( ) 0 0 or The acceleraion is zero when = 0 and when =. 7. a 6 v 6 6 c When = 0, v = 0 c 0 v 6 (6 ) of 8 /0/ ME

4 MEI M General moion Exercise soluions The vehicle comes o res again when = 6, so i reaches poin B when = 6. s 6 k When = 0, s = 0 k 0 s When = 6, s The disance AB is 6 m. A greaes speed, acceleraion is zero When =, v (6 ) 9 The greaes speed is 9 ms (i) a 6 4 v c When = 0, v = 0 c 0 v 4 s 4 k When = 0, s = 0 k 0 s (ii) When s = 0, 0 ( ) 0 0 or The paricle is a he origin when = 0 and when =. (iii) The paricle changes direcion when v = ( 4) 0 0 or The paricle does no change direcion in he firs second. When =, s so he disance ravelled in he firs second is m (i) v When v = 0, = 0 or =. 4 4 ( ) 4 s 4 4 c When = 0, s = 0 c 0 4 s 4 When he paricle is nex a res, = so The disance ravelled is 7 m. 4 s of 8 /0/ ME

5 MEI M General moion Exercise soluions (ii) a 4 ( ) By symmery he greaes acceleraion is when = The greaes acceleraion is ms -. (iii) The greaes speed is when he acceleraion is zero ( ) 0 0 or When = 0, v = 0 When =, v 4 ( ) 6. a v c When =, v = c c 0 4 Displacemen The displacemen is m.. (i) s 9 v 6 9 (ii) When v = 0, ( )( ) 0 or Since mus be posiive, he body is a res when =. 5 of 8 /0/ ME

6 MEI M General moion Exercise soluions (iii) v v a a (i) a 6 v 6 6 c When = 0, v = so v 6 c s 6 k When = 0, s = 0 k 0 s (ii) When = 5, v When = 5, s When = 5, he velociy is - ms - 70 and he displacemen is m. This means ha he paricle has passed hrough O and is on he oher side of O heading away from O.. a k( ) When =, a = 4 4 k( ) 4 8k k 6 of 8 /0/ ME

7 MEI M General moion Exercise soluions a ( ) v ( ) c When =, v = 5 5 c 5 5 c c v Iniial velociy = ms s 4 v If here is a change of direcion, hen he velociy is zero, 0 For his quadraic equaion, he discriminan b 4ac is Since he discriminan is negaive, he equaion has no real soluions and so he velociy is never zero. Therefore he paricle never changes is direcion of moion. 5. (i) s v a 6 (ii) When v = 0, (iii) v v (since mus be posiive) 7 of 8 /0/ ME

8 MEI M General moion Exercise soluions a 0 a 0 (iv) The objec moves owar P unil = A his poin i comes o res insananeously and hen sars o accelerae owar O. I passes O and when = i is acceleraing owar Q. (v) When =, s The displacemen when = is m. When = 0, s = 0, so When i comes o res, s When =, s = Toal disance ravelled.8 m. 8 of 8 /0/ ME

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