CHAPTER 9 MOTION ALONG A STRAIGHT LINE FORM 5 PAPER 2

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1 PPER. particle moves in a straight line and passes through a fixed point O, with a velocity of m s. Its acceleration, a m s, t seconds after passing through O is given by a 8 4t. The particle stops after k seconds. (a) Find (i) the maximum velocity of the particle, (ii) the value of k. [6 marks] (b) Sketch a velocity-time graph for t k. Hence, or otherwise, calculate the total distance travelled during that period. [4 marks]. particle moves along a straight line from a fixed point R. Its velocity, V m s, is given by V t t, where t is the time, in seconds, after leaving the point R. (ssume motion to the right is positive) Find (a) the maximum velocity of the particle, [ marks] (b) the distance travelled during the 4th second, [ marks] (c) the value of t when the particle passes the points R again, [ marks] (d) the time between leaving R and when the particle reverses its direction of motion. [ marks]. The following diagram shows the positions and directions of motion of two objects, and B, moving in a straight line passing two fixed points, P and Q, respectively. Object passes the fixed point P and object B passes the fixed point Q simultaneously. The distance PQ is 9 m. B P M Q 9 m The velocity of, V m s, is given V 8t t, where t is the time, in seconds, after it passes P while B travels with a constant velocity of - m s. Object stops instantaneously at point M. (ssume that the positive direction of motion is towards the right.) Find (a) the maximum velocity, in, m s, of, [ marks] (b) the distance, in m, of M from P, [4 marks] (c) the distance, in m, between and B when is at the points M. [ marks]

2 4. particle moves in a straight line and passes through a fixed point O. Its velocity, v ms, is given by v t 4t, where t is the time, in seconds, after leaving O. [ssume motion to the right is positive.] (a) Find (i) the initial velocity of the particle, (ii) the time interval during which the particle moves towards the left, (iii) the time interval during which the acceleration of the particle is positive. [ marks] (b) Sketch the velocity-time graph of the motion of the particle for t. [ marks] (c) Calculate the total distance travelled during the first seconds after leaving O. [ marks]. particle moves along a straight line and passes through a fixed point O. Its velocity, v ms, is given by v t 8t, where t is the time, in seconds, after passing through O. [ssume motion to the right is positive.] Find (a) the initial velocity, in ms, [ mark] (b) the minimum velocity, in ms, [ marks] (c) the range of values of t during which the particle moves to the left, [ marks] (d) the total distance, in m, travelled by the particle in the first seconds. [4 marks] 6. particle moves along a straight line and passes through a fixed point O. Its velocity, v ms is given by v 8 t t, where t is the time, in seconds, after passing through O. The particle stops instantaneously at point M. [ssume motion to the right is positive.] Find (a) the acceleration, in ms, of the particle at M, [ marks] (b) the maximum velocity, in ms, of the particle [ marks] (c) the total distance, in m, travelled by the particle in the first seconds, after passing through O. [4 marks],

3 7. particle moves along a straight line and passes through a fixed point O, with velocity of ms. Its acceleration, a ms, is given by a t + 8 where t is the time, in seconds, after passing through point O. The particle stops after k s. (a) Find: (i) the maximum velocity of the particle, (ii) the value of k [6 marks] (b) Sketch a velocity-time graph of the motion of the particle for t k. Hence, or otherwise, calculate the total distance travelled during that period. [4 marks] 8. particle moves along a straight line. Its velocity, v ms -, from a fixed point, O, is given by v t t + 4 where t is the time, in seconds, after passing through point O. [ssume motion to the right is positive.] Find (a) the initial velocity of the particle, [ marks] (b) the minimum velocity of the particle, [ marks] (c) the range of values of t during which the particle moves to the left, [ marks] (d) the total distance travelled by the particle in the first 6 seconds. [4 marks] 4

4 NSWERS PPER. a 8 4t V ( 8-4t ) 4t 8t c 8 t t c V, t. 8 () () c c V 8t t a) (i) maximum velocity, a 8 4t 8 4t t s V max imum 8() () 8 ms - (ii) particle stops: V 8t t t 8t t 4t (t + )(t ) t k b) V 8t t V t V 8 8 Total Distance V 8 t t 4t t t t

5 4( ) ( ) () 66 m. a) V t t dv a 4t Maximum velocity, a 4t 4t t V max () ( ) ms - b) s v t t t t c s, t c s t t t ; s ( ) ( ) 7m t 4; s (4 ) (4 ) 7 m Distance travelled during the fourth second m c) t point R again, s t t t ( t) t t t 6

6 d) Maximum displacement, V t t t ( t) t Time seconds. a) V 8t t a 8 4t V maximum when a 8 4t 8 4t t V max 8() ( ) 8 ms - b) Object at M when V Distance M from P 8t t t 8t t 4t t t t v 8t t t 4t t () 4( ) ( ) 66 m c) When t, distance B from Q v t m Distance between and B m # 7

7 4. a) i) v t 4t t, v 4() ms - # ii) particle moves to the left, v < t 4t < t t < t < t < # iii) v t 4t dv a t 4 a is positive, a > t 4 > t > t > # b) v t 4t v t v t c) Total distance v v t t ( t 4t t t 4 ) t t t t m #. a) v t 8t t, v 8() ms # 8

8 b) v t 8t dv a t 8 dv When, t 8 t 8 t 4 When t 4, v min 4 8(4) + - ms - # c) Particle moves to the left, v < t 8t t t t d) Total distance v < t < v t ( t 8t t 8 ) t 4t t t 4t t m # 6. a) v 8 t t When v, 8 t t t t 8 (t + ) (t 4) t 4 dv a t t M, a (4) 6 ms # 9

9 b) Maximum velocity, a t t t When t ; v max imum 8 + () 9 ms # c) Total distance, s v 8 t t t 8t t c When t, s c s s t 4 s t t t 8t t 4 8(4) 4 6 m 8() m t 4 m t 6 m Total distance in the first seconds (6 ) 6 m # 7. a) i) a t + 8 V t + 8 t 8t c When t, V : t 8t c V t 8t When maximum velocity, a - t + 8 t 8 t 4 V max imum (4) 8(4) 4

10 6 ms # (ii) Particle stops, V t 8t t 8t (t + )(t ) t k V 6 b) V t 8t t 4 V 6 4 t Distance ( t 8t ) t [ 4t t ] 4( ) () 66 m # 8. a) v t t 4 When t, initial velocity, v () 4 4 ms # b) dv a t When minimum velocity, a t - t v max imum () 4 ms # 4

11 c) When moves to the left, v t t 4 ( t 4 )( t 6 ) 4 6 t 4 t 6 d) Distance, s ( t t 4 ) t t 4t c When t, s c s t t 4t t, s t 4, s (4) (4) 4(4) 7 m t 6, s (6) (6) 4(6) 6 m t 6 t Total distance travelled during the first 6 seconds m # 6 t 4 7 m 4

an expression, in terms of t, for the distance of the particle from O at time [3]

an expression, in terms of t, for the distance of the particle from O at time [3] HORIZON EDUCATION SINGAPORE Additional Mathematics Practice Questions: Kinematics Set 1 1 A particle moves in a straight line so that t seconds after passing through O, its velocity v cm s -1, is given