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

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1 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 by v = t 2 6t + 5. The particle comes to instantaneous rest, firstly at A and then at B. (i) the acceleration of the particle where t = 3 [2] an expression, in terms of t, for the distance of the particle from O at time t [3] (iii) the distance AB [3] (iv) the total distance travelled in the first 5 seconds after passing through O. [1] 2 A vehicle moves along a straight road with velocity, v m s -1, given by the equation v = 8t 1 2 t2 + 4, where t is the time after passing a fixed point M. On reaching its maximum velocity, the vehicle starts to decelerate constantly at 6 ms -1, become coming to rest at the point N. For the journey from M to N, (a) find the maximum velocity reached by the vehicle, [3] (b) sketch and label clearly the velocity-time graph of the vehicle, [3] (c) find the total distance travelled by the vehicle. [3] 3 A particle P moves in a straight line so that, t seconds after passing through a fixed point O, its displacement s m is given by s = t 3 9t t. (a) the maximum speed attained by the particle P. (b) the total distance travelled by P during the first 6 seconds. (c) Sketch the velocity-time graph of P for the first 6 seconds. [7] 4 A particle start from rest travels in a straight line so that, t seconds after passing a fixed point O, its velocity v m/s, is given by v = 12 3t 2. The particles comes to instantaneous rest at A. (a) the acceleration of particles at A, [3] (b) the greatest velocity of the particle, [2] (c) the total distance travelled by the particle between t = 0 and t = 5. [4] Page 1 of 6

2 5 A particle moves in a straight line so that at time t seconds after leaving a fixed point O, its velocity v m s -1, is given by v = rt 2 + st, where r and s are constants. Given that at the instant when t = 1, the acceleration of the particle is zero and its velocity is 3 m s -1, calculate (i) the values of r and of s. [4] the total distance travelled by the particles during the interval from t = 3 to t = 3. [3] 6 A particle moving in a straight line passes after a fixed point O on the line with a velocity 15 m s -1. The acceleration, a m s -2, of the particles, t seconds after passing O is given by a = 17 8t. (a) the time taken to reach the maximum distance from O in the direction of the initial motion, [3] (b) the value of this maximum distance, [1] (c) the total distance travelled in the first 6 seconds. [2] 7 A particle moves in a straight line from a point O such that, t seconds after leaving O, its velocity v m/s is given by v = 5 e 2t. (i) the velocity of the particles when its acceleration is 0.5 m/s 2, [3] the distance travelled by the particle in the first 6 seconds, [3] (iii) the maximum possible velocity that the particle will tend to attain when t is large. Sketch the velocity-time graph to justify your answer. [3] 8 A particle moves in a straight line from a point O such that t seconds after leaving O, its velocity v m s -1, is given by v = 4 e 2t. (i) the initial velocity of the particle and state the value v tends to as t becomes very large. [2] Show that the acceleration of the particle is always positive. [1] (iii) the distance travelled by the particle in the first two seconds. [2] (iv) Sketch the velocity time graph of the particle. [2] Page 2 of 6

3 9 A particle P moves in a straight line so that its displacement, s m from a fixed point A is given by s = 4t 2 t 3, where t is the time in seconds after leaving A. (i) the time when it is next to A, [1] the value of t when it is instantaneously at rest, [2] (iii) the acceleration when t = 5, [2] (iv) the total distance travelled in the first 5 seconds, [2] (v) the maximum velocity of P. [2] 10 A particle starts from rest at a fixed point O and moves in a straight line so that its velocity, v m/s, is given by v = e 2t 4. (a) Show that the distance, s, from O is such that s = 1 2 e2t 4t 1 2. (b) the time when the particle is instantaneously at rest and hence find the total distance travelled during the first 2 seconds. [6] 11 A particle passes a fixed point P and moves in a straight line with a velocity, v = 8 (1 e 1 2 t ) m s -1, where t is the time in seconds after leaving P., (i) the velocity when its acceleration is 2 m s -2, [3] the distance travelled during the fourth second of motion, [3] (iii) state the value which v approaches as t becomes very large. [1] (iv) sketch the velocity-time curve. [2] 12 A particle, moving in straight line, starts from rest from a fixed point O. The acceleration, a m s -2, of the particle, t seconds after passing through O, is given by a = 4e 0.5t. (i) the velocity of the particle when t = 6. [3] Sketch the velocity-time graph for the motion for t 0, and use your graph to explain why the particle will never return to O. [2] (iii) the average speed of the particle over the first five seconds. [4] Page 3 of 6

4 13 A particle P moves in a straight line so that, t seconds after passing a fixed point O, its velocity v m s -1, is given by v = 1 2 t2 3t + 5. (i) the speed when its acceleration is 1 m/s 2, the values of t when P is instantaneously at rest, (iii) the distance travelled in the first 3 seconds. [8] 14 A particle moves in a straight line such that t seconds after passing through a fixed point O, the velocity v m/s is given by v = 12t 3t 2 9. (i) its initial acceleration (iii) the values of t when it is instantaneously at rest the interval of time during which its speed is decreasing (iv) the distance travelled in the first 2 seconds [10] 15 A particle moves in a straight line so that t seconds after passing through a fixed point O, its velocity, v cm/s, is given by v = e 3t 7e 2t. (a) the time at which the particle is instantaneously at rest, [3] (b) an expression for the displacement S, [3] (c) the initial velocity and acceleration of the particle. [3] 16 A particle moves along a straight line form a fixed point P. Its velocity, V m/s is given by V = 2t(6 t), where t is the time, in seconds, after leaving the point P. (Assume motion to the right is positive). : (a) the maximum velocity of the particle, [3] (b) the distance travelled during the third second, [3] (c) the value of t when the particle passes the point P again, [2] (d) the time between leaving P and when the particle reverses its direction of motion. [2] Page 4 of 6

5 17 A particle P travels in a straight line so that its displacement, s cm, from a fixed point O is given by s = 3t + 12, where t is the time in seconds after leaving the t+1 starting point A. (i) the initial velocity of P, [2] the velocity of P when it is next at A, [2] (iii) the minimum displacement of P from O, [2] (iv) the total distance travelled during the first 3 seconds. [2] 18 A particle moves in a straight line from a point O such that, t seconds after leaving O, its velocity v m s -1, is given by v = 10(1 e t ). (i) the acceleration of the particle when t = 2, [2] the displacement of the particle from O when t = 2, [3] (iii) the maximum possible velocity that the particle will tend to attain when t is very large. [1] 19 A particle travelling in a straight line, passes a fixed point A on the line with a speed of 5 m s 1. The acceleration, a m s 2, of the particle, t s after passing A, is given by a = 4e 2t. At t = 3, the particle travels at a constant deceleration until it comes to rest 5 seconds later at B. (i) the velocity of the particle when t = 3. [3] Sketch the velocity-time graph of the particle for 0 t 8. [2] (iii) the distance AB. [4] 20 A particle moves in a straight line so that, t seconds after leaving a fixed point O, its velocity, v m s 1, is given by v = t 3 7t t. (i) the values of t when the particle is instantaneously at rest, [2] the total distance travelled in the interval 0 t 4. [4] Page 5 of 6

6 21 A particle passes a fixed point O with a velocity of 7 m s 1 and moves in a straight line with an acceleration of 2(3 t) m s 2, where t is the time in seconds after passing O. (a) an expression in terms of t for its (i) its velocity, [2] its displacement, [2] (b) the value of t at which the particle is instantaneously at rest, and [2] (c) the distance travelled when the velocity is again at 7 m s 1. [3] END Page 6 of 6

Created by T. Madas CALCULUS KINEMATICS. Created by T. Madas

Created by T. Madas CALCULUS KINEMATICS. Created by T. Madas CALCULUS KINEMATICS CALCULUS KINEMATICS IN SCALAR FORM Question (**) A particle P is moving on the x axis and its acceleration a ms, t seconds after a given instant, is given by a = 6t 8, t 0. The particle