A particle moves in a straight line with initial speed 5 m/s and constant acceleration 3 m/s2. Find its speed after 4 seconds.
(Total for Question 1 is 1 mark)
2
A ball is thrown vertically upwards with speed 19.6 m/s. Using g = 9.8 m/s2, find the time taken to reach its greatest height.
(Total for Question 2 is 1 mark)
3
A particle passes point A with speed 6 m/s and accelerates uniformly at 2 m/s2. Find its speed 5 seconds later, at point B.
(Total for Question 3 is 1 mark)
4
A particle P moves along the x-axis with displacement x = t2 - 5t metres at time t seconds. Find the velocity of P when t = 0.
(Total for Question 4 is 1 mark)
5
A particle moves in a straight line with initial speed 4 m/s and constant acceleration 1.5 m/s2. Find the distance it travels in the first 6 seconds.
(Total for Question 5 is 2 marks)
6
A ball is thrown vertically upwards from ground level with speed 24.5 m/s. Using g = 9.8 m/s2, find the greatest height reached, giving your answer to 3 significant figures.
(Total for Question 6 is 2 marks)
7
Two runners pass the same point at the same instant, moving in the same direction along a straight track. Runner A moves at a constant speed of 5 m/s. The distance runner B has covered t seconds after passing the point is t2 metres. Find the value of t (other than t = 0) at which B catches up with A.
(Total for Question 7 is 2 marks)
8
A particle P moves along the x-axis. Its displacement from the origin at time t seconds (t ≥ 0) is x = t3 - 12t metres. Find the value of t at which P is instantaneously at rest.
(Total for Question 8 is 2 marks)
9
A particle has velocity v = 3sin(t) m/s at time t seconds. Find an expression for the acceleration of the particle at time t.
(Total for Question 9 is 2 marks)
10
A particle P moves in a straight line with velocity v = 5 - 2t m/s. State the initial velocity of P (at t = 0).
(Total for Question 10 is 1 mark)
11
A ball is thrown vertically upwards from a point 2 m above the ground with speed 12 m/s. Using g = 9.8 m/s2, find the greatest height above the ground reached by the ball, giving your answer to 3 significant figures.
(Total for Question 11 is 2 marks)
12
A particle P moves along the x-axis with acceleration a = 6t - 4 m/s2 at time t seconds. Given that P has velocity 3 m/s when t = 0, find an expression for the velocity of P at time t.
(Total for Question 12 is 2 marks)
13
A particle P moves along the x-axis. At time t seconds, the velocity of P is v = t2 - 4t m/s. Find the acceleration of P when t = 5 seconds.
(Total for Question 13 is 2 marks)
14
A particle decelerates from 20 m/s to rest in a straight line, taking 8 seconds, with constant deceleration. State the magnitude of the deceleration.
(Total for Question 14 is 1 mark)
15
A cyclist passes point A with speed 4 m/s and constant acceleration 0.6 m/s2. She passes point B, 10 m from A in the direction of motion, T seconds after passing A. Show that 3T2 + 40T - 100 = 0, and hence find the value of T, giving your answer to 3 significant figures.
(Total for Question 15 is 3 marks)
16
A stone is thrown vertically upwards from a point 5 m above the ground with speed 16 m/s. Using g = 9.8 m/s2, find the time taken for the stone to hit the ground, giving your answer to 3 significant figures.
(Total for Question 16 is 3 marks)
17
Two cars, A and B, pass the same road sign at the same instant, travelling in the same direction along a straight road. Car A passes the sign with speed 15 m/s and travels at this constant speed. Car B passes the sign from rest with constant acceleration 3 m/s2. Let t seconds be the time after the cars pass the sign. Find the value of t (other than t = 0) at which car B catches up with car A, and find the speed of car B at this instant.
(Total for Question 17 is 4 marks)
18
A particle P moves along the x-axis. At time t seconds (t ≥ 0), the velocity of P is v = t2 - 6t + 8 m/s. Given that P is at the origin when t = 0, find the values of t at which P is instantaneously at rest, and hence find the total distance travelled by P in the interval 0 ≤ t ≤ 5.
(Total for Question 18 is 4 marks)
19
A particle is projected vertically upwards from the top of a tower 20 m high, with initial speed 15 m/s. Taking the ground at the foot of the tower as the origin, with distances measured vertically upwards, and using g = 9.8 m/s2, show that the times at which the particle is at a height of 22 m above the ground satisfy 4.9t2 - 15t + 2 = 0, and hence find both times, giving your answers to 3 significant figures.
(Total for Question 19 is 4 marks)
Mark scheme · M5D Mechanics: Kinematics Depth: Fluency and Exam Drill
Question 1
B1 17 cao
Answer: 17 m/s
Question 2
B1 2 cao
Answer: 2 s
Question 3
B1 16 cao
Answer: 16 m/s
Question 4
B1 -5 cao
Answer: -5 m/s
Question 5
M1 s = ut + 1/2 a t2 with u=4, a=1.5, t=6
A1 51 m cao
Answer: 51 m
Question 6
M1 use v2 = u2 - 2gs with v=0, u=24.5
A1 awrt 30.6 m
Answer: 30.6 m (3 s.f.)
Question 7
M1 set the two distances equal: t2 = 5t
A1 t = 5 cao (rejecting t=0)
Answer: t = 5 s
Question 8
M1 differentiate and set v = 3t2 - 12 = 0
A1 t = 2 cao (rejecting the negative root as t ≥ 0)
Answer: t = 2 s
Question 9
M1 differentiate v with respect to t
A1 a = 3cos(t) cao
Answer: a = 3cos(t)
Question 10
B1 5 cao
Answer: 5 m/s
Question 11
M1 use v2 = u2 - 2gs with v=0, u=12 to find height risen above the point of projection, then add 2 m
A1 awrt 9.35 m
Answer: 9.35 m (3 s.f.)
Question 12
M1 integrate a with respect to t, then use v(0)=3 to find the constant
A1 v = 3t2 - 4t + 3 cao
Answer: v = 3t2 - 4t + 3
Question 13
M1 differentiate: a = 2t - 4, substitute t=5
A1 6 m/s2 cao
Answer: 6 m/s2
Question 14
B1 2.5 cao
Answer: 2.5 m/s2
Question 15
M1 use s = ut + 1/2 a t2 with u=4, a=0.6, s=10 to form 10 = 4T + 0.3T2, then multiply by 10 to give the printed equation
M1 apply the quadratic formula to 3T2 + 40T - 100 = 0 and reject the negative root
A1 awrt 2.15 s
Answer: T = 2.15 s (3 s.f.)
Question 16
M1 use s = ut - 1/2 g t2 with s=-5 (taking upward as positive, ground 5 m below the start): -5 = 16t - 4.9t2
M1 form 4.9t2 - 16t - 5 = 0 and apply the quadratic formula, rejecting the negative root
A1 awrt 3.55 s
Answer: 3.55 s (3 s.f.)
Question 17
M1 write distances travelled: A is 15t, B is 1.5t2 (using s = 1/2 a t2 with u=0, a=3)
M1 set 1.5t2 = 15t and solve, rejecting t=0
A1 t = 10 s cao
A1 speed of B = 3 x 10 = 30 m/s cao
Answer: t = 10 s; speed of B = 30 m/s
Question 18
M1 set v=0 and solve: (t-2)(t-4)=0
A1 t = 2 and t = 4 cao
M1 integrate v to find x = t3/3 - 3t2 + 8t (using x(0)=0), and evaluate at t=0,2,4,5
A1 total distance = |20/3 - 0| + |16/3 - 20/3| + |20/3 - 16/3| = awrt 9.33 m
Answer: t = 2 s and t = 4 s; total distance = 9.33 m (3 s.f.)
Question 19
M1 height above ground = 20 + 15t - 4.9t2, set equal to 22
A1 cso: rearrange 15t - 4.9t2 = 2 to give 4.9t2 - 15t + 2 = 0