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AS Mechanics · Edexcel 8MA0 · free A-level maths worksheet

Variable acceleration

Differentiating and integrating displacement, velocity and acceleration in one dimension, maximum velocity, distance travelled.

12 original questions, 88 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

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The questions

1

A particle P moves along a straight line. At time t seconds, t ≥ 0, the displacement of P from a fixed point O on the line is x metres, where

x = 2t3 − 9t2 + 12t + 1

  1. Find an expression for the velocity of P at time t seconds.
  2. Find the two times at which P is instantaneously at rest.
  3. Find the acceleration of P when t = 3
  4. Find the displacement of P from O when t = 2
6 marks
Show answer
a) v = 6t2 - 18t + 12 b) t = 1 and t = 2 c) 18 m s-2 d) 5 m
2

A particle moves along the x-axis. At time t seconds the velocity of the particle is v m s−1 in the positive x direction, where

v = 3t2 − 4t + 2,   t ≥ 0

When t = 0 the particle is at the origin O.

  1. Find the displacement of the particle from O when t = 3
  2. Find the acceleration of the particle when t = 3
  3. Show that the particle is never at rest.
6 marks
Show answer
a) 15 m b) 14 m s-2 c) b2 - 4ac = -8 < 0, so v is never 0 (shown)
3

A particle moves in a straight line. At time t seconds its acceleration is a m s−2, where

a = 6 − 2t,   t ≥ 0

When t = 0 the velocity of the particle is 4 m s−1.

  1. Find an expression for the velocity of the particle at time t seconds.
  2. Find the maximum velocity of the particle.
  3. Find the time at which the particle comes to instantaneous rest, giving your answer to 3 significant figures.
6 marks
Show answer
a) v = 4 + 6t - t2 b) 13 m s-1 c) t = 3 + √13 = 6.61 s
4

A cyclist rides along a straight road between two sets of traffic lights. She starts from rest at the first set and stops at the second set. Her velocity t seconds after leaving the first set is v m s−1, where

v = 8t − 2t2,   0 ≤ t ≤ 4

  1. Sketch a velocity-time graph for the motion of the cyclist on the axes below.
  2. Find the maximum speed of the cyclist.
  3. Find the distance between the two sets of traffic lights.
012340123456789t (s)v (m/s)
7 marks
Show answer
a) n-shaped curve from (0, 0) to (4, 0), maximum at (2, 8) b) 8 m s-1 c) 643 = 21.3 m
5

A particle P moves along a straight line. At time t seconds the velocity of P is v m s−1, where

v = t2 − 5t + 4,   t ≥ 0

When t = 0, P is at the point O.

  1. Find the values of t at which P is instantaneously at rest.
  2. Find the displacement of P from O when t = 5
  3. Find the total distance travelled by P in the interval 0 ≤ t ≤ 5
7 marks
Show answer
a) t = 1 and t = 4 b) -56 m c) 496 = 8.17 m
6

A car moves along a straight road. It starts from rest at time t = 0. For 0 ≤ t ≤ 4 its velocity is v m s−1, where

v = 6t − 34t2

For t > 4 the car moves at a constant speed equal to its speed when t = 4

  1. Show that the acceleration of the car is zero when t = 4
  2. Find the constant speed of the car for t > 4
  3. Find the distance travelled by the car in the first 10 seconds.
7 marks
Show answer
a) a = 6 - 1.5(4) = 0 (shown) b) 12 m s-1 c) 104 m
7

A particle P moves along a straight line through the fixed point O. At time t seconds the displacement of P from O is x metres, where

x = 12t − t3,   t ≥ 0

  1. Find the greatest positive displacement of P from O.
  2. Find the exact value of t when P returns to O.
  3. Find the speed of P as it passes through O at this time.
  4. Find the total distance travelled by P in the first 4 seconds.
8 marks
Show answer
a) 16 m b) t = 2√3 c) 24 m s-1 d) 48 m
8

A particle P moves along a straight line. At time t seconds the acceleration of P is (6t − 8) m s−2. When t = 0, P passes through the fixed point O with velocity 4 m s−1.

  1. Find an expression for the displacement of P from O at time t seconds.
  2. Show that when P first returns to O it is instantaneously at rest.
  3. Find the greatest distance of P from O in the interval 0 ≤ t ≤ 2
8 marks
Show answer
a) x = t3 - 4t2 + 4t b) x = t(t - 2)2 = 0 at t = 2, and v(2) = 0 (shown) c) 3227 m
9

A lift moves vertically upwards from rest at ground level. It comes to rest again at the top floor 6 seconds later. While moving, the velocity of the lift t seconds after it starts is v m s−1, where

v = kt(6 − t),   0 ≤ t ≤ 6

and k is a positive constant. The top floor is 24 m above ground level.

  1. Find the value of k.
  2. Find the maximum speed of the lift.
  3. Find the magnitude of the acceleration of the lift when t = 5
7 marks
Show answer
a) k = 23 b) 6 m s-1 c) 83 = 2.67 m s-2
10

Two cyclists, Priya and Tom, ride along a straight cycle track. At time t = 0 Priya passes a marker post O with speed 2 m s−1, and she moves with constant acceleration 2 m s−2. At the same moment Tom starts from rest at O and rides in the same direction. Tom's velocity at time t seconds is 3t2 m s−1. The model is valid for 0 ≤ t ≤ 3

  1. Find, in terms of t, the distance of each cyclist from O.
  2. Find the time at which Tom catches up with Priya, and how far they are from O at that moment.
  3. Find the time at which the two cyclists have the same speed, giving your answer to 3 significant figures.
8 marks
Show answer
a) Priya: t2 + 2t, Tom: t3 b) t = 2 s, 8 m from O c) 1.22 s
11

A train leaves a station from rest at time t = 0. For the first 30 seconds its acceleration is a m s−2, where

a = 0.6 − 0.02t,   0 ≤ t ≤ 30

  1. Find the speed of the train when t = 30
  2. Find the distance travelled by the train in the first 30 seconds.

The train then travels at this speed for 2 minutes. It then decelerates uniformly at 0.25 m s−2 until it comes to rest at the next station.

  1. Find the distance between the two stations.
  2. Find the average speed of the train between the two stations.
9 marks
Show answer
a) 9 m s-1 b) 180 m c) 1422 m (1420 m to 3 s.f.) d) 7.65 m s-1
12

A particle P moves along the x-axis. At time t seconds the velocity of P is v m s−1 in the positive x direction, where

v = t2 − 6t + 8,   t ≥ 0

  1. Find the interval of time during which P is moving in the negative x direction.
  2. Find the values of t at which the speed of P is 3 m s−1, and explain why there are only two such values.
  3. Find the total distance travelled by P in the interval 0 ≤ t ≤ 5
9 marks
Show answer
a) 2 < t < 4 b) t = 1 and t = 5; the least velocity is -1, so v = -3 never happens c) 283 = 9.33 m

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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