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

Further kinematics

Constant acceleration with vectors in 2D, variable acceleration with vectors (differentiating and integrating i j expressions), relative position, velocity and acceleration.

12 original questions, 86 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or 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

[In this question i and j are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin O.]

A particle P moves with constant acceleration (2i − 3j) m s−2. At time t = 0 its velocity is (4i + j) m s−1.

  1. Find the velocity of P when t = 3
  2. Find the speed of P when t = 3, giving your answer as an exact surd.
4 marks
Show answer
a) (10i - 8j) m s-1 b) 2√41 m s-1 (12.8)
2

[In this question i and j are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin O.]

A particle moves with constant acceleration (0.5i + j) m s−2. At time t = 0 it is at the point with position vector (2i + 5j) m and has velocity (3i − 2j) m s−1.

  1. Find the position vector of the particle when t = 4
  2. Find the distance of the particle from O when t = 4
4 marks
Show answer
a) (18i + 5j) m b) √349 = 18.7 m
3

[In this question i and j are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin O.]

At time t seconds, the position vector of a particle P is r metres, where

r = (t3 − 4t)i + (2t2 + 1)j

  1. Find the speed of P when t = 2
  2. Find the acceleration of P when t = 2
5 marks
Show answer
a) 8√2 m s-1 (11.3) b) (12i + 4j) m s-2
4

[In this question i and j are horizontal unit vectors due east and due north respectively, and position vectors are given relative to a fixed origin O.]

A particle P moves with constant acceleration. At time t = 0 the velocity of P is (−3i + 6j) m s−1 and at time t = 4 seconds its velocity is (5i − 2j) m s−1.

  1. Find the acceleration of P.
  2. Find the value of t when P is moving parallel to i.
  3. Find the value of t when P is moving north-east.
7 marks
Show answer
a) (2i - 2j) m s-2 b) t = 3 c) t = 9/4
5

[In this question i and j are horizontal unit vectors due east and due north respectively, and position vectors are given relative to a fixed origin O.]

A particle P moves with constant acceleration (−2i + j) m s−2. At time t = 0, P is at the point with position vector (−10i + 4j) m and is moving with velocity (7i − 2j) m s−1.

  1. Find the position vector of P at time t seconds.
  2. Find the two values of t for which P is due north of O.
  3. Show that at the first of these times P is moving due east.
7 marks
Show answer
a) r = (-10 + 7t - t2)i + (4 - 2t + ½t2)j b) t = 2 and t = 5 c) v = 3i at t = 2, so due east
6

[In this question i and j are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin O.]

At time t seconds, where t ≥ 0, a particle P has velocity v m s−1, where

v = (t2 − 4t + 3)i + (2t + 1)j

  1. Find the values of t when P is moving parallel to j.
  2. Find the magnitude of the acceleration of P when t = 4
  3. Find the displacement of P from t = 0 to t = 3
8 marks
Show answer
a) t = 1 and t = 3 b) 2√5 m s-2 (4.47) c) 12j m
7

[In this question i and j are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin O.]

At time t seconds, where t ≥ 0, a particle P has acceleration a m s−2, where

a = 3t12i − 2j

When t = 0, P is at O and is moving with velocity (4i + 5j) m s−1.

  1. Find the speed of P when t = 4
  2. Find the position vector of P when t = 4
8 marks
Show answer
a) √409 = 20.2 m s-1 b) (41.6i + 4j) m
8

[In this question i and j are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin O.]

Two particles A and B move on a smooth horizontal surface with constant velocities. At time t = 0, A is at the point with position vector (2i + j) m and B is at the point with position vector (10i − 3j) m. The velocity of A is (3i + j) m s−1 and the velocity of B is (−i + 3j) m s−1.

  1. Find the vector AB in terms of t.
  2. Show that A and B collide, and find the position vector of the point where they collide.
  3. Find the exact distance between A and B when t = 0.5
7 marks
Show answer
a) AB = (8 - 4t)i + (2t - 4)j b) AB = 0 at t = 2; (8i + 3j) m c) 3√5 m (6.71 m)
9

[In this question i and j are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin O.]

A particle P moves with constant acceleration (2i − j) m s−2. At time t = 0, P passes through O with velocity (−4i + 2j) m s−1. At time T seconds, where T > 0, P passes through the point C with position vector (λi − 6j) m, where λ is a constant.

  1. Find the value of T.
  2. Find the value of λ.
  3. Find the exact speed of P as it passes through C.
8 marks
Show answer
a) T = 6 b) λ = 12 c) 4√5 m s-1
10

[In this question i and j are horizontal unit vectors due east and due north respectively, and position vectors are given relative to a fixed origin O.]

At time t seconds, where t ≥ 0, the position vector of a particle P is r metres, where

r = (t3 − 3t)i + (5t − t2)j

  1. Find the velocity of P at time t.
  2. Find the value of t when P is moving north-east.
  3. Find the distance of P from O when P is due east of O.
  4. Find the magnitude of the acceleration of P at this time.
9 marks
Show answer
a) v = (3t2 - 3)i + (5 - 2t)j b) t = 4/3 c) 110 m d) √904 = 30.1 m s-2
11

[In this question i and j are horizontal unit vectors due east and due north respectively, and position vectors are given relative to a fixed origin O.]

At noon, ship A is at O and ship B is at the point with position vector (20i + 4j) km. Ship A moves with constant velocity (6i + 8j) km h−1 and ship B moves with constant velocity (2i + 10j) km h−1. At time t hours after noon the position vectors of A and B are rA km and rB km.

  1. Show that the position vector of B relative to A is (20 − 4t)i + (4 + 2t)j
  2. Find the time at which B is due north of A, and the distance between the ships at that time.
  3. Find, using calculus, the time at which the ships are closest together and the least distance between them.
9 marks
Show answer
a) r_B - r_A = (20 - 4t)i + (4 + 2t)j b) t = 5 (5 pm), 14 km c) t = 3.6 (3:36 pm), 12.5 km
12

[In this question i and j are horizontal unit vectors due east and due north respectively, and position vectors are given relative to a fixed origin O.]

At time t seconds, where t > 0, a particle P has velocity v m s−1, where

v = 6t12i + (2t − 5)j

  1. Find the magnitude of the acceleration of P when t = 4
  2. P is at the point A when t = 1 and at the point B when t = 4. Find the exact distance AB.
  3. When t = 1, the position vector of P is (−2i + 3j) m. Find the position vector of P at time t.
  4. Show that P is never due east of O.
10 marks
Show answer
a) 2.5 m s-2 b) 28 m c) r = (4t3/2 - 6)i + (t2 - 5t + 7)j d) t2 - 5t + 7 has discriminant -3 < 0, so the j component is never 0

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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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