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Grade 9 · Higher · free GCSE maths worksheet

Calculus in kinematics

Edexcel IGCSE

Velocity and acceleration by differentiating displacement, and greatest values.

10 original questions, 34 marks. Type your answers below for an instant mark and a score, 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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How to do it

  1. Velocity is the gradient of displacement: v = ds/dt, so differentiate s
  2. Acceleration is the gradient of velocity: a = dv/dt, so differentiate again
  3. For a value at a time, substitute t into the right expression (s, v or a)
  4. At rest means v = 0. Back at O means s = 0. Greatest value: set the derivative to 0
  5. Units: s in m, v in m/s, a in m/s^2. A negative v or a just means the opposite direction

Watch out: Substitute the time into v or a, not into s, and remember acceleration needs two differentiations, not one.

Worked example

A particle moves along a straight line through a fixed point O.

The displacement, s metres, of the particle from O at time t seconds is given by

s = t3 − 6t2 + 5t   for 0 ≤ t ≤ 5

  1. Find an expression for the velocity, v m/s, of the particle at time t seconds.
  2. Find the velocity of the particle when t = 3.
  3. Find the acceleration of the particle when t = 1.

a) v = dsdt, differentiate each term

v=3t2 - 12t + 5

v = 3t2 - 12t + 5

b) put t = 3 into v

v=3 × 32 - 12 × 3 + 5
=27 - 36 + 5

v = -4 m/s

c) a = dvdt, differentiate v

a=6t - 12
a=6 × 1 - 12

a = -6 m/s2

The questions

1

A particle moves along a straight line. At time t seconds its displacement, s metres, from a fixed point O is given by

s = 3t2 + 4t

  1. Find an expression for the velocity, v m/s, of the particle at time t seconds.
  2. Find the velocity of the particle when t = 2.
2 marks
Show answer
a) 6t + 4 b) 16
2

A particle moves along a straight line. At time t seconds its velocity, v m/s, is given by

v = 12t − 2t2

  1. Find an expression for the acceleration, a m/s2, of the particle at time t seconds.
  2. Find the acceleration of the particle when t = 5.
2 marks
Show answer
a) 12 - 4t b) -8
3

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

s = t3 − 6t2 + 9t + 2

  1. Find an expression for the velocity, v m/s, of P at time t seconds.
  2. Find an expression for the acceleration, a m/s2, of P at time t seconds.
  3. Find the acceleration of P when t = 4.
4 marks
Show answer
a) 3t2 - 12t + 9 b) 6t - 12 c) 12
4

A particle moves along a straight line. At time t seconds its displacement, s metres, from a fixed point O is given by

s = 2t3 − 15t2 + 24t   for 0 ≤ t ≤ 6

Find the values of t for which the particle is momentarily at rest.

3 marks
Show answer
t = 1 or t = 4
5

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

s = t3 − 9t2 + 5t   for t ≥ 0

  1. Find an expression for the acceleration, a m/s2, of P at time t seconds.
  2. Find the value of t when the acceleration of P is zero.
3 marks
Show answer
a) 6t - 18 b) t = 3
6

The number of bacteria, N, in a dish t hours after the start of an experiment is given by

N = 400 + 30t2 − t3   for 0 ≤ t ≤ 20

  1. Find an expression for dNdt.
  2. Find the rate at which the number of bacteria is increasing, in bacteria per hour, when t = 2.
3 marks
Show answer
a) 60t - 3t2 b) 108
7

Priya throws a ball straight up into the air. The height, h metres, of the ball above the ground t seconds after it leaves her hand is given by

h = 30t − 5t2

  1. Find the value of t when the ball reaches its greatest height.
  2. Find the greatest height, in metres, that the ball reaches.
4 marks
Show answer
a) t = 3 b) 45
8

A train moves along a straight track. At time t seconds its velocity, v m/s, is given by

v = 24t − 3t2   for 0 ≤ t ≤ 8

  1. Find an expression for the acceleration, a m/s2, of the train at time t seconds.
  2. Find the value of t when the velocity of the train is greatest.
  3. Find the greatest velocity of the train, in m/s.
4 marks
Show answer
a) 24 - 6t b) t = 4 c) 48
9

A particle P moves along a straight line. At time t seconds the displacement, s metres, of P from a fixed point O is given by

s = 6t2 − t3

P starts at O when t = 0 and later returns to O.

  1. Find the value of t when P returns to O.
  2. Find the velocity of P, in m/s, at the instant it returns to O.
4 marks
Show answer
a) t = 6 b) -36
10

A particle moves along a straight line through a fixed point O. At time t seconds the displacement, s metres, of the particle from O is given by

s = 2t3 − 21t2 + 60t   for 0 ≤ t ≤ 6

  1. Find the values of t for which the velocity of the particle is zero.
  2. Find the greatest displacement, in metres, of the particle from O during the 6 seconds.
5 marks
Show answer
a) t = 2 or t = 5 b) 52

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Every question here is original and written by Robinson Tuition. This topic is on the Pearson Edexcel International GCSE (4MA1) and is not examined on the UK GCSE; the grade is our estimate against the Edexcel grade descriptors. This is not an exam-board paper.

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