Impulse and force–time graphs

Impulse is the effect of a force acting for a time: force × time. It equals the change in momentum. Spreading a change in momentum over a longer time means a smaller force, which is the idea behind airbags, crumple zones, crash mats and bending your knees when you land.

Key facts

  • Impulse = force × time = FΔt (N s).
  • Impulse = change in momentum: FΔt = mv − mu. (1 N s = 1 kg m/s.)
  • Force = rate of change of momentum: F = Δp / Δt.
  • For the same change in momentum, a longer contact time gives a smaller force.
  • Impulse = area under a force–time graph.
  • Rebounds: the velocity reverses, so the change in velocity is u + v.

Worked examples

Example 1

A racket pushes a 0.058 kg ball from rest to 40 m/s in 0.005 s. Find the impulse and the average force.

  1. After the hit the ball moves at 40 m/s. Δp = 0.058 × 40 − 0 = 2.32 N s
  2. Force = impulse ÷ time. F = 2.32 ÷ 0.005 = 464 N

Answer: 2.32 N s; 464 N

Example 2

A 70 kg driver stops from 15 m/s. Compare the force if he stops in 0.02 s (hitting the dashboard) with 0.15 s (airbag).

  1. The change in momentum is the same either way: it is the area under each force–time graph. Δp = 70 × 15 = 1050 N s
  2. Dashboard: a big force for a short time. 1050 ÷ 0.02 = 52 500 N
  3. Airbag: the same area, spread over 7.5 times as long, so 7.5 times less force. 1050 ÷ 0.15 = 7000 N

Answer: 52 500 N vs 7000 N

Example 3

A force–time graph for a kick is a triangle with a peak of 600 N lasting 0.010 s. The ball has mass 0.45 kg and starts at rest. Find its speed.

  1. Impulse = area of the triangle. ½ × 0.010 × 600 = 3.0 N s
  2. Impulse = change in momentum, from rest. v = 3.0 ÷ 0.45 = 6.67 m/s

Answer: 6.7 m/s