Impulse and Force-Time Relationship

Impulse connects force with the change in momentum of an object. Newton’s second law can be written as:

$F = \frac{mv - mu}{t}$

Rearranging:

$Ft = mv - mu$

The product $Ft$ is called the impulse. It equals the change in momentum.

Symbol	Quantity	Unit
$F$	Resultant force	N
$t$	Time of contact	s
$mv - mu$	Change in momentum	kg m/s

Safety applications

The equation shows that for a given change in momentum, increasing the time of impact reduces the force. This is the principle behind:

Safety feature	How it works
Crumple zones	Car body deforms slowly, increasing impact time
Airbags	Inflate to slow the passenger over a longer time
Helmets	Padding compresses to extend the stopping time
Bending knees on landing	Increases the time over which the body decelerates

Worked example

A 0.40 kg ball hits a wall at 12 m/s and bounces back at 8.0 m/s. The contact time is 0.050 s. Calculate the average force on the ball.

Taking the initial direction as positive:

$F = \frac{mv - mu}{t} = \frac{0.40 \times (-8.0) - 0.40 \times 12}{0.050}$ $F = \frac{-3.2 - 4.8}{0.050} = \frac{-8.0}{0.050} = -160\text{ N}$

The force is 160 N in the direction of the bounce (away from the wall).

Common errors and how to correct them

Forgetting the direction change in a bounce. The velocity reverses, so $mv - mu$ involves a sign change, which makes the change in momentum larger than if the ball simply stopped.

Confusing impulse (force x time) with moment (force x distance). Impulse changes momentum; moment causes rotation. They have different units.

How to apply this in an exam

When explaining safety features, state: the feature increases the time of impact, and for the same change in momentum this reduces the force ($F = \Delta p / t$). This two-step reasoning is required for full marks.