Impulse and force-momentum changeEdexcel International A Level Physics: Revision notes

The momentum of an object is $p = mv$, a vector in kg m s⁻¹. When a force acts, momentum changes. The impulse of a force is the force multiplied by the time it acts:

$\text{impulse} = Ft = \Delta p$

Impulse is measured in N s, which is the same as kg m s⁻¹.

Newton's second law in terms of momentum states that the resultant force equals the rate of change of momentum: $F = \dfrac{\Delta p}{\Delta t}$.

It follows from $F = ma$ with $a = \dfrac{v-u}{t}$:

$F = \dfrac{m(v-u)}{t} = \dfrac{mv - mu}{t}$, so $Ft = mv - mu = \Delta p$

This is the form used when the force is only known as an average over a collision.

For a fixed change of momentum $\Delta p$, the average force is $F = \Delta p / t$. Increasing the time of impact reduces the force. Safety features use this:

• Airbags and seat belts stretch the stopping time
• Crumple zones extend the time over which a car stops
• Cushioned landing surfaces and bending the knees reduce force on the body

In all cases $\Delta p$ is the same; only $t$ changes.

A 0.057 kg ball arrives at 25 m s⁻¹ and leaves in the opposite direction at 30 m s⁻¹. Contact lasts 5.0 ms.

• $\Delta p = 0.057 \times (30 + 25) = 3.1$ kg m s⁻¹
• $F = \Delta p / t = 3.135 \div 0.0050 = 6.3\times10^{2}$ N

The racket force is directed opposite to the initial motion of the ball.

A trolley on a runway is pulled by a hanging weight on a string. Two light gates (or a motion sensor) and a data logger give the speeds $u$ and $v$ and the time $t$ between them.

1. Calculate $\Delta p = m(v - u)$.
2. Change the force by moving masses between the trolley and the hanger, keeping the total mass constant.
3. Keep the time $t$ the same each run.
4. Plot $\Delta p$ against $F$: a straight line through the origin shows $\Delta p \propto F$ and the gradient is $t$.

Tilt the runway slightly to compensate for friction.