# Momentum & impulse

Momentum is a quantity of motion defined by

$\displaystyle \vec{p} = m \vec{v}$,

measured in kilogram-metres per second (kg⋅m/s). A change in momentum is called impulse, and it is defined by

$\displaystyle \Delta{} \vec{p} = m \Delta{} \vec{v} = \vec{F} \Delta{} t$.

A lightweight, fast-moving object can have the same momentum as a heavy, slow-moving object because mass and velocity are multiplied. Similarly, a large force applied over a short time interval can deliver the same impulse as a small force applied over a long time interval.

Just as the area under an acceleration-time graph represents $\displaystyle \Delta{} \vec{v}$, so too the area under a force-time graph represents $\displaystyle \Delta{} \vec{p}$.

## Example

What average force is needed to stop a 34 kg ball in 2.5 s if the initial speed of the ball is 19 m/s [fwd]?

Since $\displaystyle \vec{F} \Delta{} t = m \Delta{} \vec{v}$,

$\displaystyle \vec{F} = \frac{m \Delta{} \vec{v}}{\Delta{} t} = \frac{\vec{v}_{2} - \vec{v}_{1}}{\Delta{} t} = \frac{\left ( 34 \, \text{kg} \right ) \left ( 0 - 19 \, \text{m/s} \right )}{2.5 \, \text{s}} = - 258.4 \, \text{N}$,

therefore an average force of 260 N [bwd] is required.