Momentum & impulse

Momentum is a quantity of motion defined by

$\overrightarrow{p}=m\overrightarrow{v}\text{,}$

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

$\mathrm{\Delta }\overrightarrow{p}=m\mathrm{\Delta }\overrightarrow{v}=\overrightarrow{F}\mathrm{\Delta }t\text{.}$

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 $\mathrm{\Delta }\overrightarrow{v}\text{,}$ so too the area under a force-time graph represents $\mathrm{\Delta }\overrightarrow{p}\text{.}$

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 $\overrightarrow{F}\mathrm{\Delta }t=m\mathrm{\Delta }\overrightarrow{v}\text{,}$

$\overrightarrow{F}=\frac{m\mathrm{\Delta }\overrightarrow{v}}{\mathrm{\Delta }t}=\frac{{\overrightarrow{v}}_2-{\overrightarrow{v}}_1}{\mathrm{\Delta }t}=\frac{(34\text{kg})(0-19\text{m/s})}{2.5\text{s}}=-258.4\text{N}\text{,}$

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