Momentum & impulse

Momentum is a quantity of motion defined by

p⃗=mv⃗\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

Δp⃗=mΔv⃗=F⃗Δt\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 Δv⃗\displaystyle \Delta{} \vec{v}, so too the area under a force-time graph represents Δp⃗\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 F⃗Δt=mΔv⃗\displaystyle \vec{F} \Delta{} t = m \Delta{} \vec{v},

F⃗=mΔv⃗Δt=v⃗2−v⃗1Δt=(34 kg)(0−19 m/s)2.5 s=−258.4 N\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.