Momentum and Impulse

Momentum is $p=mv$, measured in $\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-1}$ (equivalently N s). It is a $\text{vector}$, so direction matters: an object moving left has momentum of opposite sign to one moving right.

Choosing a positive direction and applying it consistently is essential in every momentum problem. A ball bouncing off a wall reverses its momentum, so its change is $\text{twice}$ its magnitude, not zero.

Newton’s second law in its general form is $F=\frac{\Delta p}{\Delta t}$: the resultant force equals the rate of change of momentum.

For constant mass this reduces to $F=ma$, since $\Delta p/\Delta t=m\Delta v/\Delta t$. But the momentum form also covers changing mass, such as a rocket expelling fuel or water hitting a surface.

$\text{Impulse}$ is force multiplied by the time it acts: $F\Delta t$. Rearranging the second law shows impulse equals change in momentum.

When the force varies, impulse is the $\text{area under a force–time graph}$. That is how sports scientists analyse a kick or a jump from force-plate data.

The same change in momentum can come from a large force over a short time or a small force over a long time. Safety features — airbags, crumple zones, crash mats, bending knees on landing — all increase $\Delta t$ to reduce the force for the same $\Delta p$.

In a closed system — no external resultant force — the total momentum before an interaction equals the total momentum after.

It follows from Newton’s third law: during a collision each object exerts an equal and opposite force on the other for the same time, so their momentum changes are equal and opposite and cancel in the total.

It applies to explosions too. A stationary system that breaks apart has zero total momentum, so the fragments move off with equal and opposite momenta — which is why a gun recoils.