Impulse and momentum

Momentum as vector motion

Linear momentum is defined as

ec{p}=mec{v}

for a particle of mass $m$ and velocity ec{v}. Momentum is a vector, so direction matters. A fast object and a massive object both have large momentum.

Momentum is useful because Newton's second law is most generally written in terms of momentum:

ec{F}_{net}=rac{dec{p}}{dt}

For constant mass, this reduces to ec{F}_{net}=mec{a}. But the momentum form emphasizes that forces change momentum.

Impulse

Impulse is the accumulated effect of a force over a time interval:

ec{J}=int_{t_i}^{t_f}ec{F}(t),dt

If the force is approximately constant, this becomes

ec{J}=ec{F}_{avg}Delta t

Impulse has the same units as momentum: $Ncdot s$ or $kgcdot m/s$.

Impulse-momentum theorem

Integrating Newton's second law over time gives

int_{t_i}^{t_f}ec{F}*{net},dt=int*{t_i}^{t_f}rac{dec{p}}{dt},dt

Therefore

ec{J}_{net}=Deltaec{p}

This is the impulse-momentum theorem. It says net impulse equals change in momentum.

Force-time graphs

Impulse is the area under a force-time graph. In a collision, force may vary rapidly and irregularly. Even if the exact force is complicated, the impulse can be found from the area under the curve or from the momentum change.

Average force is defined so that

ec{F}_{avg}Delta t=Deltaec{p}

This is very useful when the detailed time dependence of force is unknown.

Safety applications

The impulse-momentum theorem explains many safety devices. To stop a moving person, car, or ball, the momentum must change. If the stopping time is increased, the average force decreases:

F_{avg}=rac{Delta p}{Delta t}

Airbags, helmets, padded mats, seat belts, and crumple zones increase the time and distance over which momentum changes. The change in momentum may be the same, but the force becomes less extreme.

Bouncing versus sticking

A bouncing object often experiences a larger momentum change than an object that simply stops. If a ball moving toward a wall rebounds with similar speed, its momentum changes direction. The magnitude of $Delta p$ can be larger than if the ball came to rest.

This is why catching a ball and bringing it to rest gently requires less force than bouncing it sharply back.

Momentum and kinetic energy

Momentum and kinetic energy are related but different:

K=rac{p^2}{2m}

for a particle of mass $m$. Momentum is a vector and can cancel in a system. Kinetic energy is a scalar and cannot be negative. In collisions, momentum is conserved under broader conditions than kinetic energy.

The big idea

Momentum measures vector motion. Impulse measures force accumulated over time. The impulse-momentum theorem connects them: net impulse equals change in momentum. This provides a powerful way to understand collisions, impacts, sports, vehicle safety, and force-time behavior.