Impulse

Impulse ($\vec{J}$) is the change in Momentum caused by a force acting over time.

For a constant force:

$$\vec{J}=\vec{F}\Delta t$$

More generally:

$$\vec{J}={\int }_{t_i}^{t_f}\vec{F}(t)dt=\Delta \vec{p}$$

This is the impulse-momentum theorem. It is useful when the force is brief or varies quickly, such as a bat hitting a ball, a foot striking the ground, or a car crumple zone during a crash.

Intuition

The same change in momentum can be produced by:

• a large force for a short time, or
• a smaller force for a longer time.

That is why padding helps in collisions: it increases stopping time, reducing average force for the same momentum change.

Graph view

On a force-time graph, impulse is the area under the curve.

Unit: Newton-Second, the same as momentum.

Problem habit

For collision questions, define the system before doing algebra. If two objects collide and external forces are negligible during the short impact, total Momentum is conserved even though each individual object’s momentum changes. Impulse is best for one object; conservation of momentum is best for the whole interacting system.