Dynamics

Impulse Equals the Change in Momentum

Impulse J=F·Δt=Δp, momentum conservation, F=J/Δt (airbag), collision

Two objects collide. The impulse, force built up over the time it acts, equals the change in momentum mv, and with no outside force the total momentum is conserved.

Drag the time Δt. The impulse J = F·Δt is the area under the F-t graph, growing the longer the force acts.

Apply impulse J and the momentum p = mv changes by exactly that much (J = Δp). Drag J and watch the momentum bar grow.

Two carts push each other apart with a spring. With no outside force the total momentum stays zero, so the heavier one moves slower (m₁v₁ = m₂v₂). Drag the mass ratio.

Deliver the same impulse in a short time and the force is large; in a long time, small (F = JΔt). Drag the contact time Δt. This is how airbags and bending your knees soften an impact.

Pick a collision. Whether they stick or bounce, the total momentum before and after is the same. But the kinetic energy drops if they stick (inelastic) and is conserved if perfectly elastic.

In PracticeTo sum up, impulse J = F·Δt = Δp is force accumulated over time, equal to the change in momentum p = mv. If energy was force times distance, momentum is force times time. In a system with no outside force the total momentum is conserved, making it a powerful tool that links before and after a collision or explosion. For the same impulse, lengthening the contact time lowers the force (the airbag). Solve collisions like this: momentum is always conserved when there is no outside force, while kinetic energy is conserved if elastic and lost if inelastic; using both gives the speeds afterward. In the next lesson we dig deeper into collisions and quantify how elastic they are with the coefficient of restitution.

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