Change in Momentum and Impulse

Impulse is what a net force does over time: J = ∫F dt, the area under the F-t graph, which is just F·Δt when the force holds steady. The impulse-momentum theorem makes it exactly the change in momentum, J = Δp, with N·s and kg·m/s the same unit. Because it equals the change, a ball that bounces straight back off a wall at the same speed takes twice the impulse of one that sticks: the reversal runs the momentum all the way from +mv to -mv, a change of 2mv against mv. (Rebound at a different speed, and the impulse is just m(v_f - v_i).)

Three things trip students on impulse. They mistake an ingredient for the whole — the force alone, the time alone, or force times distance (that is work, a different quantity) — standing in for J = FΔt = Δp, so a bounce gets scored equal to a stick because both look the same. They imagine a cushion shrinks Δp, when an airbag changes nothing about the momentum the crash must remove; F = Δp/Δt only trades a longer time for a smaller force. And they cut corners on a varying force, when impulse is the area under F-t (half the peak-times-time rectangle for a triangle) and the net force at each instant is the slope dp/dt of the momentum graph, not a value read off it.