Momentum and impulse

Impulse records a force history

Momentum changes when force acts over time. Impulse is useful when the force varies rapidly or is difficult to model point by point but its time integral can be measured.

Integrate Newton’s law in time

Newton’s law gives F = dp/dt. Multiply by dt and integrate from t₁ to t₂: J = ∫F dt = ∫dp = p₂ − p₁ = Δp. For constant mass, J = m(v₂ − v₁). The area under a force–time graph is the impulse.

A 0.15 kg ball changes velocity from −20 to +30 m/s during a collision. Δp = 0.15(50) = 7.5 N·s. If contact lasts 0.012 s, the average force is 625 N; increasing contact time lowers peak force for the same momentum change.

Impulse does not equal average force without a duration

A triangular force pulse with peak 1000 N lasting 0.02 s has impulse ½(0.02)(1000) = 10 N·s, not 20 N·s. Shape and sign of the force-time curve matter.

Work it: Estimate the impulse from a force that rises linearly from 0 to 300 N in 8 ms and falls to 0 in 4 ms. What velocity change does it give a 0.20 kg object?