3.6 Momentum

For a direct impact, the coefficient of restitution e is relative speed of separation divided by relative speed of approach, measured along the line of impact. For ordinary passive impacts 0≤e≤1.

Choose one positive direction and write velocities immediately before and after impact. Combine the restitution equation with conservation of momentum when external impulse is negligible.

If two particles approach at 5 m s⁻¹ relative speed and separate at 2 m s⁻¹, e=2/5=0.4. The individual velocities still depend on their masses and momentum.

Restitution does not conserve kinetic energy except in the elastic case e=1; it also does not mean each particle reverses direction.

For a system of colliding particles, total momentum before impact equals total momentum after impact when the external impulse during the collision is negligible: Σmu=Σmv.

Keep signed velocities, identify the system, and use restitution only as a second equation. For an explosion or separation, the same momentum principle applies even though kinetic energy may increase.

A 2 kg trolley at 3 m s⁻¹ collides with a stationary 1 kg trolley and they move together: their common speed is (2×3)/3=2 m s⁻¹.

Momentum is a vector and can cancel; kinetic energy is not generally conserved in an inelastic impact.