Momentum

Linear momentum — the product of a body's mass and its velocity:

p = mv

Measured in kg m s⁻¹ or, equivalently, N s. Momentum is a vector, taking the direction of the velocity, and keeping track of its sign is where most collision problems are won or lost.

The principle of conservation of momentum — for a system of interacting bodies with no external resultant force, the total momentum before an interaction equals the total momentum after it.

Why it holds — by the third law the forces the bodies exert on each other are equal and opposite, and they act for the same length of time. So the changes in momentum are equal and opposite, and they cancel in the total.

Working in one dimension — choose a positive direction first and give every velocity its sign accordingly. An object moving the other way has negative momentum, and forgetting that sign produces an answer that looks plausible and is wrong.

Example: a 0.15 kg ball at 12 m s⁻¹ has momentum 1.8 kg m s⁻¹. If it rebounds at the same speed, its momentum is now −1.8 kg m s⁻¹, so the change in momentum is −3.6 kg m s⁻¹, not zero. A reversal is the largest momentum change available.

Elastic collision — kinetic energy is conserved as well as momentum. Truly elastic collisions are rare outside gas particles and subatomic interactions.

Inelastic collision — kinetic energy is not conserved; some is transferred to internal energy, sound or permanent deformation. Momentum is still conserved. Most everyday collisions are inelastic.

Perfectly inelastic collision — the bodies stick together and move off with a common velocity. This is the maximum possible loss of kinetic energy consistent with conserving momentum.

The test to apply — momentum is conserved in every collision with no external force. Kinetic energy is conserved only in elastic ones. To decide which kind a collision is, calculate the total kinetic energy before and after and compare; if it has fallen, the collision was inelastic.