Is momentum always conserved in a collision?

Yes, as long as no outside force acts during the collision, which is a very good approximation because the collision is so short. The forces the two objects exert on each other are equal and opposite, so their momentum changes cancel.

What is the difference between elastic and inelastic collisions?

In an elastic collision the total kinetic energy after is the same as before. In an inelastic collision some kinetic energy turns into other forms such as heat and sound. Momentum is conserved in both.

What is the coefficient of restitution?

e = speed of separation ÷ speed of approach. e = 1 is perfectly elastic, e = 0 means the objects stick together, and real collisions lie in between: about 0.9 for a snooker ball and 0.5 for a tennis ball on a hard court.

Why do equal snooker balls leave at 90°?

For equal masses in an elastic collision, conservation of momentum makes the three velocity vectors form a triangle and conservation of energy makes it obey Pythagoras, so the two final velocities must be at right angles. Try it in Advanced with the table and equal masses.

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Momentum and collisions

In every collision the total momentum stays the same. What changes from one crash to the next is how much kinetic energy survives.

Momentum of each body and the total vs time

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What's happening

Momentum is mass times velocity, p = mv, and it has a direction. When two objects collide they push on each other with equal and opposite forces for the same short time, so whatever momentum one gains the other loses: the total before equals the total after. Kinetic energy is a different story. In an elastic collision (e = 1) none is lost and the objects separate as fast as they approached. In an inelastic collision some becomes heat, sound and bent metal, and they separate more slowly. If they stick together (e = 0) the loss is as big as momentum allows. The coefficient of restitution e is the ratio of the separation speed to the approach speed. In two dimensions momentum is conserved along each axis, and two equal masses in an elastic glancing collision always leave at 90° to each other.

p = mvm₁u₁ + m₂u₂ = m₁v₁ + m₂v₂e = (v₂ − v₁) / (u₁ − u₂)ΔKE = ½ μ (1 − e²)(u₁ − u₂)², μ = m₁m₂ / (m₁ + m₂)

GCSE Physics (AQA, Edexcel, OCR): momentum and its conservation. A-Level Physics and Maths Mechanics: elastic and inelastic collisions, impulse, Newton's law of restitution, oblique collisions.

Work through the numbers with Physics Formulas and SUVAT Solver.

Challenge

Predict first: a 2 kg cart at 3 m/s hits a 1 kg cart coming the other way at 1 m/s, and they stick together. What is their common velocity, and how much kinetic energy is lost? Choose Stick together, type your velocity in and check.

FAQ

Is momentum always conserved in a collision?: Yes, as long as no outside force acts during the collision, which is a very good approximation because the collision is so short. The forces the two objects exert on each other are equal and opposite, so their momentum changes cancel.
What is the difference between elastic and inelastic collisions?: In an elastic collision the total kinetic energy after is the same as before. In an inelastic collision some kinetic energy turns into other forms such as heat and sound. Momentum is conserved in both.
What is the coefficient of restitution?: e = speed of separation ÷ speed of approach. e = 1 is perfectly elastic, e = 0 means the objects stick together, and real collisions lie in between: about 0.9 for a snooker ball and 0.5 for a tennis ball on a hard court.
Why do equal snooker balls leave at 90°?: For equal masses in an elastic collision, conservation of momentum makes the three velocity vectors form a triangle and conservation of energy makes it obey Pythagoras, so the two final velocities must be at right angles. Try it in Advanced with the table and equal masses.