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
Readouts
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.
- Momentump = mv
- Conservation of momentumm₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
- Coefficient of restitutione = (v₂ − v₁) / (u₁ − u₂)
- Kinetic energy lostΔKE = ½ μ (1 − e²)(u₁ − u₂)², μ = m₁m₂ / (m₁ + m₂)
On the syllabus
- GCSE Physics AQA, Edexcel, OCRmomentum and its conservation
- A-Level Physics and Maths Mechanicselastic and inelastic collisions, impulse, Newton's law of restitution, oblique collisions
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.
Total momentum is 2 × 3 + 1 × (−1) = 5 kg m/s, so v = 5 / 3 = 1.67 m/s. Kinetic energy falls from 9.50 J to ½ × 3 × 1.67² = 4.17 J, so 5.33 J is lost, which matches ½ μ (u₁ − u₂)² = ½ × 0.667 × 4² = 5.33 J.
Learn this properly
Lessons from GCSE Combined and Separate Science on this topic: worked steps, then exam-style questions with new numbers every time.
- Momentum, p = mvMomentumFree
- Conservation of momentum in collisionsMomentumFree
- Collisions and kinetic energyMomentumPlus
See the whole courseThe full course, topic tests and mock papers come with Plus. See Plus
FAQ
How do I find the velocities after a collision?
Use conservation of momentum together with the coefficient of restitution: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ and v₂ − v₁ = e(u₁ − u₂). A 2 kg cart at 3 m/s hitting a 1 kg cart at −1 m/s with e = 0.5 gives v₁ = 1 m/s and v₂ = 3 m/s, and 4 J of kinetic energy is lost. Pick Velocities after under Solve for to see each step.
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.