In any interaction with no external resultant force, the total momentum afterwards equals the total before. It holds for gentle bounces, crunching crashes, recoiling rifles and exploding fireworks alike. Kinetic energy is fussier: it survives only the elastic collisions.
The principle of conservation of momentum: for a system with no external resultant force, the total momentum stays constant. In a collision, m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. Momentum is a vector, so directions and signs matter, and in two dimensions it is conserved separately along each axis. Kinetic energy is conserved only in an elastic collision; in an elastic collision the relative speed of approach equals the relative speed of separation.
Set the masses and velocities of two trolleys and choose elastic or inelastic. Run the collision and compare the total momentum readout before and after. It matches every time. Watch how the kinetic energy total only stays the same when the collision is elastic.
| Quantity | Elastic | Inelastic |
|---|---|---|
| total momentum | conserved | conserved |
| total kinetic energy | conserved | not conserved (some becomes heat, sound, deformation) |
| relative speed | approach = separation | separation is less than approach |
A perfectly inelastic collision is the extreme case where the objects stick together and move with one common velocity v = (m₁u₁ + m₂u₂) / (m₁ + m₂). An explosion or recoil is a collision run in reverse: the total momentum is zero before and remains zero after, so the fragments fly apart with equal and opposite momenta.
Four quick checks on conservation, elastic versus inelastic, and recoil. Each correct answer earns XP and lights this skill on your star map.
In the simulator, comparing the total momentum just before and just after the collision, you find it is:
What distinguishes an elastic collision from an inelastic one?
A 2.0 kg trolley moving at 3.0 m s⁻¹ collides with and sticks to a stationary 1.0 kg trolley. Their common velocity afterward is:
A stationary rifle fires a bullet forward. Before firing the total momentum is zero, so afterward the rifle must:
Two traps recur. First, momentum is a vector: a head-on collision needs opposite signs for the two velocities, or you will get a wrong total. Second, do not assume kinetic energy is conserved. Momentum is conserved in every collision with no external force, but kinetic energy is conserved only in an elastic one; in a real crash some kinetic energy becomes heat, sound and deformation. Stating that "energy is destroyed" is also wrong: total energy is always conserved, it just changes form.
Unlocks once the four checks above are done. Worth more XP, written to AS Paper 1 and 2 standard.
A 3.0 kg trolley moving right at 4.0 m s⁻¹ collides head-on with a 2.0 kg trolley moving left at 1.0 m s⁻¹. They stick together. Their common velocity is:
A 4.0 kg rifle fires a 0.020 kg bullet forward at 300 m s⁻¹. The recoil speed of the rifle is:
In a one-dimensional elastic collision, ball A approaches ball B at a relative speed of 6.0 m s⁻¹. After the collision, the relative speed of separation is:
A trolley of mass 2.0 kg at 5.0 m s⁻¹ collides with a stationary 3.0 kg trolley and they stick together. How much kinetic energy is lost in the collision?
This skill is now lit gold on your star map. You have finished the lessons of Topic 3; the Paper 1 set awaits.