Collisions
Find impulse from a force–time graph, use conservation of momentum for collisions and explosions in one dimension, and compare elastic and inelastic collisions.
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Before you begin
A collision is a brief interaction in which two bodies push on each other with large forces. The lessons start with impulse, the area under a force–time graph, and how it changes a body’s momentum. They then use conservation of momentum to find velocities after collisions and explosions, add the relative-speed rule for perfectly elastic collisions, and finish by working out how much kinetic energy a collision transfers.
Be comfortable with: Newton’s laws of motion, especially momentum p = mv and the second law as rate of change of momentum.
Learning goals
- Use impulse and momentum conservation in one-dimensional elastic and inelastic collisions.
Syllabus statements covered
- recall that impulse is given by the area under the force–time graph for a body and use this to solve problems
- state the principle of conservation of momentum
- apply the principle of conservation of momentum to solve simple problems including inelastic and (perfectly) elastic interactions between two bodies in one dimension (knowledge of the concept of coefficient of restitution is not required)
- show an understanding that, for a (perfectly) elastic collision between two bodies, the relative speed of approach is equal to the relative speed of separation
- show an understanding that, whilst the momentum of a closed system is always conserved in interactions between bodies, some change in kinetic energy usually takes place
Lessons
Work through them in order.
Core lessons
- Impulse and force–time areaFind impulse as the area under a force–time graph and use it to find a change in momentum or an average force.
- Momentum conservation and one-dimensional interactionsState the principle of conservation of momentum and use it for collisions and explosions in one dimension.
- Perfectly elastic collisionsUse conservation of momentum with relative speed of approach = relative speed of separation to find both velocities after an elastic collision.
- Kinetic-energy change in collisionsFind the velocity of bodies that stick together, calculate the kinetic energy lost and explain where it goes.
Practise and check
Or choose
Topic reference
Quick reference
| Idea | Relation | Notes |
|---|---|---|
| momentum | p = mv | vector; unit N s or kg m s⁻¹ |
| impulse | area under a force–time graph | equals the change in momentum, mv-mu |
| constant force | impulse = FΔ t | |
| average force | F = (Δ p)/(Δ t) | longer contact time, smaller force |
| conservation of momentum | m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂ | no resultant external force; signed velocities |
| bodies that stick | m₁u₁ + m₂u₂ = (m₁ + m₂)v | largest loss of kinetic energy |
| perfectly elastic collision | u₁-u₂ = v₂-v₁ | total kinetic energy unchanged |
| kinetic energy | Eₖ = (1/2)mv² | never negative |
Core knowledge to remember
- Impulse is the area under a force–time graph. The impulse of the resultant force on a body equals its change in momentum.
- Principle of conservation of momentum: the total momentum of a system of bodies remains constant, provided no resultant external force acts on the system.
- Colliding bodies exert equal and opposite forces on each other for the same time, so their changes in momentum are equal and opposite.
- In a perfectly elastic collision, total kinetic energy is conserved and the relative speed of approach equals the relative speed of separation.
- In an inelastic collision, momentum is still conserved, but some kinetic energy is transferred, mainly to internal energy and sound.
Common mistakes
- Using speeds instead of signed velocities when a body rebounds or the bodies move in opposite directions.
- Counting area below the time axis of a force–time graph as positive impulse.
- Assuming kinetic energy is conserved because momentum is.
- Using the relative-speed rule for a collision that is not perfectly elastic.
- Saying an airbag reduces the change in momentum, rather than the average force.