This note covers momentum, the law of conservation of momentum in collisions and explosions, and impulse (the change in momentum caused by a force acting for some time). It explains how collisions work and why crumple zones and airbags protect people.
Key ideas
Momentum
Momentum is mass multiplied by velocity. It measures how hard it is to stop a moving object. A lorry and a bicycle at the same speed have very different momenta because their masses differ. Momentum is a vector, so it has a direction, the same as the velocity. In one-dimensional problems, choose one direction as positive and give motion the opposite way a negative sign.
Momentum is measured in kg m/s (or equivalently N s).
Conservation of momentum
In a closed system, where no external force acts, the total momentum before an interaction equals the total momentum after it. This holds for collisions, for objects that stick together, and for explosions or recoils.
- Collision, objects join: m1u1 + m2u2 = (m1 + m2)v.
- Collision, objects bounce: m1u1 + m2u2 = m1v1 + m2v2.
- Explosion or recoil: the objects start at rest with zero total momentum, so after the explosion the momenta in opposite directions are equal in size.
Kinetic energy may or may not be conserved. In an elastic collision, kinetic energy is conserved. In an inelastic collision, some kinetic energy is transferred to thermal energy and sound, so the total kinetic energy afterwards is lower. Momentum is conserved in both types.
Force and impulse
A resultant force changes momentum. Starting from F = m × a and a = (v − u) ÷ t, the resultant force equals the rate of change of momentum:
F = (mv − mu) ÷ t = Δp ÷ t.
Rearranging gives impulse: F × t = Δp. Impulse is the force multiplied by the time it acts, and it equals the change in momentum. Its unit is N s, which equals kg m/s.
Using impulse in safety
For a given change in momentum, a longer contact time means a smaller average force, because F = Δp ÷ t. This is the idea behind many safety features:
- Crumple zones in cars collapse gradually, which increases the time of the collision and reduces the force on the passengers.
- Airbags and seat belts spread the stopping over a longer time.
- A cricketer pulls the hands backwards when catching a hard ball, to lengthen the time over which the ball stops.
- Soft landing surfaces and bending the knees when you jump down do the same job.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| p = m × v | Momentum (kg m/s) from mass (kg) and velocity (m/s). |
| Conservation of momentum | Total momentum before = total momentum after, if no external force acts. |
| F = (mv − mu) ÷ t | Force = change in momentum ÷ time taken. |
| Impulse = F × t = Δp | Impulse (N s) equals the change in momentum. |
| Elastic collision | Kinetic energy is conserved. |
| Inelastic collision | Kinetic energy is not conserved; some is transferred to other forms. |
| KE = ½ m v2 | Kinetic energy (J), used to check whether a collision is elastic. |
Worked examples
Example 1 A 2.0 kg trolley moving at 3.0 m/s hits a stationary 1.0 kg trolley. They stick together. Find their common velocity, and the kinetic energy lost.
Step 1: momentum before = 2.0 × 3.0 + 1.0 × 0 = 6.0 kg m/s.
Step 2: momentum after = (2.0 + 1.0) × v = 3.0v. Set 3.0v = 6.0, so v = 2.0 m/s.
Step 3: KE before = ½ × 2.0 × 3.02 = 9.0 J.
Step 4: KE after = ½ × 3.0 × 2.02 = 6.0 J.
Step 5: KE lost = 9.0 − 6.0 = 3.0 J. This was transferred as thermal energy and sound, so the collision is inelastic.
Answer: 2.0 m/s; 3.0 J of kinetic energy lost.
Example 2 A 60 kg skater at rest on smooth ice throws a 3.0 kg ball forwards at 8.0 m/s. Find the recoil velocity of the skater.
Step 1: total momentum before = 0.
Step 2: take forwards as positive. Momentum of ball = 3.0 × 8.0 = +24 kg m/s.
Step 3: total after = 0, so skater's momentum = −24 kg m/s.
Step 4: velocity of skater = −24 ÷ 60 = −0.40 m/s.
Answer: 0.40 m/s in the opposite direction to the ball.
Example 3 A tennis ball of mass 0.057 kg arrives horizontally at 40 m/s and is hit back at 30 m/s in the opposite direction. The contact with the racket lasts 0.0050 s. Find the change in momentum and the average force.
Step 1: take the incoming direction as positive. Initial momentum = 0.057 × 40 = +2.28 kg m/s.
Step 2: final momentum = 0.057 × (−30) = −1.71 kg m/s.
Step 3: change in momentum = −1.71 − 2.28 = −3.99 kg m/s. The size is 3.99 kg m/s.
Step 4: F = Δp ÷ t = 3.99 ÷ 0.0050 = 798 N.
Answer: impulse of 3.99 N s; average force 798 N, in the direction of the ball's return.
Common errors
- Ignoring direction. If two objects move towards each other, one velocity must be negative.
- Saying momentum is lost in an inelastic collision. Momentum is conserved; kinetic energy is not.
- Mixing up momentum (kg m/s) with force (N) and impulse (N s).
- Forgetting to add both masses in the momentum equation when objects stick together.
- Explaining a crumple zone by saying that it reduces the momentum change. The change in momentum is the same; the time is longer so the force is smaller.
Exam tip: Define which direction is positive before you start, write "momentum before = momentum after", and give the sign of any final velocity in your answer.
Check yourself
- Find the momentum of a 1200 kg car moving at 15 m/s.
Answer
p = 1200 × 15 = 18 000 kg m/s. - A 0.50 kg trolley at 4.0 m/s collides with a 1.5 kg trolley at rest and they stick together. Find their velocity.
Answer
Momentum before = 0.50 × 4.0 = 2.0 kg m/s. After: (0.50 + 1.5) × v = 2.0, so v = 1.0 m/s. - A force of 500 N acts on a trolley for 0.20 s. What is the change in momentum?
Answer
Δp = F × t = 500 × 0.20 = 100 kg m/s. - Explain why bending your knees when you land from a jump reduces the force on your legs.
Answer
Your momentum must fall to zero either way. Bending the knees makes the stopping time longer, and since F = Δp ÷ t, the average force is smaller.