- Use p = mv with direction signs.
- Apply conservation of momentum in one dimension.
- Use impulse FΔt = Δp.
- Explain safety features by increasing collision time and reducing force.
10.1 Momentum
Momentum is mass × velocity: p = mv. The unit is kg m s⁻¹. Momentum is a vector, so direction matters. In one-dimensional calculations, assign one direction positive and use negative velocities for the opposite direction.
A large momentum can result from large mass, high velocity or both. Momentum is not the same as kinetic energy: momentum depends on v while kinetic energy depends on v².
10.2 Conservation of momentum
In an isolated system with no external resultant force, total momentum remains constant. Total momentum before an interaction equals total momentum after. Internal forces during a collision are equal and opposite, so their effects on total system momentum cancel.
For objects that stick together, use m₁u₁ + m₂u₂ = (m₁+m₂)v. For objects that separate, include separate final terms. A negative answer indicates motion opposite to the chosen positive direction.
10.3 Impulse and force
Impulse is force × time and equals change in momentum: FΔt = Δp. More generally, for a changing force, the area under a force–time graph represents impulse. Resultant force is rate of change of momentum: F = Δp/Δt.
For the same momentum change, increasing the collision time reduces the average force. Airbags, seat belts, crumple zones, gym mats and bending the knees on landing increase the time over which momentum changes.
10.4 Safety explanations
A complete safety explanation follows a chain: the passenger’s momentum must fall to zero; the safety feature increases stopping time; F = Δp/Δt; therefore the average force is reduced, decreasing injury risk. Do not say the feature reduces momentum change if the person still comes to rest from the same speed.
Seat belts also spread force over a larger body area, reducing pressure, and prevent impact with hard interior surfaces. Airbags provide cushioning and increase both stopping time and contact area.
Worked examples
A 2.0 kg trolley at 4.0 m s⁻¹ hits a stationary 3.0 kg trolley and they stick. Initial momentum = 8.0 kg m s⁻¹. v = 8.0/5.0 = 1.6 m s⁻¹.
Take right positive. A 0.5 kg ball at +6 m s⁻¹ rebounds at −4 m s⁻¹. Δp = 0.5(−4−6) = −5.0 kg m s⁻¹.
If the change in momentum magnitude is 5.0 kg m s⁻¹ over 0.020 s, average force magnitude = 250 N. Increasing time to 0.10 s reduces it to 50 N.
Practical focus
Use two low-friction trolleys and video or light gates to measure speeds before and after a collision. Compare total momentum and discuss external forces, track slope, timing uncertainty and whether the collision is elastic or inelastic.
Examination guidance
- Choose and state a positive direction before using momentum conservation.
- Conservation applies to total momentum of a system, not necessarily to each object.
- Safety devices reduce force by increasing time for the same Δp.
Check your understanding
- What is the SI unit of momentum?
- A stationary object explodes into two parts. What is total momentum immediately after?
- Why does a crumple zone reduce force?
- kg m s⁻¹.
- Zero, if external impulse is negligible.
- It increases the collision time for the same change in momentum.
