Momentum (Higher Tier)
Momentum is a measure of how hard it is to stop a moving object. It is a vector quantity (has direction).
$p = mv$
- $p$ — momentum, in kg m/s.
- $m$ — mass (kg).
- $v$ — velocity (m/s, with direction).
Conservation of momentum
In any closed system (no external forces), the total momentum before equals the total momentum after.
This applies to all collisions and explosions.
✦Worked example— Worked example 1 — collision
A 2 kg trolley moving at 3 m/s east collides with a stationary 4 kg trolley. They stick. Find their common velocity after.
- Momentum before: 2 × 3 + 4 × 0 = 6 kg m/s.
- Momentum after: (2 + 4) × v = 6.
- v = 6/6 = 1 m/s east.
✦Worked example— Worked example 2 — explosion
A 0.5 kg gun fires a 0.01 kg bullet at 200 m/s. Find the recoil velocity of the gun.
- Initial momentum = 0.
- Final: bullet momentum + gun momentum = 0.
- 0.01 × 200 + 0.5 × v_gun = 0.
- v_gun = −4 m/s (i.e. 4 m/s in the opposite direction).
Force = rate of change of momentum
For sustained interactions (HT):
$F = \dfrac{\Delta p}{\Delta t} = \dfrac{m\Delta v}{\Delta t} = ma$
So Newton's second law is really a statement about momentum.
Safety devices and momentum
Crumple zones, airbags, seatbelts, helmets, crash mats:
- All work by increasing the time over which momentum changes.
- Larger $\Delta t$ → smaller $F$ for the same $\Delta p$.
- Safer outcomes — less force on body.
In a car crash, you go from (say) 30 m/s to 0 m/s. The question is over what time. With no airbag: 0.05 s — huge force. With airbag: 0.5 s — tenfold less force.
⚠Common mistakes
- Forgetting momentum is a vector — must sign directions consistently.
- Saying conservation only applies to elastic collisions. It applies to ALL collisions and explosions, regardless of energy.
- Confusing momentum (kg m/s) with KE (J) — they're different.
- Forgetting to convert g to kg, or km/h to m/s.
AI-generated · claude-opus-4-7 · v3-deep-physics