P5.5 Momentum
📖Definition— Definition and formula
Momentum is a property of any moving object. It depends on both the mass and the velocity:
p = mv
p = momentum (kg·m/s), m = mass (kg), v = velocity (m/s)
Momentum is a vector — it has both magnitude and direction. Direction is crucial when two objects collide.
Conservation of momentum
Law of conservation of momentum: In a closed system (no external forces), the total momentum before a collision equals the total momentum after.
p_before = p_after
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
This applies to:
- Elastic collisions: Objects bounce apart; kinetic energy is conserved.
- Inelastic collisions: Objects stick together or some KE is lost to thermal energy.
✦Worked example— Worked example — two objects colliding
A 3 kg trolley moving at 4 m/s collides with a stationary 1 kg trolley. They stick together.
Before: p = 3×4 + 1×0 = 12 kg·m/s
After: (3+1) × v = 4v = 12
v = 3 m/s (in same direction as original motion)
✦Worked example— Worked example — explosion / recoil
A gun (mass 2 kg) fires a bullet (mass 0.01 kg) at 400 m/s. What is the recoil velocity of the gun?
Before: total p = 0 (both stationary)
After: p_bullet + p_gun = 0
0.01 × 400 + 2 × v_gun = 0
v_gun = −4/2 = −2 m/s (recoil — opposite direction to bullet)
Force and change in momentum
Newton's Second Law in terms of momentum:
F = Δp/Δt = (mv − mu)/t
A larger force produces a larger rate of change of momentum. Extending the time over which a force acts reduces the force for the same momentum change — this is the principle behind crumple zones, seatbelts, airbags and gym crash mats.
Impulse
Impulse = FΔt = Δp
Impulse in N·s (equivalent to kg·m/s)
Impulse = area under a force–time graph.
Common exam errors
- Forgetting direction — momentum is a vector; set a positive direction clearly.
- Conservation only applies when no external resultant forces act (closed system).
- Confusing mass and momentum — momentum includes velocity.
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