Momentum — Quick Summary

Mr Toogood's Physics · Mechanics

AQA 3.4.1.6
p = mv
Momentum
m₁u₁+m₂u₂=m₁v₁+m₂v₂
Conservation of momentum
F = Δp/Δt
Force
FΔt = Δp
Impulse

Momentum & its conservation

p = mv

Momentum measures an object's "un-stoppability". It's conserved in every interaction — collisions, explosions, anything — provided no external resultant force acts:

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Why it's conserved: by Newton's 3rd law, interacting objects exert equal, opposite forces on each other for exactly the same time — so they experience equal, opposite impulses, and therefore equal, opposite changes in momentum. Unlike energy, momentum can never be "lost" to heat or sound.

Method (momentum is a vector — always assign a direction first):

  1. Choose a positive direction.
  2. Write total momentum before the interaction.
  3. Write total momentum after the interaction.
  4. Set the two equal.
Two trolleys of different mass colliding and sticking together

A collision: momentum before = momentum after the trolleys join.

Explosions

Two stationary trolleys pushed apart by a released spring

Starting from zero momentum, the trolleys must move apart with equal and opposite momenta.

Starting from rest, total initial momentum is zero, so mv = −(3/2)mV for unequal masses. The lighter object always moves off faster; the minus sign confirms the two move in opposite directions.

Elastic vs. inelastic collisions

MomentumKinetic energyTotal energy
ElasticConservedConservedConserved
InelasticConservedNot conservedConserved
Momentum is always conserved in both cases — only KE differs. Objects separating (rebounding) after a collision is a strong sign it's elastic, or close to it; objects sticking together means KE was lost (inelastic).

Impulse

The change in momentum, Δp = mΔv = m(v−u), divided by time gives force — a restatement of Newton's 2nd law:

Δp/t = F

Rearranged, the product of force and time is the impulse:

Impulse = FΔt = Δp
Force against time graph, area under the line representing impulse

Area under a force–time graph = impulse = change in momentum.

Momentum (Ft) and energy (Fs) have different dimensions — time vs. length — which is exactly why momentum can be conserved in a collision while kinetic energy isn't.

Momentum & safety

Rearranging: F = Δp/Δt. For a given Δp, increasing Δt reduces the average force needed.

  • Crumple zones deform over a longer time, reducing peak force.
  • Airbags & seatbelts extend the time a passenger's momentum changes over.
  • Cushioned trainers, mats, packaging all extend contact/stopping time.
Ethical design: safety features add cost, weight, and complexity, and protecting occupants can shift more risk onto pedestrians or cyclists — real transport design balances physics against cost, environment, and everyone's safety, not just those inside the vehicle.

Exam essentials

Key equations

  • p=mv
  • m₁u₁+m₂u₂=m₁v₁+m₂v₂
  • F=Δp/Δt   Impulse=FΔt=Δp

Elastic vs. inelastic

  • Momentum: always conserved.
  • Elastic: KE also conserved (objects separate).
  • Inelastic: KE lost (often stick together).

Common slips

  • Assign a direction before calculating — momentum is a vector.
  • Don't assume objects simply "swap" or "reverse" velocities in an elastic collision — only true for equal masses.
  • Impulse units: N s = kg m s⁻¹ (equivalent, both accepted).