Newton's Laws, Momentum and Impulse

A-Level Physics · Mechanics

Newton's laws of motion

First law: an object stays at rest or moves at constant velocity unless acted on by a resultant force (inertia).
Second law: the resultant force equals the rate of change of momentum:

F = ma        (or more generally  F = Δp ÷ Δt)

Third law: for every action there is an equal and opposite reaction — the two forces act on different objects, are the same type, and are equal in size, opposite in direction.

Momentum (p) = mass × velocity (a vector), in kg m/s:

p = mv

In a closed system (no external forces), the total momentum before = total momentum after a collision or explosion.

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

This applies to all collisions and explosions (momentum is always conserved).

Elastic: kinetic energy is conserved (as well as momentum). Rare in the everyday world.
Inelastic: momentum conserved, but kinetic energy is not (some is transferred to heat/sound/deformation). If objects stick together, it's perfectly inelastic.

Impulse = force × time = change in momentum:

impulse = FΔt = Δp = mv − mu

Impulse is the area under a force–time graph.
A longer contact time reduces the force for the same change in momentum — the principle behind crumple zones, airbags and catching a ball with "give".

A 0.15 kg ball hits a wall at 20 m/s and rebounds at 15 m/s. Find the impulse on the ball.

Taking towards the wall as positive: Δp = m(v − u) = 0.15 × (−15 − 20) = 0.15 × (−35) = −5.25 kg m/s (i.e. 5.25 N s away from the wall). ✓ (Remember to include the direction change.)

Forgetting momentum and velocity are vectors — include direction (signs) in collisions.
Saying kinetic energy is always conserved — only in elastic collisions; momentum is always conserved.
Applying Newton's third law to two forces on the same object (they act on different objects).

Use F = Δp/Δt and impulse = FΔt = Δp; link long contact time to smaller force (safety features).
In collisions, define a positive direction and use signs consistently.
State whether kinetic energy is conserved to classify a collision as elastic/inelastic.

Newton's laws: 1 inertia, 2 F = ma (= Δp/Δt), 3 equal & opposite forces on different objects.
Momentum p = mv is conserved in a closed system; elastic collisions also conserve kinetic energy, inelastic don't.
Impulse = FΔt = Δp (area under a force–time graph); longer contact time → smaller force (crumple zones, airbags).