IMAT Physics · Dynamics, Work and Energy

Momentum, Impulse and Collisions

Momentum is mass times velocity; a force acting for a time changes it, and in any collision with no outside force the total momentum stays the same.

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Why this matters

The 2023 ministry paper asked for both velocities after an elastic collision between a moving mass and a heavier one at rest. Momentum is otherwise new ground for the paper, so expect the standard cases: sticking together, recoil and rebound.

Concept 1 of 2: Momentum and impulse

A slow lorry and a fast bullet are both hard to stop: momentum combines mass and velocity into one measure of how much motion a body carries. To change it you need a force acting for some time. The same change of momentum can come from a big force for a short time or a small force for a long time, which is why airbags and crumple zones save lives.

Definition

Momentum $p = mv$ : a vector, in $\text{kg m/s}$ , in the direction of the velocity.
Impulse = force × time = change of momentum: $F\Delta t = \Delta p$ . Unit: N s, the same as kg m/s.
This is the second law in its general form: $F = \Delta p/\Delta t$ .
For a rebound, the change of velocity includes the reversal: from $+u$ to $-v$ is a change of $u + v$ .
Stretching the stopping time lowers the average force for the same change of momentum.

Momentum and impulse

p = mv \qquad F\,\Delta t = \Delta p = mv - mu

$p$ momentum, in kg m/s
$F$ average force, in N
$\Delta t$ time the force acts, in s

Worked example

A 0.15 kg ball reaches a bat at 20 m/s and leaves it at 30 m/s in the opposite direction. The contact lasts 0.010 s. Find the impulse and the average force.

Practice this conceptself-check · 3 quick reps

A rebound changes the momentum by more than a stop does

Momentum is a vector. A ball arriving at 20 m/s and leaving at 30 m/s the other way changes its velocity by 50 m/s, not 10 m/s. Bouncing back always needs a bigger impulse than simply stopping.

Concept 2 of 2: Conservation of momentum in collisions and explosions

When two bodies collide, each pushes the other with an equal and opposite force for the same time, so whatever momentum one gains the other loses. With no outside force, the total momentum is the same before and after. Kinetic energy is a different story: some is usually turned into heat, sound and dents. Only in an elastic collision does all of it survive.

Definition

Conservation of momentum: in an isolated system (no external resultant force), the total momentum before = total momentum after. This holds for every collision and explosion.

Elastic collision: kinetic energy is also conserved.
Inelastic collision: some kinetic energy is lost. Perfectly inelastic: the bodies stick together, and the loss is the largest possible.
Explosion or recoil from rest: total momentum stays zero, so the pieces move off in opposite directions with equal momentum.
Elastic head-on collision, mass $m_1$ at speed $u$ hitting $m_2$ at rest: $v_1 = \dfrac{m_1 - m_2}{m_1 + m_2}u$ and $v_2 = \dfrac{2m_1}{m_1 + m_2}u$ . Equal masses simply swap velocities.

Conservation of momentum

m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2

$u_1, u_2$ velocities before the collision, with signs
$v_1, v_2$ velocities after the collision, with signs

Worked example

A 2.0 kg trolley moving at 6.0 m/s hits a stationary 1.0 kg trolley and they stick together. Find their common velocity and the kinetic energy lost.

Practice this conceptself-check · 4 quick reps

Momentum is always conserved; kinetic energy only in elastic collisions

In any collision with no external force the total momentum stays the same. Kinetic energy is conserved only when the collision is elastic. An option saying both are always conserved, or that momentum is lost when bodies stick together, is wrong.

Velocities after a collision carry signs

A light body bouncing off a heavier one at rest moves backwards. Writing every speed as positive in the momentum equation gives an answer that breaks conservation of momentum.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Momentum and impulse
Momentum and impulse
$p = mv \qquad F\,\Delta t = \Delta p = mv - mu$
Conservation of momentum in collisions and explosions
Conservation of momentum
$m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$

Watch out for (3)

A rebound changes the momentum by more than a stop does
→ Momentum and impulse
Momentum is always conserved; kinetic energy only in elastic collisions
→ Conservation of momentum in collisions and explosions
Velocities after a collision carry signs
→ Conservation of momentum in collisions and explosions

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