Conservation of Momentum — Collisions Explained

What momentum is

Momentum ($\vec{p}$) measures how much motion an object carries — its mass times its velocity:

$$\vec{p} = m\vec{v}$$

It is a vector: direction matters. A truck rolling north and an identical truck rolling south at the same speed have momenta that are equal in size but opposite — and can cancel each other in a head-on collision.

Conservation: the universe's bookkeeping

Here is the payoff. In an isolated system — no net external force acting during the interaction — the total momentum never changes:

$$\vec{p}_{\text{before}} = \vec{p}_{\text{after}}$$

Why? Newton's third law: during a collision, A pushes B and B pushes A with equal, opposite forces for the same time. Equal opposite impulses mean equal opposite momentum changes — they cancel exactly. What one gains, the other loses. The universe's books always balance.

Impulse: force over time

Impulse ($J$) is momentum's delivery mechanism — force multiplied by the time it acts:

$$J = F\Delta t = \Delta p$$

This is just $F = ma$ rewritten ($a = \Delta v/\Delta t$). Its power is practical: the same momentum change can be a sledgehammer blow (huge $F$, tiny $\Delta t$) or a gentle push (small $F$, long $\Delta t$). Airbags don't reduce the impulse of a crash — they stretch $\Delta t$ so the peak $F$ drops.

Elastic vs inelastic collisions

Collisions come in two flavors:

• Elastic: momentum and kinetic energy conserved. Idealized — nearly true for billiard balls and gas molecules, never perfect in the macroscopic world.
• Inelastic: momentum conserved, kinetic energy not — some becomes heat, sound, or permanent deformation. Cars crumpling and clay balls sticking are inelastic.

The extreme case, perfectly inelastic, is when objects stick together — maximum kinetic energy lost, momentum still fully conserved.

Worked mini-example

A 1200 kg car moving at $15\ \mathrm{m/s}$ rear-ends a stationary 900 kg car, and they lock together (perfectly inelastic). What is their joint velocity?

Momentum before: $p = 1200 \times 15 + 900 \times 0 = 18{,}000\ \mathrm{kg·m/s}$.

After: $(1200 + 900)v = 2100v$. Setting equal:

$$v = \frac{18{,}000}{2100} \approx 8.6\ \mathrm{m/s}$$

Note what we never needed: the impact force, the crumple distance, the collision duration. Conservation sidesteps all of it.

Variables and units

Common mistakes

Real-world applications

• Crumple zones and airbags extend collision time Δt, shrinking the peak force for the same momentum change — the same impulse, gentler delivery.

• Rocket propulsion is momentum conservation in action: momentum thrown backward in exhaust appears as forward momentum of the rocket.

• Forensic analysts reconstruct crash speeds from post-collision motion using momentum conservation, since external impulses during the brief impact are negligible.