Momentum and Its Conservation
Mass in motion, and the conservation law that lets you solve collisions without knowing the forces.
Momentum is one of the most useful quantities in physics because it is conserved: in any isolated system, the total momentum before an event equals the total momentum after. This single principle lets you predict the outcome of collisions, explain how rockets accelerate, and understand recoil — often without knowing any of the messy details of the forces involved.
What momentum is
The momentum of an object is the product of its mass and its velocity:
Momentum is a vector: it has both magnitude and direction. A heavy truck moving slowly can carry the same momentum as a light motorcycle moving fast. Because velocity has direction, momentum does too — a ball moving east has momentum pointing east. This directionality is essential when objects collide from different angles.
Why momentum is conserved
Conservation of momentum follows directly from Newton's third law. When two objects interact, they push on each other with equal and opposite forces for exactly the same length of time. The impulse — force multiplied by time — that one object delivers to the other is therefore equal and opposite. Whatever momentum one object gains, the other loses. The total stays fixed.
Collisions: elastic and inelastic
Physicists divide collisions into two types. In an elastic collision, both momentum and kinetic energy are conserved — the objects bounce apart cleanly, like billiard balls. In an inelastic collision, momentum is still conserved but some kinetic energy is lost to heat, sound, or deformation. The extreme case is a perfectly inelastic collision, where the objects stick together and move as one.
Momentum conservation holds in both cases. That is what makes it so powerful: even when energy is messily lost, the momentum bookkeeping always balances.
A worked example
Imagine a 2 kg ball moving at 3 m/s strikes a stationary 1 kg ball and they stick together. Before the collision the total momentum is 2 times 3 plus 1 times 0, which is 6 kg·m/s. After the collision the combined 3 kg mass moves at some velocity v, so 3v must equal 6, giving v = 2 m/s. The stuck-together pair glides off at 2 metres per second — no knowledge of the collision forces required.
p = m*v to compute momentum from mass and velocity.Impulse: changing momentum
To change an object's momentum you must apply a force over time. This product is called impulse, and it equals the change in momentum. The insight has real safety consequences: airbags, crumple zones and padded surfaces all work by extending the time over which momentum changes, which reduces the peak force. The same change in momentum spread over a longer time means a gentler force — the difference between a bruise and a broken bone.
Rockets and recoil
Conservation of momentum explains how a rocket works even in the vacuum of space, where there is nothing to push against. The rocket expels exhaust gas in one direction; to keep total momentum constant, the rocket must move in the opposite direction. The same principle explains the kick of a fired gun and the backward step you take when you throw something heavy.
Key takeaways
- Momentum is mass times velocity, and it is a vector with direction.
- In an isolated system, total momentum is always conserved.
- Conservation holds in both elastic and inelastic collisions, even when kinetic energy is lost.
- Impulse — force times time — equals the change in momentum, the principle behind airbags and crumple zones.