How Momentum Is Conserved in Collisions Explained

Momentum is one of the most useful quantities in physics because it is conserved. In any collision where no outside force acts on the system, the total momentum before the collision equals the total momentum after it. That single fact lets you predict the velocities of colliding billiard balls, cars in a crash test, or fragments from an explosion, without tracking every contact force in detail.

This post defines momentum, explains why it is conserved, works through a collision with numbers, and compares elastic and inelastic collisions. Energy is also examined, because whether kinetic energy is conserved is what separates the two collision types.

What momentum is

Momentum is the product of mass and velocity, written p = m v. It is a vector, so direction matters. A 2 kg object moving at 3 m/s to the right has momentum 6 kg·m/s to the right. The SI unit is kg·m/s, which is the same as newton-seconds.

Momentum measures how hard it is to stop an object. A truck at 10 m/s has far more momentum than a bicycle at 10 m/s because the truck has much more mass. Stopping either one requires an impulse, which is force multiplied by the time over which it acts. The impulse equals the change in momentum, so the same momentum change can be produced by a large force over a short time or a small force over a long time.

Why momentum is conserved

Newton's second law can be written as F = dp/dt, meaning the rate of change of momentum equals the net external force. If no external force acts on a system, the net force is zero, so dp/dt = 0 and the total momentum does not change.

The key point is the word "external." Forces between objects inside the system come in equal and opposite pairs, by Newton's third law. Those internal forces move momentum from one object to another but cannot change the total. Only forces from outside the system, such as friction with the ground or gravity acting on a system that is not in free fall, can change total momentum.

This is why momentum conservation is so powerful. You do not need to know the detailed contact force during an impact. You only need to know the masses and velocities before and after, and to verify that outside forces are negligible during the collision time.

A worked collision

Consider a 2 kg ball moving at 3 m/s that strikes a 1 kg ball initially at rest, on a frictionless table, in one dimension.

1. Total momentum before: p = 2 × 3 + 1 × 0 = 6 kg·m/s.
2. If the balls stick together, the combined mass is 3 kg. Conservation gives 3 kg × v = 6 kg·m/s, so v = 2 m/s.
3. If the collision is elastic instead, kinetic energy is also conserved. Solving both equations gives the first ball rebounding at 1 m/s to the right and the second ball moving at 4 m/s.

Check the elastic result. Momentum after is 2 × 1 + 1 × 4 = 6 kg·m/s, matching the start. Kinetic energy before is ½ × 2 × 3² = 9 J. After it is ½ × 2 × 1² + ½ × 1 × 4² = 1 + 8 = 9 J, so energy is also conserved.

The sticky collision gives kinetic energy after of ½ × 3 × 2² = 6 J. The missing 3 J became heat, sound and deformation of the balls.

Elastic versus inelastic collisions

An elastic collision conserves both momentum and kinetic energy. Ideal billiard ball collisions are close to elastic, and so are collisions between hard steel spheres. The bodies bounce apart without permanent deformation.

An inelastic collision conserves momentum but not kinetic energy. Some kinetic energy becomes internal energy, through deformation, heat or sound. A perfectly inelastic collision is the extreme case in which the objects stick together and move as one. Car crashes are mostly inelastic, which is why crumple zones are designed to absorb energy.

The table below compares the two types for the example above.

Quantity	Before	Elastic after	Perfectly inelastic after
Total momentum (kg·m/s)	6	6	6
Total kinetic energy (J)	9	9	6
Ball 1 velocity (m/s)	3	1	2
Ball 2 velocity (m/s)	0	4	2

Coefficient of restitution

Real collisions sit between the two extremes. The coefficient of restitution e measures the ratio of separation speed to approach speed. It is 1 for a perfectly elastic collision and 0 for a perfectly inelastic one. A steel ball dropped on a steel plate might have e near 0.9, while a lump of clay has e near 0.

Restitution can be measured by dropping a ball and comparing the height of the bounce to the drop height. The square root of that height ratio gives e, because energy scales with the square of speed.

Momentum in two dimensions

The same rules apply to each direction separately. In a two-dimensional collision you write one momentum equation for the x component and one for the y component. Pool players use this when a cue ball strikes an object ball at an angle. The total momentum vector before the strike equals the vector sum after, and the 90-degree split for equal-mass elastic billiard balls comes straight from these equations.

Checking your answer

Two checks catch most errors in collision problems. First, confirm that total momentum before and after agree, including sign. A leftward velocity must carry a negative sign in every term. Second, confirm that kinetic energy after is not larger than kinetic energy before, since a collision cannot add energy by itself. If the answer violates the energy check, either the collision is not isolated or a mistake was made in the setup.

Units offer a third check. Momentum has units of kg·m/s and kinetic energy has units of joules, which is kg·m²/s². Writing units next to every term catches many algebra slips before they reach the final number.

Momentum outside collisions

Conservation of momentum also explains recoil and rocket propulsion. A gun fires a bullet forward, and the gun moves backward so that total momentum stays zero. A rocket throws exhaust backward and gains forward momentum, with no need for air to push against. Both cases use the same conservation law as a billiard table.

Key takeaways

Momentum is mass times velocity, and in any system with no net external force, total momentum stays constant. Internal forces redistribute momentum but cannot change the total. Elastic collisions also conserve kinetic energy, while inelastic collisions lose some of it to heat and deformation. Using momentum conservation first lets you solve many collision problems with two simple equations before worrying about the messy details of contact.