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Conservation of Momentum

Classical Mechanics

Momentum is a measure of how much motion an object carries, and it has a remarkable property: in a closed system, the total momentum never changes. Collisions, explosions, and recoils all obey this rule. That is why physicists can predict what happens after two objects crash into each other without knowing the details of the impact itself.

What momentum is

Momentum is defined as mass times velocity: p = m v. It is a vector, so it has a direction as well as a size, and its SI unit is the kilogram metre per second (kg·m/s). A heavy lorry moving slowly can have the same momentum as a light car moving quickly. Because velocity is a vector, momentum in one direction must be treated separately from momentum in the opposite direction, one is positive and the other negative in a calculation. The total momentum of a system is the vector sum of the momenta of all its parts.

The conservation rule

The principle of conservation of momentum states that the total momentum of a system stays constant if no external force acts on it. In a collision between two objects, momentum may be transferred from one to the other, but the total before the collision equals the total after it. External forces such as friction or gravity can change the total, so the rule works best when those forces are small or act for a very short time. This is why the rule is so useful for collisions: the impact forces are large and brief, and external forces have little time to matter.

Collisions and recoil

Conservation of momentum explains both collisions and recoil. In a perfectly elastic collision, kinetic energy is conserved as well as momentum, like ideal billiard balls bouncing apart. In an inelastic collision, the objects may stick together and kinetic energy is lost as heat and sound, but momentum is still conserved. Recoil is the same idea in reverse: when a gun fires a bullet forward, the gun recoils backward so that the total momentum stays zero. Rockets work the same way, throwing gas backward to gain forward momentum.

Common mistakes

• Treating momentum as a scalar and ignoring direction. Correction: momentum is a vector, give opposite directions opposite signs in your calculation.
• Thinking kinetic energy is always conserved in a collision. Correction: momentum is conserved in all collisions, but kinetic energy is conserved only in elastic ones.
• Forgetting momentum before an explosion is zero. Correction: if a stationary object splits, the parts must have equal and opposite momenta so the total stays zero.

Practice questions

1. What is the momentum of a 2 kg ball moving at 3 m/s?
   Show answer

   p = m v = 2 x 3 = 6 kg·m/s in the direction of motion.

2. A 0.2 kg ball moving at 10 m/s hits a stationary 0.2 kg ball and they stick together. What is their speed after?
   Show answer

   Momentum before = 0.2 x 10 = 2 kg·m/s. Total mass = 0.4 kg, so v = 2 / 0.4 = 5 m/s.

3. A gun of mass 4 kg fires a 0.01 kg bullet at 400 m/s. What is the recoil speed of the gun?
   Show answer

   Bullet momentum = 0.01 x 400 = 4 kg·m/s. Gun recoil momentum is 4 kg·m/s backward, so v = 4 / 4 = 1 m/s backward.

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