What is the conservation of momentum?

The law of conservation of momentum says that the total momentum of a closed system stays constant if no external force acts on it.

Understanding momentum

Momentum is the “quantity of motion” of an object. It equals mass times velocity (p = m × v). Because velocity has direction, momentum also has direction. When two objects interact—like colliding carts—their momenta can change individually, but the total momentum of both together stays the same as long as no outside force (like friction with the ground) pushes on the system.

Formal statement

In an isolated system (no net external force), the vector sum of momenta before an interaction equals the vector sum after the interaction:

p_{total before} = p_{total after}

Simple example

• Cart A (1 kg) moves at +3 m/s. Its momentum is +3 kg·m/s.
• Cart B (2 kg) is at rest. Momentum is 0.
• Total before = +3 kg·m/s. After a perfectly inelastic collision, they stick. Total mass = 3 kg.
• Final velocity = total momentum / total mass = 3 / 3 = 1 m/s. Total after = 3 kg × 1 m/s = +3 kg·m/s (same as before).

Elastic vs inelastic collisions

Vector nature and multi-direction cases

Because momentum is a vector, you must conserve it in each direction (x, y, and z). In two-dimensional collisions, add momenta using components. Even when objects fly off at angles, the sums along each axis are equal before and after if no external force acts.

Where the law is used

• Engineering: Crash safety design and recoil management.
• Space: Rocket propulsion (exhaust gases carry momentum one way; rocket moves the other way).
• Sports: Billiards, football tackles, and bat-ball impacts.

Common mistakes

• Forgetting external forces like friction; if they are significant, the system is not isolated.
• Mixing up momentum and kinetic energy; energy may change forms, momentum of the system still stays the same.
• Ignoring direction; signs and angles matter.