Conservation of momentum

Conservation of momentum states that the total momentum of an isolated system never changes. Momentum is mass times velocity (p = mv), a vector quantity, and "isolated" means no net external force acts on the system. Objects within the system can push on each other however violently — collide, explode, stick together — but those internal forces come in equal and opposite pairs (Newton's third law), so their effects on total momentum cancel exactly.

The law does the heavy lifting in collision problems. Setting total momentum before equal to total momentum after — m₁v₁ + m₂v₂ = m₁v₁′ + m₂v₂′ — lets you solve for unknown velocities without knowing anything about the complicated forces during impact. A 2 kg cart moving at 3 m/s that hits and sticks to a stationary 4 kg cart must leave the pair moving at 1 m/s, because 6 kg·m/s of momentum has to survive the collision.

Collisions come in flavors distinguished by kinetic energy. Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve momentum but lose kinetic energy to heat and deformation, with perfectly inelastic collisions (objects stick together) losing the most. Momentum conservation also governs recoil and explosions: a system starting at rest must end with vector momenta summing to zero. The related concept of impulse — force times time, equal to the change in momentum — connects forces to momentum changes when systems aren't isolated.

Conservation of momentum is a pillar of the AP Physics 1 exam, which tests collision analysis, impulse, and qualitative reasoning about isolated systems, and it appears on the FE Civil exam within dynamics, where impulse-momentum methods complement work-energy approaches.