Conservation of momentum

Isolated system and internal forces

A system is said to be isolated (or pseudo-isolated) if the sum of the external forces acting on it is zero or negligible. Internal forces (between parts of the system, for example during a collision) have, by the principle of reciprocal actions, no effect on the total momentum: they cancel each other out in pairs.

Conservation theorem

If the resultant of the external forces is zero, then dP/dt = 0, so:

P_before = P_after = constant

This is the law of conservation of momentum, which holds for any isolated system, regardless of the type of internal interaction (collision, explosion, chemical reaction, etc.).

Example: recoil of a firearm

A rifle of mass M = 4 kg fires a bullet of mass m = 10 g at a velocity v = 400 m/s. Before the shot, P = 0. After the shot:

MV + mv = 0 -> V = -(mv)/M = -(0.01400)/4 = -1 m/s

The rifle recoils at 1 m/s in the opposite direction to the bullet.

Summary table

Situation	Is P conserved?	Condition
Collision between two marbles on a frictionless table	Yes	Sum of external forces is zero (weight balanced by the normal reaction)
Free fall of two bodies colliding	No, vertically	Weight is a non-zero external force
Explosion of a shell in flight	Yes, as an approximation	Very large internal forces outweigh the weight during the very short duration of the explosion

Classic pitfall

Conservation applies to the system as a whole, not to each part separately: the momentum of each sub-part may change over time; only the total sum remains constant.