For a system of particles, the total linear momentum is

$$\mathbf P=\sum_i\mathbf p_i.$$

Forces between parts of the chosen system are internal. They can redistribute momentum among those parts, but they do not change the system's total momentum. The change in total momentum is determined by the net external impulse:

$$\Delta\mathbf P=\mathbf J_{\mathrm{ext}}.$$

Therefore, if the net external impulse is zero,

$$\mathbf J_{\mathrm{ext}}=\mathbf0,$$

then

$$\mathbf P_i=\mathbf P_f.$$

This is conservation of linear momentum.

Choosing the system

Whether momentum is conserved depends on the system boundary. An interaction between two objects is internal only if both objects are included in the system. A force exerted by something outside the system contributes external impulse.

Momentum can also be approximately conserved over a short interval when the external impulse is not exactly zero but is negligible compared with the internal impulses. This is why momentum conservation is often useful during impacts, explosions, and other brief interactions.

Example: two skaters push apart

Two skaters initially stand at rest together on nearly frictionless ice. Their masses are $40,\mathrm{kg}$ and $60,\mathrm{kg}$. After they push apart, the $40,\mathrm{kg}$ skater moves at $3.0,\mathrm{m/s}$ to the right.

The initial total momentum is zero. Taking right as positive,

$$0=(40)(3.0)+(60)v_2.$$

Thus

$$v_2=-2.0,\mathrm{m/s}.$$

The heavier skater moves left more slowly so that the two momenta remain equal in magnitude and opposite in direction.

Momentum conservation is a system statement

The individual skaters do not conserve their own momentum: each changes momentum because the other exerts a force. The combined system conserves momentum because those interaction forces are internal and the external impulse is negligible.

The same reasoning applies to collisions and separations. Additional information, such as whether kinetic energy is conserved, is needed to determine the detailed outcome of a particular collision.