Law of Conservation of Total Linear Momentum

There are three fundamental conservation laws in mechanics — of total energy, total linear momentum, and total angular momentum. The law of conservation of linear momentum follows directly by combining Newton's second and third laws.

Derivation. Consider two particles interacting only with each other (an isolated system, with no external force). By the third law, the force particle 2 exerts on particle 1, $\vec F_{12}$, and the force particle 1 exerts on particle 2, $\vec F_{21}$, satisfy $\vec F_{12}=-\vec F_{21}$. By the second law, $\vec F_{12}=d\vec p_1/dt$ and $\vec F_{21}=d\vec p_2/dt$. Substituting:

Statement. If no external force acts on a system, its total linear momentum $\vec p_{tot}$ is a constant vector — it is conserved in time. Individual particle momenta $\vec p_1,\vec p_2,\ldots$ may still change, as long as their vector sum stays fixed. The internal forces between the particles of the system (a third-law pair) can never, by themselves, change the system's total momentum.

Why it's useful. Applying the second law directly requires knowing the exact force involved throughout an interaction — often difficult (e.g. the brief, complicated contact force during a collision). Momentum conservation sidesteps this entirely: it only needs the momenta before and after, never the (possibly very complicated) forces during.

Worked examples: …