Conservation of Momentum

Theorem: Conservation of Momentum

If the vector sum of all external forces acting on a physical system is zero, then the total momentum in the system remains constant.

$$\sum {\mathbf{F}}_{\text{ext}}=0⟹\sum \mathbf{p}=\text{const}$$

PROOF

We consider an interaction between any two objects $A$ and $B$ in the physical system. According to Newton’s third law of motion, this interaction manifests as two forces ${\mathbf{F}}_{A\text{ on }B}$ and ${\mathbf{F}}_{B\text{ on }A}$ with

$${\mathbf{F}}_{A\text{ on }B}=-{\mathbf{F}}_{B\text{ on }A}$$

According to Newton’s second law of motion the force ${\mathbf{F}}_{A\text{ on }B}$ is equal to the change $\frac{\mathrm{d}{\mathbf{p}}_B}{\mathrm{d}t}$ in the momentum of $B$ and the force ${\mathbf{F}}_{B\text{ on }A}$ is equal to the change $\frac{\mathrm{d}{\mathbf{p}}_A}{\mathrm{d}t}$ in the momentum of $A$. This means that

$$\frac{\mathrm{d}{\mathbf{p}}_A}{\mathrm{d}t}=-\frac{\mathrm{d}{\mathbf{p}}_B}{\mathrm{d}t}$$ $$\frac{\mathrm{d}{\mathbf{p}}_A}{\mathrm{d}t}+\frac{\mathrm{d}{\mathbf{p}}_B}{\mathrm{d}t}=0$$

Therefore, the change in the total momentum of $A$ and $B$ is zero and is thus unaffected by internal forces. This is true for any pair of objects in the physical system and so it must be true for the total momentum of the entire system. This means that the total momentum is only affected by external forces and if there are no such forces, then it is not affected by anything and therefore remains unchanged.