Momentum Conservation Requires Newton's Third Law

The conservation of momentum and Newton's third law of action-reaction forces are shown to be conceptually and mathematically equivalent within a closed system. When no external forces act on a system, the total momentum ($\vec{p}$) remains conserved. If the system is divided into two parts ($\vec{p}_1$ and $\vec{p}_2$), the conservation condition, $\frac{d \vec{p}_1}{d t}+\frac{d \vec{p}_2}{d t}=0$, directly yields the relationship $\vec{F}_1=-\vec{F}_2$, which is explicitly defined as Newton's third law. This dynamic relationship means that if one subsystem experiences a net force ($\vec{F}_1$), the other must simultaneously experience an equal and opposite net force ($-\vec{F}_1$). Furthermore, visualizing a two-body collision demonstrates that the law of conservation of momentum is a direct consequence of Newton's third law. The conservation of total momentum is guaranteed because the internal forces between the objects are consistently equal in magnitude and opposite in direction, ensuring that the net internal force is zero at every instant, which thus maintains a zero total rate of change of momentum.

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