5.7 Conservation of momentum

CONSERVATION OF MOMENTUM

The second and third laws of motion lead to an important consequence: the law of conservation of momentum. Let’s assume an example a bullet is fired from a gun. If the force on the bullet by the gun is \(\vec{F}\), the force on the gun by the bullet is \(-\vec{F}\), according to the third law. The two forces act for a common interval of time \(\Delta t\). According to the second law, \(\vec{F} \Delta t\) is the change in momentum of the bullet, and \(-\vec{F} \Delta t\) is the change in momentum of the gun. Thus if \(\vec{p}_{b}\) is the momentum of the bullet after firing and \(\vec{p}_{g}\) is the recoil momentum of the gun, \(\vec{p}_{g}=-\vec{p}_{b}\) i.e. \(\vec{p}_{b}+\vec{p}_{g}=0\). That is, the total momentum of the (bullet + gun) system is conserved.

Thus in an isolated system (i.e. a system with no external force), mutual forces between pairs of particles in the system can cause momentum change in individual particles, but since the mutual forces for each pair are equal and opposite, the momentum changes cancel in pairs and the total momentum remains unchanged. This fact is known as the law of conservation of momentum.

The total momentum of an isolated system of interacting particles is conserved.

An important example of the application of the law of conservation of momentum is the collision of two bodies. Consider two bodies A and B, with initial momenta \(\mathbf{p}_A\) and \(\mathbf{p}_B\). The bodies collide, get apart, with final momenta \(\mathbf{p}_A^{\prime}\) and \(\mathbf{p}_B^{\prime}\) respectively. By the Second Law
\(
\begin{aligned}
& \mathbf{F}_{A B} \Delta t=\mathbf{p}_A^{\prime}-\mathbf{p}_A \text { and } \\
& \mathbf{F}_{B A} \Delta t=\mathbf{p}_B^{\prime}-\mathbf{p}_B
\end{aligned}
\)
(where we have taken a common interval of time for both forces i.e. the time for which the two bodies are in contact.)
Since \(\mathbf{F}_{A B}=-\mathbf{F}_{B A}\) by the third law,
\(
\mathbf{p}_A^{\prime}-\mathbf{p}_A=-\left(\mathbf{p}_B^{\prime}-\mathbf{p}_B\right)
\)
i.e. \(\mathbf{p}_A^{\prime}+\mathbf{p}_B^{\prime}=\mathbf{p}_A+\mathbf{p}_B\)
which shows that the total final momentum of the isolated system equals its initial momentum. Notice that this is true whether the collision is elastic or inelastic. In elastic collisions, there is a second condition that the total initial kinetic energy of the system equals the total final kinetic energy.