The law of conservation of momentum

The proof of the law of conservation of momentum is as follows.

Let us consider a system of two objects in motion as shown in the figure. We take, $\vec p_1$ and $\vec p_2$ are the momenta of the object 1 and object 2. These are the momenta of the objects before the collision. The two objects involve in a collision, and after the collision, let the momenta of the object 1 and the object 2 as $\vec p_1'$ and $\vec p_2'$.

Now, let us look at what happens when the two objects collide. During the collision, object 1 exerts a force on object 2 and object 2 exerts a force on object 1. These two forces are equal and opposite according to Newton's third law. Let us take, $\vec F_{21}$ is the force on object 2 by object 1 and, $\vec F_{12}$ is the force on object 1 by object 2. Since these forces are equal and opposite, we have

$\vec F_{21}=-\vec F_{12}$.

You learned that a force on an object change the momentum of the object, according to the equation,

$\vec F=\dfrac{\Delta \vec p}{\Delta t}$

So, the force $\vec F_{21}$ on the object 2 changes the momentum of this object from $\vec p_2$ to $\vec p_2'$, therefore,

$\vec F_{21}=\dfrac{\Delta \vec p_2}{\Delta t}=\dfrac{\vec p_2'-\vec p_2}{\Delta t}$

Likewise, the force, $\vec F_{12}$ on object 1 changes the momentum of this object, from $\vec p_1$ to $\vec p_1'$, therefore

$\vec F_{12}=\dfrac{\Delta \vec p_1}{\Delta t}=\dfrac{\vec p_1'-\vec p_1}{\Delta t}$

Substituting the forces in the equation, $\vec F_{21}=-\vec F_{12}$, we get,

$\dfrac{\vec p_2'- \vec p_2}{\Delta t}=-\dfrac{\vec p_1'-\vec p_1}{\Delta t}$

Canceling, $\Delta t$ and rearranging, we get,

$\vec p_1+\vec p_2=\vec p_1'+\vec p_2'$.

Left hand side is the total momentum of the objects before the collision and the right hand side is the total momentum of the objects after the collision. i.e.,Total momentum of the system before the collision is equal to total momentum after the collision. Thus, we proved the law of conservation of momentum.