Derive the mathematical formula of conservation of momentum.

To explain conservation of momentum, let us take the following example. Consider two balls A and B having masses m₁ and m₂, respectively. Let the initial velocity of ball A be u₁, and that of ball B be u₂ (u₁ > u₂). Their collision takes place for a very short interval of time t and after that A and B start moving with velocities v₁ and v₂ (now v₁ < v₂) respectively as shown in Figure.

The momentum of ball A before and after the collision is m₁u₁ and m₁v₁ respectively. If there are no external forces acting on the body, then the rate of change of momentum of ball A, during the collision will be

= {𝑚_1(𝑣_1− 𝑢_1) \over 𝑡}

and, similarly the rate of change in momentum of ball B

= {𝑚_2(𝑣_2− 𝑢_2) \over 𝑡}

Let F₁₂ be the force exerted by ball A on B and F₂₁ be the force exerted by ball B on A.

Then,

according to Newton’s second law of motion

F₁₂ = {𝑚_1(𝑣_1− 𝑢_1) \over 𝑡}  and   F₂₁ = {𝑚_2(𝑣_2− 𝑢_2)\over 𝑡}

According to Newton’s third law of motion, we have

F₁₂ = – F₂₁

Or, {𝑚_1(𝑣_1− 𝑢_1)\over 𝑡}  = – {𝑚_2(𝑣_2− 𝑢_2)\over 𝑡}

Or, m₁v₁ – m₁u₁ = – m₂v₂ + m₂u₂

Or, m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

i.e., Total momentum before collision = Total momentum after collision

Thus, we find that in a collision between the two balls the total momentum before and after the collision remains unchanged or conserved provided no net force acts on the system. This result is law of conservation of momentum.