# Derive the mathematical formula of conservation of momentum.

## Answer

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, m1v1m1u1 = – 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.

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