The

law of conservation of linear momentumstates that if no external forces act on the system of two colliding objects, then the vector sum of the linear momentum of each body remains constant and is not affected by their mutual interaction.

Alternatively, it states that if net external force acting on a system is zero, the total momentum of the system remains constant.

Proof:

Let us consider a particle of mass ‘m’ and acceleration ‘a’. Then, from 2

^{nd}law of motion, If no external force acts on the body then, F=0,

Therefore, ‘P’ is constant or conserved.

(

*Note: If the derivative of any quantity is zero, it must be a constant quantity.*)**Deduction of Law of Conservation of linear momentum for two colliding bodies**

Let us consider two bodies of masses m

_{1}and m_{2}moving in straight line in the same direction with initial velocities u_{1}and u_{2}. They collide for a short time ∆t. After collision, they move with velocities v_{1}and v_{2}.From 2

^{nd}law of motion,Force applied by A on B = Rate of change of momentum of B

F

_{AB}= (m_{2}v_{2}-m_{2}u_{2})/∆tSimilarly,

Force applied by B on A = Rate of change of momentum of A

F

_{BA}= (m_{1}v_{1}-m_{1}u_{1})/∆tFrom Newton’s 3

^{rd}law of motion,F

_{AB}= -F_{BA}Or, (m

_{2}v_{2}-m_{2}u_{2})/∆t = -(m_{1}v_{1}-m_{1}u_{1})/∆tOr, m

_{2}v_{2}-m_{2}u_{2}= -m1v1+m1u1Or, m

_{1}u_{1}+ m_{2}u_{2}= m_{1}v_{1}+ m_{2}v_{2}

This means the total momentum before collision is equal to total momentum after collision. This proves the principle of co conservation of linear momentum.