State and prove conservation of linear momentum in case of collision of two bodies.

Explanation

Statement : In an isolated system of collision of bodies, the total linear momentum before impact is equal to the total linear momentum after impact.

Let `m_(1)` and `m_(2)` be the masscs of two bodies moving along `vecv_(1i)` and `vecv_(2i)`. Let `vecv_(1f)` and `vecv_(2f)` be the final velocities after the impact.
At the time of impact, the force of action acts on the body `B` and the force of reaction acts on `A`.
Applying Newton's `III` law of motion
`|"Force of action on"B|=-|"Force of reaction on"A|`
But, `vecF_(AB)Deltat=vecp_(A)'-vecp_(A)` and `vecF_(BA)Deltat=vecp_(B)'-vecp_(B)` (from Newton's second law)
i.e. `vecp_(A)'-vecp_(A)=-(vecp_(B)'-vecp_(B))`
i.e. `vecp_(A)+vecp_(B)=vecp_(A)'+vecp_(B)'`.
i.e. `m_(1)vecv_(1i)+m_(2)vecv_(2i)=m_(1)vecv_(1f)+m_(2)vecv_(2f)`
This shows that the total final linear momentum of the isolated system equals its total initial momentum.