Prove the law of conservation of linear momentum use it to find the recoil velocity of a gun when a bullet is fired from it

Explanation

When two particles interact with each other they exert equal and opposite forces on each other . The particle 1 exerts force `vec(F ) _(21)` on particle 2 and particle 2 exerts an exactly equal and opposite force `vec(F ) _(u)` on particle 1 According to Newton's third law.
`vec(F ) = (vec(dp)_(1))/( dt ) = vec(F )_(21)= (vec(dp)_(1))/(dt)`
Here `vec(p)_(1)` is teh momentum of particle 1 which changes due to the force `vec(F )_(21)` exerted by particle 2. further `vec(p)_(2)` the momentum of particle 2. This changes due to `vec(F )_(21)` exerted by particle 1 .
`(vec(dp)_(1))/(dt)= (vec(d p)_(2))/( dt)`
`(d ) /(dt ) (vec(p) _(1)+ vec(p)_(2)) = 0`
It is implied that `vec(p)_(1) + vec(P)_(2)` constant vector.
From this result the law of conservation of linear momentum can be stated as follows .
If there are no external forces acting on the system then the total linear momentum of the system `(vec(p)_(mas))` is always a constant vector.
Let us consider the firing of gun .When the gun and bullet are at rest initially . the total linear momenturn of the system is zero before firing Since initially both are at rest .

Total momenturn before firing the gun is zero `vec(p)_(1) + p_(2) = 0`
the law of conservation states that of linear momenturn total , `vec(p)_(1) " to " vec(p)_(1)` To conserve the total linear momenturn of the system the momenturn of the gun must also change from `vec(p)_(1) " to " vec(p)_(2)`
Due to the conservation of linear momenturn `vec(p)_(1) + vec(p)_(2)` . It implies that `vec(p)_(1) = vec(p)_(2)` . The momenturn of the gun is exactly equal but in the opposite direction to the momenturn of teh bullet.