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

(i) The force on each particle ( Newton's second law) can be written as
` vecF_(12) = ( dvecp_(1))/(dt) and vecF_(21) = ( dvecP_(2))/(dt)`
(ii) Here ` vecP_(1)` is the momentum of particle 1 which changes due to the force ` vecF_(12)` exerted by particle 2. Further ` vecP_(2)` is the momentum of particle 2.
This changes due to ` vecF_(21)` exerted by particle 1 .
` ( dvecp_(1))/(dt) = - (dvecp_(2))/( dt)`
` ( dvecP_(1))/( dt) + ( d vecP_(2))/(dt) = 0`
` d/( dt) ( vecP_(1) + vecP_(2)) = 0`
(iii) It implies that ` vecP_(1) + vecP_(2)` = constant vector ( always)
(iv) `vecP_(1) + vecP_(2) ` is the total linear momentum of the two particles ` ( vecP_(" tot" = vecP_(1) + vecP_(2))` . It is also called as total linear momentum of the system. Here , the two particles constitute the system.
(v) If there are no external forces acting on the system, then the total linear momentum of the system ` (vecP_(" tot"))` is always a constant vector .