Derive the law of conservation of linear momentum from Newton's third law of motion.

Explanation

Consider two bodies A and B of masses `m_(1)`, and `m_(2)` moving in same direction along a straight line with velocities `u_(1)` and `u_(2)` respectively (`u_(1) gt u_(2)` ).
They collide for time `Deltat`. After collision, let their velocities be `v_(1)` and `v_(2)` respectively.

During collision, body A exerts force `F_(BA)` on body B. From Newton.s third law, the body B exerts force `F_(AB)` on A.
From Newton.s third law `vecF_(AB) = - vecF_(BA)`
Multiplying both sides by `Deltat` of equation (i), we have `vecF_(AB) (Deltat)=- vecF_(BA)(Deltat)`
Impulse of `vecF_(AB)=` -Impulse of `vecF_(BA)` or change in momentum.of A= change in momentum of B
`(m_(1)vecv_(1) - m_(1)vecu_(1))= (m_(2)vecv_(2) - m_(2)vecu_(2)) implies m_(1)vecv_(1)- m_(1)vecu_(1)= -m_(2)vecv_(2)+ m_(2)vecu_(2)`
`m_(1)vecu_(1)+ m_(2)vecu_(2)= m_(1)vecv_(1)+ m_(2)vecv_(2)`
Therefore, total linear momentum before collision = total linear momentum after collision.