Law of conservation of momentum states that total momentum of the system remains conserved in the absence of external force.
Proof: Consider two bodies of mass m1 and m2 moving with initial velocity u1 and u2 respectively. The two bodies collide with each other for a time interval ‘t’. The velocity after collision be v1 & v2 respectively. Let F12 be the force applied by m1 on m2 and F21 be the force applied by m2 on m1.
Momentum of mass m1 before collision = m1 u1
Momentum of mass m2 before collision = m2 u2
Momentum of mass m1 after collision = m1 v1
Momentum of mass m2 after collision = m2 v2
Impulse = force × time = change in momentum
For mass m1 :
F12 t = m1 v1 - m1 u1 …………….(1)
For mass m2:
F21t = m2 v2 – m2 u2 ……………..(2)
Adding equation (1) & (2)
F12 t+ F21t = (m1 v1 - m1 u1) + (m2 v2 – m2 u2)
(F12 + F21 ) t = (m1 v1 + m2 v2) - (m1 u1 + m2 u2)
According to Newton’s third law:
F12 = - F21
F12 + F21 = 0
(m1 v1 + m2 v2) = (m1 u1 + m2 u2)
Final momentum = Initial momentum
Hence momentum is conserved.