State the law of conservation of momentum. Derive it by using Newton’s third law of motion.

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.


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