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

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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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