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 m_1.

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.