Question

State and prove law of conservation of linear momentum?

Answer 1

Dear Student,

The principle of conservation of momentum states that states that if two objects collide, then the total momentum before and after the collision will be the same if there is no external force acting on the colliding objects.

Consider an isolated system of two objects m₁ and m₂ whereby m₁ is striking m₂. (An isolated system is one which has no external forces acting on it.)
Mathematically, we can deduce the principle of conservation of linear momentum by applying Newton’s second and third laws to the colliding bodies as illustrated below:
From Newton’s third law, the contact forces of impact are action and reaction pair
F₂₁ = – F₁₂

If the two bodies are in contact for a time of Δt,
F₂₁ Δt = – F₁₂ Δt ——(1)

impulse on m₁ = F₂₁ Δt ( impulse = change in momentum)
F₁₂ Δt = m₁v₁ – m₁u₁
impulse on m₂ = F₁₂Δt
F₂₁ Δt = m₂v₂ – m₂u₂
Substitute into (1),

(m₁v₁– m₁u₁) = – (m₂v₂ – m₂u₂)

Rearranging => m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Thus, total linear momentum before collision = total linear momentum after collision,

which proves the principle of conservation of linear momentum.

Regards.