Total momentum of a body remains unchanged before and after the collision. Explain.

Consider two marbles A and B having masses m₁ and m₂ respectively. Let the initial velocity of marble A be u₁ and that of marble B be u₂. Suppose the two marbles collide with each other head on and their collision lasts for t seconds, then the velocities of marbles A and B become v₁ and v₂ respectively after the collisions, then the momentum of marble A before and after the collisions is m₁u₁ and m₁v₁ respectively. The rate of change of momentum during the collision will be . Similarly the rate of change of momentum of marble B will be . If the force exerted by marble A and B is F₁₂ and that by B on A is F₂₁, then according to the second law of motion
F₁₂ = ----(1)
and  F₂₁ = ----(2)
According to the third law of motion, the force F₁₂ exerted by marble A on marble B and the force F₁₂ exerted by the marble B on the marble A should be equal and opposite to each other.Therefore,
F₁₂ = -F₂₁
Substituting, the value of F₁₂ and F₂₁ from equations (1) and (2), we get
=
m₁(v₁ – u₁) = m₂(v₂ – u₂)
m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂
Now (m₁u₁ + m₂v₂) is the total momentum of the two marbles before the collision and (m₁v₁ + m₂v₂) is their total momentum after the collision. Hence we find that in a collision between the two marbles the total momentum before and after the collision remains unchanged.