Derivation of Conservation of Momentum

Let us consider two bodies A and B with initial velocities u₁ and u₂ respectively. After some time, they undergo collision. After the collision, the velocities of A and B will become v₁ and v₂. Let the time of contact between two bodies is t and m₁ and m₂ be the masses of bodies A and B respectively.

Let’s calculate the change in momentum for body A and it is represented by p_A and is given by,

p_A = m₁ (v₁ – u₁)

Let’s calculate the change in momentum for body B and it is represented by p_B and is given by,

p_B = m₂ (v₂ – u₂)

Now, from Newton’s Third Law of Motion it can be written as,

F_AB = -F_BA                                                                                                                                                                                                 ……(1)

where F_AB is the force acting on A due to B and F_BA is the force acting on B due to A.

Now, from Newton’s Second Law of Motion it can be written as,

F_AB = m₁ a₁  and                                                                                                                                                                                       ……(2)

F_BA = m₂ a₂                                                                                                                                                                                              ……(3)

where a₁ and a₂ are the acceleration of bodies A and B.

Therefore, substitute equation (2) and (3) in equation (1) as, 

m₁ a₁ = m₂ a₂ 

Therefore, this implies that,

m₁ (v₁ – u₁) / t = m₂ (v₂ – u₂) / t

m₁ (v₁ – u₁) = m₂ (v₂ – u₂)

m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂                                                                                                                                                               ……(4)

this equation (4) represents the equation of law of conservation of momentum.

where L.H.S represents the total momentum of bodies A and B before the collision and R.H.S represents the total momentum of bodies A and B after the collision.

Thus, from this it is concluded that momentum is conserved.

Assume a fast truck collides with a stopped automobile, causing the automobile to begin moving. What exactly is going on behind the scenes? In this case, as the truck’s velocity drops, the automobile’s velocity increases, and therefore the momentum lost by the truck is acquired by the automobile. What do you think? Let’s learn more about the momentum and its discussion below: