Question

State and prove the law of conservation of linear momentum.

Answer 1

According to the law of conservation of momentum 

When two or more bodies act upon each other their total momentum remains constant provided no external forces are acting 

•So, Momentum is never created or destroyed. 

•When this law is applied for a collision between two bodies, the total momentum of the colliding bodies before the collision is equal to the total momentum after the collision. 

•We can apply this law for a collision between two vehicles. This law is applicable for all types of collisions. 

•Consider two particles say A and B of mass m₁ and m₂ collide with each other and forces acting on these particles are only the ones they exert on each other. 

•Let u₁ and v₁ be the initial and final velocities of particle A and similarly, u₂ and v₂ for particle B. Let the two particles be in contact for a time t. 

So, Change in momentum of A = m₁ (v₁-u₁) Change in the momentum of B = m₂ (v₂-u₂) 

•During the collision, let A impart an average force equal to F_BA on B and let B exert an average F_AB on A. 

We know that from third law of motion F_BA = F_AB eq. (4) 

F_BA = m₂ x a₂ = \(\frac{m_2(v_2-u_2)}{t}\)

 F_AB = m₁ x a₁ = \(\frac{m_1(v_1-u_1)}{t}\)

Canceling t on both sides and rearranging the equation we get m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂ eq.(5) 

Now, m₁ u₁ + m₂ u₂ represents the total momentum of particles A and B before collision and m₁ v₁ + m₂ v₂ represents the total momentum of particles after the collision. This means that 

Total momentum before collision = total momentum after the collision 

•Equation 5 which m₁ u₁ + m₂ u₂ = m₁ v₁ + m₂ v₂, is known as the law of conservation of momentum.