Question

Explain how momentum gets conserved in collision of two bodies?

Answer 1

Momentum

1. The momentum of any object is the product of its mass and velocity.
2. According to the law of conservation of momentum, when two objects or bodies collide with each other the total momentum before the collision is equal to the total momentum after the collision.
3. The mass of the system must remain constant during the interaction.

Conservation of momentum

1. Let us consider two objects A and B (as shown in the figure) with mass m_A and m_B respectively moving with initial velocities u_A and u_B.
2. If these two objects collide with each other for time ‘t’, object A applies force on object B, that is F_AB. Similarly, object B applies force on A, that is F_BA.
3. Here F_AB is an action whereas F_BA is a reaction. According to Newton’s third law, every action has an equal and opposite reaction .$F_{AB}=-F_{BA}$ -------- (1)
4. If v_A and v_B are the velocities of the objects A and B after collision respectively, then according to Newton's second law, a force to be equal to change in momentum (mass times velocity) per change in time. $F_{AB}=\frac{m_A\left(v_A-u_A\right)}{t}$ and $F_{BA}=\frac{m_B\left(v_B-u_B\right)}{t}$
5. Hence Eq. (1) can be written with the values of F_AB and F_BA as

$$\begin{gathered} \frac{m_A\left(v_A-u_A\right)}{t}=-\frac{m_B\left(v_B-u_B\right)}{t} \\ or{m}_A\left(v_A-u_A\right)=-m_B\left(v_B-u_B\right) \\ or{m}_A{v}_A+m_B{v}_B=m_A{u}_A+m_B{u}_B \end{gathered}$$

The above equation shows that momentum gets conserved in a collision of two bodies.