Question

Consider the following statements $ A $ and $ B $ . Identify the correct choice in the given answers.
A. In a one dimensional perfectly elastic collision between two moving bodies of equal masses, the bodies merely exchange their velocities after collision.
B. If a lighter body at rest suffers perfectly elastic collision with a very heavy body moving with a certain velocity, then after collision both travel with same velocity.

• A. $ A $ and $ B $ are correct
• B. Both $ A $ and $ B $ are wrong
• C. $ A $ is correct, $ B $ is wrong
• D. $ A $ is wrong, $ B $ is correct

Answer 1

Answer: (C)

For $ A $ :
Using the laws of conservation of linear momentum and energy, the velocities of two bodies after perfectly elastic collision are given by
$ v_{1} = \frac{\left(m_{1}-m_{2}\right)u_{1}}{m_{1}+m_{2}}+\frac{2m_{2}u_{2}}{m_{1}+m_{2}} \quad\ldots\left(i\right) $
$ v_{2} =\frac{2m_{1}u_{2}}{m_{1}+m_{2}}+\frac{\left(m_{2}-m_{1}\right)u_{2}}{m_{1}+m_{2}}\quad \ldots \left(ii\right) $
where $ m_1 $ and $ m_2 $ are the masses of two bodies and $ u_1 $ and $ u_2 $ are the velocities of two bodies before collision.
If $ m_1 = m_2 $ , then
From $ (i) $ and $ (ii) $ , we get
$ v_1 = u_2 $ and $ v_2 = u_1 $
i.e., in perfectly elastic collision of two moving bodies of equal masses, the bodies merely exchange their velocities after collision. Thus, statement $ A $ is correct.
For $ B $ :
In Eqs. $ (i) $ and $ (ii) $ ,
If $ m_2 < < m_1 $ and $ u_2 = 0 $ , then
$ v_1 = u_1 $ and $ v_2 = 2u_1 $
Thus, statement $ B $ is wrong.