Question

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A: Body $‘P’$ having mass $M$ moving with speed ‘$u$’ has head-on collision elastically with another body ‘$Q$’ having mass ‘$m$’ initially at rest. If $m<<M$, body ‘$Q$’ will have a maximum speed equal to ‘$2u$’ after collision.

Reason R: During elastic collision, the momentum and kinetic energy are both conserved.

In the light of the above statements, choose the most appropriate answer from the options given below:

• A. A is correct but R is not correct.
• B. Both A and R are correct but R is NOT the correct explanation of A
• C. A is not correct but R is correct.
• D. Both A and R are correct and R is the correct explanation of A.

Answer 1

The correct option is D

Both A and R are correct and R is the correct explanation of A.

Step 1. Given data :

The mass of a body$P$ is $M$.

The mass of a body $Q$ is$m$.

Where $m<<M$.

The velocity of body$P$ is $u$ before the collision

Step 2. Solving for the maximum velocity of $Q$ after collision

An elastic collision is a collision in which there is no net loss in kinetic energy in the system as a result of the collision.

Both momentum and kinetic energy are conserved quantities in elastic collisions.

From momentum conservation, second body at rest initially we have

$m_1{v}_1=m_{1}v{\text{'}}_1+m_{2}v{\text{'}}_2$

Kinetic energy we have$\frac{1}{2}{m}_1{v^2}_1=\frac{1}{2}{m}_{1}v{{\text{'}}^2}_1+\frac{1}{2}{m}_{2}v{{\text{'}}^2}_2$

The velocity of second body after collision

$$\begin{gathered} v_2\text{'}=\frac{2m_1}{m_1+m_2}{v}_1 \\ \Rightarrow v_2\text{'}=\frac{2M}{M+m}u \end{gathered}$$

now since $M>>m$, we have

$v_2\text{'}=\frac{2M}{M}u=2u$

Where $M$ is mass of Body $‘P’$ having with initial velocity $u$ and body ‘$Q$’ having mass ‘$m$’ initially at rest. After collision body $P$ have velocity $v{\text{'}}_1$ and body $Q$ have velocity $v_2\text{'}$ .

So, From conservation of momentum and kinetic energy equation was derived which gave velocity of second particle after collision to be $2u$. So, both the assertion and reason are correct and reason explains the assertion.

Hence, option D is correct.