Question

A body of mass $m_1$ moving with certain velocity $V_0$ collides head-on with another body of mass $m_2$ at rest. List-1 gives relation between $m_1$ and $m_2$ and List-2 gives velocity of $m_2$ after collision.

If $e=\frac{1}{3}$ Match List - 1 with List - 2

$\begin{array}{cccc}& \text{List-1}& & \text{List-2}\\ \text{(I)}& m_2=m_1& \text{(P)}& V_0\\ \text{(II)}& m_2=4m_1& \text{(Q)}& \frac{2V_0}{3}\\ \text{(III)}& m_2=2m_1& \text{(R)}& \frac{2V_0}{5}\\ \text{(IV)}& m_2=9m_1& \text{(S)}& \frac{4V_0}{9}\\ & & \text{(T)}& \frac{4V_0}{15}\\ & & \text{(U)}& \text{None}\end{array}$

• A. I $\to$ R, II $\to$ S, III $\to$ T, IV $\to$ U
• B. I $\to$ S, II $\to$ T, III $\to$ U, IV $\to$ R
• C. I $\to$ Q, II $\to$ S, III $\to$ T, IV $\to$ R
• D. I $\to$ Q, II $\to$ T, III $\to$ S, IV $\to$ U

Answer 1

The correct option is D I $\to$ Q, II $\to$ T, III $\to$ S, IV $\to$ U


By COM $\Rightarrow m{V}_0=m_2{V}_2-m_1{V}_1\dots \dots \left(1\right)$

By Vel of seperation $=e\times$ Vel of approach,
$\frac{V_0}{3}=V_2+V_1\dots \dots \left(2\right)$

I) $\begin{array}{c}{V}_0=V_2-V_1\\ \frac{V_0}{3}=V_2+V_1\end{array}\}\Rightarrow \frac{4}{3}{V}_0=2V_2\Rightarrow V_2=\frac{2V_0}{3}$

II) $\begin{array}{c}{V}_0=4V_2-V_1\\ \frac{V_0}{3}=V_2+V_1\end{array}\}\Rightarrow \frac{4}{3}{V}_0=5V_2\Rightarrow V_2=\frac{4}{15}{V}_0$

III) $\begin{array}{c}{V}_0=2V_2-V_1\\ \frac{V_0}{3}=V_2+V_1\end{array}\}\Rightarrow \frac{4}{3}{V}_0=3V_2\Rightarrow V_2=\frac{4}{9}{V}_0$

IV) $\begin{array}{c}{V}_0=9V_2-V_1\\ \frac{V_0}{3}=V_2+V_1\end{array}\}\Rightarrow \frac{4}{3}{V}_0=10V_2\Rightarrow V_2=\frac{2}{15}{V}_0$