Question

A point mass $M$ moving with a certain velocity collides with a stationary point mass $M/2$. The collision is elastic and in one-dimension. Let the ratio of the final velocities of $M$ and $M/2$ be $x$. The value of $x$ is

• A. $2$
• B. $3$
• C. $1/2$
• D. $1/4$

Answer 1

Answer: (D)

$1/4$

Answer 2

As collision is elastic both linear momentum and kinetic energy are conserved.
We have, following given condition,

Momentum conservation gives,
$M u_{1}=M v_{1}+\frac{M}{2} v_{2}$
$\Rightarrow 2 u_{1}=2 v_{1}+v_{2} \ldots (i)$
Energy conservation gives,
$\frac{1}{2}Mu_{1}^{2} =\frac{1}{2}Mv_{1}^{2} + \frac{1}{2}\frac{M}{2} v_{2}^{2}$
$\Rightarrow\,\,\, 2u_{1}^{2} =2u_{1}^{2}+u_{2}^{2} \dots(ii)$
Now, substituting for $u_{1}$ from E$(i)$ in E $(ii),$ we get
$2\left(\frac{2u_{1}+u_{2}}{2}\right)^{2} =2u_{1}^{2}+u_{2}^{2}$
$\Rightarrow 4v_{1}^{2}+v_{2}^{2}+4v_{1}v_{2}=4v_{1}^{2}+2v_{2}^{2}$
$\Rightarrow v_{2}^{2}=4v_{1}v_{2}$
or $\frac{v_{1}}{v_{2}}=\frac{1}{4}$
$\therefore$ Ratio of final velocities of $M$ and$\frac{M}{2}$ is $\frac{1}{4}$ .