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Question

$(d)$ As collision is elastic both linear momentum and kinetic energy are conserved.

We have, following given condition,

Momentum conservation g

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$(d)$ As collision is elastic both linear momentum and kinetic energy are conserved.

We have, following given condition,

Momentum conservation gives,

$M u_{1}=M v_{1}+\frac{M}{2} v_{2} \Rightarrow 2 u_{1}=2 v_{1}+v_{2} \ldots 	ext { (i) }$

Energy conservation gives,

$\frac{1}{2} M u_{1}^{2} =\frac{1}{2} M v_{1}^{2}+\frac{1}{2} \frac{M}{2} v_{2}^{2}$

$\Rightarrow \quad 2 u_{1}^{2} =2 v_{1}^{2}+v_{2}^{2}\ldots 	ext { (ii) }$

Now, substituting for $u_{1}$ from Eq. $(i)$ in Eq. $(ii)$, we get

$2\left(\frac{2 v_{1}+v_{2}}{2}ight)^{2}=2 v_{1}^{2}+v_{2}^{2}$

$\Rightarrow 4 v_{1}^{2}+v_{2}^{2}+4 v_{1} v_{2}=4 v_{1}^{2}+2 v_{2}^{2}$

$\Rightarrow v_{2}^{2}=4 v_{1} v_{2}$

$	ext { or } \frac{v_{1}}{v_{2}}=\frac{1}{4}$

$	herefore$ Ratio of final velocities of $M$ and $\frac{M}{2}$ is $\frac{1}{4}$.

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

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