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(C) Let the mass of each ball be $m$. The initial velocity of the first ball is $u$ and the second ball is at rest $(0)$.The initial kinetic energy of

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$\frac{1}{\sqrt{2}}$

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(C) Let the mass of each ball be $m$. The initial velocity of the first ball is $u$ and the second ball is at rest $(0)$.The initial kinetic energy of the system is $KE_i = \frac{1}{2}mu^2$.The loss in kinetic energy during a perfectly inelastic collision between two equal masses is given by $\Delta KE = \frac{1}{4}m(1-e^2)u^2$.Given that the loss in kinetic energy is $\frac{1}{4}$ of the initial kinetic energy:$\Delta KE = \frac{1}{4} KE_i = \frac{1}{4} (\frac{1}{2}mu^2) = \frac{1}{8}mu^2$.Equating the two expressions for $\Delta KE$:$\frac{1}{4}m(1-e^2)u^2 = \frac{1}{8}mu^2$.Dividing both sides by $\frac{1}{4}mu^2$:$1-e^2 = \frac{1}{2}$.$e^2 = 1 - \frac{1}{2} = \frac{1}{2}$.$e = \frac{1}{\sqrt{2}}$.

$A$ ball impinges directly on a similar ball at rest. If $1/4^{th}$ of the kinetic energy is lost by the impact, the value of the coefficient of restitution is:

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