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(C) The loss in kinetic energy during a perfectly inelastic collision between two bodies of equal mass $m$,where one is at rest,is given by the formul

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$1 / \sqrt{2}$

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(C) The loss in kinetic energy during a perfectly inelastic collision between two bodies of equal mass $m$,where one is at rest,is given by the formula: $\Delta KE = \frac{1}{2} \frac{m_1 m_2}{m_1 + m_2} (1 - e^2) (u_1 - u_2)^2$.Given $m_1 = m_2 = m$,$u_1 = u$,and $u_2 = 0$,the formula becomes: $\Delta KE = \frac{1}{2} \frac{m^2}{2m} (1 - e^2) u^2 = \frac{1}{4} m (1 - e^2) u^2$.We are given that the loss in kinetic energy is $1/4$ of the initial kinetic energy $(KE_i = \frac{1}{2} m u^2)$.So,$\frac{1}{4} m (1 - e^2) u^2 = \frac{1}{4} (\frac{1}{2} m u^2)$.Dividing both sides by $\frac{1}{4} m u^2$,we get: $1 - e^2 = \frac{1}{2}$.Therefore,$e^2 = 1 - 1/2 = 1/2$.Taking the square root,we get: $e = 1 / \sqrt{2}$.

$A$ ball impinges directly on a similar ball of same mass at rest. If $1/4$ of the kinetic energy is lost by the impact,the value of the coefficient of restitution is

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