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(C) Let the mass of both bodies be $m$. Let the initial velocity of the first body be $u_1 = v$ and the initial velocity of the second body be $u_2 =

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$5: 7$

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(C) Let the mass of both bodies be $m$. Let the initial velocity of the first body be $u_1 = v$ and the initial velocity of the second body be $u_2 = v/2$.By the law of conservation of linear momentum: $m u_1 + m u_2 = m v_1 + m v_2$,where $v_1$ and $v_2$ are the final velocities.$v + v/2 = v_1 + v_2 \implies v_1 + v_2 = 1.5v$  --- $(1)$The coefficient of restitution $e$ is defined as $e = (v_2 - v_1) / (u_1 - u_2)$.Given $e = 0.5$,$u_1 = v$,and $u_2 = v/2$,we have $0.5 = (v_2 - v_1) / (v - v/2)$.$0.5 = (v_2 - v_1) / (0.5v) \implies v_2 - v_1 = 0.25v$ --- $(2)$Adding equations $(1)$ and $(2)$: $2v_2 = 1.75v \implies v_2 = 0.875v = (7/8)v$.Subtracting equation $(2)$ from $(1)$: $2v_1 = 1.25v \implies v_1 = 0.625v = (5/8)v$.The ratio of the velocities $v_1 : v_2 = (5/8)v : (7/8)v = 5:7$.

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