A sphere of mass m moving with velocity u col | vedclass

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(C) Let the velocities of the two spheres after collision be $v_1$ and $v_2$ respectively.By the law of conservation of linear momentum: $mu + m(0) =

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(C) Let the velocities of the two spheres after collision be $v_1$ and $v_2$ respectively.By the law of conservation of linear momentum: $mu + m(0) = mv_1 + mv_2$, which simplifies to $v_1 + v_2 = u$.By the definition of the coefficient of restitution $e$: $e = \frac{v_2 - v_1}{u_1 - u_2}$.Given $e = \frac{1}{2}$, $u_1 = u$, and $u_2 = 0$, we have $\frac{1}{2} = \frac{v_2 - v_1}{u}$, which implies $v_2 - v_1 = \frac{u}{2}$.Now, we have a system of two equations:$1) v_1 + v_2 = u$$2) -v_1 + v_2 = \frac{u}{2}$Adding the two equations: $2v_2 = \frac{3u}{2} \implies v_2 = \frac{3u}{4}$.Subtracting the equations: $2v_1 = u - \frac{u}{2} = \frac{u}{2} \implies v_1 = \frac{u}{4}$.The ratio of their velocities is $\frac{v_1}{v_2} = \frac{u/4}{3u/4} = \frac{1}{3}$.

$A$ sphere of mass $m$ moving with velocity $u$ collides with another sphere of the same mass $m$ at rest. If the coefficient of restitution is $e = \frac{1}{2}$, what is the ratio of their velocities after the collision?

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