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(C) Let $v$ be the final velocity of the centre of the sphere and $\omega$ be the final angular velocity when it starts rolling without slipping.Since

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$\frac{5}{7} v_{0}$

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(C) Let $v$ be the final velocity of the centre of the sphere and $\omega$ be the final angular velocity when it starts rolling without slipping.Since the friction force acts at the point of contact, the net torque about the point of contact is zero.Therefore, the angular momentum about the point of contact remains conserved.Initial angular momentum about the point of contact: $L_i = m v_0 r$Final angular momentum about the point of contact: $L_f = mvr + I_{cm}\omega$For a solid sphere, the moment of inertia about its centre is $I_{cm} = \frac{2}{5} mr^2$.Since it rolls without slipping, the condition is $v = r\omega$, or $\omega = \frac{v}{r}$.Equating initial and final angular momentum:$mv_0 r = mvr + (\frac{2}{5} mr^2)(\frac{v}{r})$$mv_0 r = mvr + \frac{2}{5} mvr$$v_0 = v + \frac{2}{5} v$$v_0 = \frac{7}{5} v$$v = \frac{5}{7} v_0$

$A$ solid uniform sphere resting on a rough horizontal plane is given a horizontal impulse directed through its centre so that it starts sliding with an initial velocity $v_{0}$. When it finally starts rolling without slipping, the speed of its centre is

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