Question

A billiard ball, initially at rest, is given a sharp impulse by cue. The cue is held horizontally at a distance $h$ above the central line as shown in the figure. The ball leaves the cue with a speed $v_0$. It rolls and slides while moving forward and eventually acquires a final speed $\left(9/7\right)v_0$. If $R$ is the radius of the ball, then $h$ is

• A. $\frac{2R}{5}$
• B. $\frac{3R}{5}$
• C. $\frac{4R}{5}$
• D. $\frac{R}{5}$

Answer 1

The correct option is C $\frac{4R}{5}$

$I(w-w_0)+m(v-v_0)R=0$

$\Rightarrow m{v}_{0}h=I{w}_0$

$\therefore Iw+m(v-v_0)R-m{v}_{0}h=0$

$\Rightarrow w=\frac{v}{R}$

$\Rightarrow I=\frac{2}{5}m{R}^2$

$\therefore m(v-v_0)R+\frac{2}{5}m{R}^2-m{v}_{0}h=0$

$v=\frac{9}{7}{v}_0$

$\therefore h=(\frac{9}{7}-1)R+\frac{2}{5}R\left(\frac{9}{7}\right)$

$=\frac{4}{5}R$

Hence, the answer is $\frac{4}{5}R.$