$H=\frac{u^2 \sin ^2 \theta}{2 g} \Rightarrow \sin \theta=\sqrt{\frac{2 g H}{u^2}}$
$R=\frac{2 u^2 \sin \theta \cos \theta}{g}$
$R=\frac{2 u^2}{g} \sqrt{\frac{2 g H}{u^2}} \times \sqrt{1-\frac{2 g H}{u^2}}$
$R=\frac{2 u^2}{g} \sqrt{\frac{2 g H}{u^2}} \times \sqrt{\frac{u^2-2 g H}{u^2}}$
$\frac{g R}{2 \sqrt{2 g H}}=\sqrt{u^2-2 g H}$
Squaring both the sides,
$\frac{g R^2}{4 \times 2 g H}=u^2-2 g H$
$\Rightarrow u^2=2 g H+\frac{9 R^2}{8 H}$
$u=\sqrt{2 g H+\frac{g R^2}{8 H}}$
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| List $-I$ | List $-II$ |
| $(A)$ Moment of inertia of solid sphere of radius $(R)$ about any tangent | $(I)$ $\frac{5}{3} MR ^{2}$ |
| $(B)$ Moment of inertia of hollow sphere of radius $(R)$ about any tangent | $(II)$ $\frac{7}{5} MR ^{2}$ |
| $(C)$ Moment of inertia of circular ring of radius $(R)$ about its diameter. | $(III)$ $\frac{1}{4} MR ^{2}$ |
| $(D)$ Moment of inertia of circular disc of radius $(R)$ about any diameter. | $(IV)$ $\frac{1}{2} MR ^{2}$ |
Question: Choose the correct answer from the options given below
