Three solid spheres each of mass m and radius R are released from the position shown in figure. The speed of any one sphere at the time of collision would be

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$\sqrt{G m (\frac{1}{R} - \frac{2}{d})}$

$\sqrt{G m (\frac{3}{d} - \frac{1}{R})}$

$\sqrt{G m (\frac{1}{d} - \frac{3}{R})}$

$\sqrt{G m (\frac{2}{R} - \frac{1}{d})}$

answer is D.

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Detailed Solution

From conservation of mechanical energy,

$\begin{array}{l} 3 {\frac{1}{2} m v^{2}} = 3 {\frac{G m^{2}}{2 R} - \frac{G m^{2}}{d}} \\ v^{2} = G m {\frac{1}{R} - \frac{2}{d}} \Rightarrow v = \sqrt{G m {\frac{1}{R} - \frac{2}{d}}} \end{array}$

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