First slide

Conservation of mechanical energy

Question

Three solid spheres each of mass m and radius R are placed at three corners of an equilateral triangle of side ‘d’ released. The speed of any one sphere at the time of collision would be $(d > 2 R)$ [Assume there is no external gravitational force acting on the system of three spheres].

Moderate

A

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

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

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

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

Solution

From conservation of mechanical energy
Energy of system at the instance of release = Energy of system at the instance of Collision
$∴ 3 [- \frac{G m^{2}}{d}] = 3 [- \frac{G m^{2}}{2 R}] + 3 [\frac{1}{2} m v^{2}]$
and $v = \sqrt{G m (\frac{1}{R} - \frac{2}{d})}$
Therefore, the correct answer is (D).

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