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>2R [Assume there is no external gravitational force acting on the system of three spheres].

# 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 $\left(d>2R\right)$ [Assume there is no external gravitational force acting on the system of three spheres].

1. A

$\sqrt{Gm\left(\frac{1}{d}-\frac{3}{R}\right)}$

2. B

$\sqrt{Gm\left(\frac{3}{d}-\frac{1}{R}\right)}$

3. C

$\sqrt{Gm\left(\frac{2}{R}-\frac{1}{d}\right)}$

4. D

$\sqrt{Gm\left(\frac{1}{R}-\frac{2}{d}\right)}$

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### Solution:

From conservation of mechanical energy

Energy of system at the instance of release = Energy of system at the instance of Collision

$\therefore \text{\hspace{0.17em}}3\left[-\frac{G{m}^{2}}{d}\right]=3\left[-\frac{G{m}^{2}}{2R}\right]+3\left[\frac{1}{2}m{v}^{2}\right]\text{\hspace{0.17em}\hspace{0.17em}}$

and $v=\sqrt{Gm\left(\frac{1}{R}-\frac{2}{d}\right)}$

Therefore, the correct answer is (D).