Question

$3$ identical masses are released from rest. What will be their velocities when the separation between any $2$ masses becomes $a/2$ ?

• A. $\sqrt{\frac{Gm}{a}}$
• B. $\sqrt{\frac{2Gm}{a}}$
• C. $\sqrt{\frac{Gm}{2a}}$
• D. $\sqrt{\frac{2Gm}{3a}}$

Answer 1

The correct option is B $\sqrt{\frac{2Gm}{a}}$

Initial state

Initially system is at rest, so there will be only gravitational potential energy present.

$E_i=\frac{-Gmm}{a}-\frac{Gmm}{a}-\frac{Gmm}{a}$

$\Rightarrow E_i=-\frac{3G{m}^2}{a}$


Let $v$ be the velocity of each mass at $a/2$ separation.

Finally, the system will have both kinetic energy and potential energy.

$\therefore E_f=-\frac{3G{m}^2}{a/2}+3\left(\frac{1}{2}m{v}^2\right)$

$\Rightarrow E_f=-\frac{6G{m}^2}{a}+\frac{3}{2}m{v}^2$

Since, there is no external force acting on the system. so energy remains conserved of the system.

$\therefore E_i=E_f$

$-\frac{3G{m}^2}{a}=-\frac{6G{m}^2}{a}+\frac{3}{2}m{v}^2$

$\Rightarrow \frac{Gm}{a}=\frac{v^2}{2}$

$\therefore v=\sqrt{\frac{2Gm}{a}}$

Hence, option $\left(b\right)$ is correct.