Question

Consider the two identical particles, they are released from rest and may move towards each other under the influence of mutual gravitational force. The speed of each particle, when the separation reduces to half of the initial separation is

• A. $\sqrt{\frac{Gm}{d}}$
• B. $\sqrt{\frac{2Gm}{d}}$
• C. $\sqrt{\frac{Gm}{2d}}$
• D. $none$ $of$ $these$

Answer 1

Answer: (A)

$\sqrt{\frac{Gm}{d}}$

When the 2 particles are at rest, at a separation $d$,
Potential energy of the system $=\frac{-G{m}^2}{d}$ and Kinetic energy $=$ 0
Let $V$ be the speed at half the separation.
Potential energy at $\frac{d}{2}$ is $=\frac{-2G{m}^2}{d}$ and Kinetic Energy $=2\times \frac{1}{2}m{V}^2$

From Conservation of energy principle,
$\frac{-G{m}^2}{d}=\frac{-2G{m}^2}{d}+2\times \frac{1}{2}m{V}^2$

$\frac{-Gm}{d}=\frac{-2Gm}{d}+V^2$
Hence , $V=\sqrt{\frac{Gm}{d}}$