Question

A super dense particle of mass $\frac{M}{2}$ is projected from the surface of the planet of mass M and radius R. Escape velocity for this particle of mass $\frac{M}{2}$ from surface of planet is

• A. $\sqrt{\frac{GM}{R}}$
• B. $\sqrt{\frac{2GM}{R}}$
• C. $\sqrt{\frac{3GM}{R}}$
• D. $2\sqrt{\frac{GM}{R}}$

Answer 1

The correct option is C $\sqrt{\frac{3GM}{R}}$


Say, the speed of the dense particle is $V$. Then by conservation of momentum, the speed of the planet will be $\frac{V}{2}$.
Then, by conservation of energy,
$\frac{1}{2}\frac{M}{2}{V}^2+\frac{1}{2}M\frac{V^2}{4}=\frac{GM\frac{M}{2}}{R}$
$\frac{3V^2}{4}=\frac{GM}{R}$
$V=2\sqrt{\frac{GM}{3R}}$
Escape velocity is the velocity of the object wrt planet.
Therefore, the escape velocity is $\frac{3V}{2}=\sqrt{\frac{3GM}{R}}$