Question

Two identical particles each of mass $m$ start moving towards each other from rest from infinite separation under gravitational attraction. Their relative velocity of approach at separation $r$ is :

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

Answer 1

The correct option is C $2\sqrt{\frac{GM}{r}}$

When the two masses are at a distance $r$ from each other, gravitation potential energy between these two is $-\frac{GMm}{r}$.
Since the bodies are in motion, they both have kinetic energy and the magnitude of force on each body is same, so both will have same speeds.
This total energy is equal to the total energy at infinite separation which is zero.
$\therefore -\frac{GMm}{r}+\frac{1}{2}m{v}^2+\frac{1}{2}m{v}^2=0$
$\Rightarrow m{v}^2=\frac{GMm}{r}$
$\Rightarrow v=\sqrt{\frac{GM}{r}}$
This is the speed of each body. The relative velocity will be 2$v$ as they are moving opposite to each other.
Relative velocity $=2\sqrt{\frac{GM}{r}}$