Question

Two point objects of masses $m$ and $4m$ are at rest at an infinite separation. They move towards each other under mutual gravitational attraction. If $G$ is the universal gravitational constant, then at a separation $r$

• A. the total mechanical energy of the two objects is zero
• B. their relative velocity is $\sqrt{\frac{10Gm}{r}}$
• C. the total kinetic energy of the object is $\frac{4G{m}^2}{r}$
• D. their relative velocity is zero

Answer 1

Answer: (A), (B), (C)

The correct options are
A the total mechanical energy of the two objects is zero
B their relative velocity is $\sqrt{\frac{10Gm}{r}}$
C the total kinetic energy of the object is $\frac{4G{m}^2}{r}$

By applying law of conservation of momentum
$m{v}_1-4m{v}_2=0\Rightarrow v_1=4v_2$

The total mechanical energy of the system is zero, as they were at rest earlier at infinite seperation.

By applying conservation of energy $\frac{1}{2}m{v}_1^2+\frac{1}{2}4m{v}_2^2=\frac{G\left(m\right)\left(4m\right)}{r}$
$\Rightarrow 10m{v}_2^2=\frac{G4m^2}{r}$
$\Rightarrow v_2=2\sqrt{\frac{Gm}{10r}}$
$\therefore \text{Total kinetic energy}=\frac{4G{m}^2}{r}$

Relative velocity for the particle,
$\Rightarrow v_{rel}=|\overrightarrow{v_1}-\overrightarrow{v_2}|=5v_2=\sqrt{\frac{10Gm}{r}}$