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 objects 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 objects is $\frac{4G{m}^2}{r}$

At infinity, total mechanical energy was zero, since, mechanical energy remains conserved, thus total mechanical energy will be zero through the motion of the particles.
$K.E.+P.E.=0\Rightarrow K.E.=-P.E.$
$KE=-\left(\frac{-G.4m.m}{r}\right)$
$KE=\frac{4G{m}^2}{r}$
Reduced mass : $M=\frac{m\times 4m}{m+4m}=\frac{4m}{5}$
$\Rightarrow KE=\frac{4G{m}^2}{r}$
$\frac{4G{m}^2}{r}=\frac{1}{2}M{V}_{rel}^2$
$\frac{4G{m}^2}{r}=\frac{1}{2}\times \frac{4m}{5}{V}_{rel}^2$
$\Rightarrow V_{rel}=\sqrt{\frac{10Gm}{r}}$