Question

Two point objects of masses m and 4m are at rest at an infinite separation. They move toward 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 $\frac{4G{m}^2}{r}$

$K_i=0U_i=0M{E}_i=K_i+U_i=0M{E}_i=M{E}_f=0\frac{1}{2}\mu v_{ref}^2=\frac{Gm\left(4m\right)}{r}\{where\mu =\frac{m\left(4m\right)}{m+4m}\}{v}_{rel}=\sqrt{\frac{10Gm}{r}}KE=\frac{1}{2}\left(\frac{4m}{5}\right)\frac{10Gm}{r}=\frac{4G{m}^2}{r}$