Question

Two satellites of the same mass are launched in the same orbit around the earth so as to rotate opposite to each other. If they collide inelastically and stick together as wreckage, the total energy of the system just after a collision is?

• A. $-\frac{2GMm}{r}$
• B. $-\frac{GMm}{r}$
• C. $\frac{GMm}{2r}$
• D. $\frac{GMm}{4r}$

Answer 1

The correct option is B $-\frac{GMm}{r}$

For a satellite of mass m orbiting in a circular orbit at a distance $r$ from the earth's centre:

$\frac{m{v}^2}{r}=\frac{GMm}{r^2}\Rightarrow =\frac{GM}{r}$

Kinetic energy $=\frac{1}{2}m{v}^2=\frac{GMm}{2r}$

Total energy$=P.E+K.E=-\frac{GMm}{r}+\frac{GMm}{m}=\frac{-GMm}{2r}$
For $2$ satellites with same mass and orbital radius, staying together after collision total energy $=2\left(\frac{-GMm}{2r}\right)=\frac{-GMm}{r}$