Question

Two satellites of same mass are launched in the same orbit around the earth so that they rotate opposite to each other. If they collide in-elastically, obtain the total energy of the system before and just after the collision. Describe the subsequent motion of the wreckage.

Answer 1

Potential energy of the satellite in its orbit $=-\frac{GMm}{r}$
$\Rightarrow K.E.=\frac{|U|}{2}=\frac{GMm}{2r}$ where $m$ is mass of satellite, $M$ the mass of the earth and $r$ the orbital radius.
Total energy $=K.E.+P.E.$
$=\frac{GMm}{2r}-\frac{GMm}{r}=-\frac{GMm}{2r}$
For two satellites $E=-\frac{GMm}{r}$
Let $v^{\prime }$ be the velocity after collision.
By conservation of momentum
$m\overrightarrow{v_1}+m\overrightarrow{v_2}=0=(m+m)v^{\prime }\Rightarrow v^{\prime }=0$
The wreckage of mass $\left(2m\right)$ has no kinetic energy, but it has only potential energy. So, energy after collision $=\frac{-GM\left(2m\right)}{r}$
Therefore, the wreckage falls down under gravity.