First slide

Motion of planets and satellites

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 in elastically and stick together as wreckage, the total energy of the system just after collision is

Moderate

A

$- \frac{2 GMm}{r}$
B

$- \frac{GMm}{r}$
C

$\frac{GMm}{2 r}$
D

$\frac{GMm}{4 r}$

Solution

Energy of each satellite in the orbit $= \frac{- GMm}{2 r}$
Total energy of the system before collision,
$E_{i} = E_{1} = E_{2} = 2 E = - 2 \times \frac{GMm}{2 r} = - \frac{GMm}{r}$
As the satellites of equal mass are moving in the opposite directions and collide in elastically, the velocity of the wreckage just after the collision is
$mv - mv = 2 mV, i.e., V = 0$
The energy of the wreckage just after the collision will be totally potential and will be
$E_{f} = \frac{GM (2 m)}{r} = - \frac{2 GMm}{r}$
As after collision the wreckage comes to standstill in the orbit, it will move along the radius towards the earth under gravity.

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