Consider two satellites `A` and `B` of equal mass `m`, moving in the same circular orbit about the earth, but in the opposite sense as shown in Fig. The orbital radius is `r`. The satellites undergo a collision which is perfectly inelastic. For this situation, mark out the correct statement(s). [Take mass of earth as `M`]

Options

A. The total energy of the two satellite plus earth system just before collision is `-(GMm)/r`
B. The total energy of the two satellites plus earth system just before collision is `-(2GMm)/r`
C. The total energy of two satellites plus earth system just after collision is `-(GMm)/(2r)`
D. The combined mass (two satellites) will fall towards the earth just after collision.

Answer

A, B, D

Explanation

Say the shell has acquired a mass `m` and further a mass `dM` is to be added `dW=VdM=-(GMdM)/R` or `W=-int_(0)^(M)(GMdM)/R`
`=(GM^(2))/(2R)`=self energy `=U`
Say `F` is now the attractive force per unit area. If the shell expands from `R` to `R+dR` then work done by attractive force is `-Fxx4piR^(2)dR` since this is the work done by gravitational field, this may be equal to reduction in gravitational potential energy. or, `-piR^(2)FdR=dU` or,
`F=-1/(4piR^(2))(dU)/(dR^(2))=(GM^(2))/(8piR^(4))`
Now total force experienced by one hemisphere `=int(1r)/(Fds)=FxxpiR^(2)` (vector addition adds only components perpendicular to the equation plane)
=Net gravitational force `F_(G)=(GM^(2))/(8R^(2))`