The satellites when launched from earth are not given the orbital velocity initially, in…

The satellites when launched from earth are not given the
orbital velocity initially, in practice, a multi-stage rocket
propeller carries the spacecraft up to its orbit and during each
stage rocket has been fired to increase the velocity to acquire
the desired velocity for a particular orbit. The last stage of the
rocket brings the satellite in circular/elliptical (desired) orbit.
Consider a satellite of mass 150 kg in low circular orbit, in
this orbit, we can't neglect the effect of air drag. This air drag
opposes the motion of satellite and hence total mechanical
energy of earth-satellite system decreases means total energy
becomes more negative and hence orbital radius decreases
which causes the increase in KE. When the satellite comes in
enough low orbit, the excessive thermal energy generation
due to air friction may cause the satellite to burn up.
If due to air drag, the orbital radius of earth decreases from R to R – \(\Delta R\), \(\Delta R\) << R, then the expression for increase in orbital velocity Dv is