What is the minimum energy required to launch a satellite of mass m from the surface of a planet of mass M and radius R in a circular orbit at an altitude of 2R?

Explanation

Energy of satellite on the surface of Earth
`E_(1) = - (GM m)/(R ) = - (mg R^(2))/(R ) = - mg R` ..(i)
It `upsilon` is the velocity of the satellite at the distance `2R` from the surface of the Earth, then total energy of the satellite
`E_(2) = K.E + P.E. = (1)/(2) m upsilon^(2) - (GM m)/((2R + R))` .(ii)
Orbital velocity of satellite, `upsilon = sqrt((GM)/(2R + R))`
or `upsilon^(2) = (GM)/(3R)`
So, `(1)/(2) m upsilon^(2) = (GM m)/(6R)`
`:. E_(2) = (GM m)/(6R) - (GM m)/(3R) = - (GM m)/(6R)`
`= - (m gR)/(6)`
`:.` Minimum energy required to launch the satellite is
`= E_(2) - E_(1) = - (1)/(6) mgR + mg R = (5)/(6) mg R`