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

`E_(1)= -(GMm)/(R )= -(mgR^2)/(R )= -mgR`
If v is velocity of the satellite at distance 2R, than total energy
`E_(2)= K.E. + P.E.`
`=(1)/(2)mv^(2)- (GMm)/((2R+R))`
Orbital velocity of satellite, `v=sqrt((GM)/(2R+R))" or "v^(2)=(GM)/(3R)`
So, `(1)/(2)mv^(2)= (GMm)/(6R)`
`E_(2)= (GMm)/(6R)-(GMm)/(3R)= -(GMm)/(6R)= -(mgR)/(6)`
Minimum energy required to launch the satellite is
`=E_(2)- E_(1)= -(1)/(6)mgR+mgR= (5)/(6)mgR`.