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 orbitat an altitude of 2R?

What is the minimum energy required to launch a satellite of mass m fr

1.	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 orbitat an altitude of 2R?
Answer» Solution :`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`.