The additional kinetic energy to be provided to a satellite of mass `m` revolving around a planet of mass `M`, to transfer it forms a circular orbit of radius `R_(1)` to another of radius `R_(2)(R_(2)gtR_(1))` is

Options

A. `GmM(1/(R_(1)^(2))-1/(R_(2)^(2)))`
B. `GmM(1/(R_(1))-1/(R_(2)))`
C. `2GmM(1/(R_(1))-1/(R_(2)))`
D. `1/2GmM(1/(R_(1))-1/(R_(2)))`

Answer

D

Explanation

Total energy of the orbiting satellite of mass `m` having orbital radius `r` is
`E=-(GMm)/(2r)` where `M` is the mass of the planet.
Additional kinetic energy required to transfer the satellite from a circular orbit of radius `R_(1)` to another radius `R_(2)` is
`E_(2)-E_(1)`
`=-(GMm)/(2R_(E))-(-(GMm)/(2R_(1)))=-(GMm)/(2R_(2))+(GMm)/(2R_(1))`
`=(GMm)/(2) (1/(R_(1))-1/(R_(2)))`