A `400 kg` satellite is in a circular orbit of radius `2 R_(E)` around the Earth. How much energy is required to transfer it to a circular orbit of radius `4 R_(E)`? What are the changes in the kinetic and potential energies?
Given `g = 9.81 m^(-2) , R_(E) = 6.37 xx 10^(6) m`.

Explanation

Total energy of satellite in an orbit of radius `2R_(E)` is
`E = (-GMm)/(4R_(E)) " "[because E = (-GMm)/(2r)]`
Total energy in an orbit of radius `4R_(E)` is
`E. = -(GMm)/(8R_(E))`
Energy required, `E_(R ) = E. - E`
`= -(GMm)/(8R_(E)) + (GMm)/(4R_(E))`
`= (GMm)/(8R_(E)) = ((GM)/(R_(E)^(2)))((mR_(E))/(8))`
`= (gmR_(E))/(8)`
`g = 9.8 m//s^(2)`
`m = 400 kg`
`R_(E) = 6.37 xx 10^(6) m`
`:. E_(R ) = (9.8 xx 400 xx 6.37 xx 10^(6))/(8)`
`= 3.13 xx 10^(9) J`
Change in K.E. `= -E_(R ) - -3.13 xx 10^(9)J`
Change in P.E. `= 2E = -6.26 xx 10^(9) J`