A 400 kg satellite is in a circular orbit of radius `2 R_(E)` about the Earth. How much energy is required to transfer it to a circular orbit of radius `4R_(E)` ? What are the changes in the kinetic and potential energies ?

Explanation

`implies` Mass of satellite m = 400 kg
Initially energy `E_(i) = - (GM_(E) m)/(2(2R_E))=-(GM_(E) m)/(4R_E)`
Final energy `E_f =-(GM_Em)/(2(4R_(E)))=-(GM_(E)m)/(8R_E)`
Change in total energy `DeltaE = E_(f) -E_i`
`=-(GM_Em)/(8R_E) -(-(GM_Em)/(4R_E))`
`= (GM_Em)/(4R_E)-(GM_Em)/(8R_E)`
`=(GM_Em)/(8R_E)`
`:. DeltaE=(GM_(E)m)/((R_E)^2)xx(mR_E)/8`
`= (gmR_E)/8 [ :. (GM_E)/((R_E)^2) =g]`
`= (9.81 xx 400xx (6.37xx10^6))/8`
`:. DeltaE = 3.124 xx 10^(9) J`
`:.` Reduced in kinetic energy Denoted by `DeltaK`
`DeltaE=-3.124 xx 10^(9) J`
Change in potential energy of satellite
`DeltaU=- 2DeltaE=- 2 xx (3.125 xx 10^(9))=-6.25xx 10^(9)J`