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

Inttlally,
`E_(1) = - (GM_(E)m)/(4R_(E))`
whtle finally
`E_(f) = - (GM_(E)m)/(8 R_(E))`
The change in the total energy is
`Delta E = E_(f) - E_(i)`
`=(GM_(E)m)/(8 R_(E)) = ((GM_(E))/(R_(E)^(2))) (mR_(E))/(8)`
`Delta E = (gmR_(E))/(8) = (9.81 xx 400 xx 6.37 xx 10^(6))/(8) = 3.13 xx 10^(9)` J
The kinetic energy is reduced and it mimics `Delta`E, namely ,` Delta K = K_(f) - K_(i) = - 3.13 xx 10^(9) J. `
The change in potential energy is twice the change n the total energy , namely
`Delta V = V_(f) - V_(i) = - 6.25 xx 10^(9) `J