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 orbit at an altitude of 3R ?

Options

A. `(7"GmM")/(8R)`
B. `(2"GmM")/(3R)`
C. `("GmM")/(2R)`
D. `("GMm")/(2R)`

Answer

A

Explanation

As we know,
Gravitatinal potential energy =` (-"GMm")/(r)`
and orbital velocity, `v_(0) = sqrt(GM//R+h)`
`E_(f) = (1)/(2) mv_(0)^(2) - ("GMm")/(3R) = (1)/(2) m (GM)/(4R) - ("GMm")/(4R)`
` = ("GMm")/(4R) ((1)/(2) - 1) = (-"GMm")/(8R)`
`E_(i) = (-"GMm")/(R) + K`
`E_(i) = E_(f)`
therefore minimum required energy, K = `(7"GMm")/(8R)`