In a two stage launch of a satellite, the first stage bringe the satlilte to a height of `500 km` and the second stage given it the necessary critical speed to put it in circular orbit around the Earth. Which stage requires more expenditude of fuel?
(Neglect damping due to air resistance, especially in the first stage).
Mass of the Earth `= 6.0 xx 10^(24) kg`,radius of Earth `= 6400 km, G = 6.67 xx 10^(-11) Nm^(2) kg^(-2)`.

Explanation

Work done on the satellite in first stage
`W_(1) = P.E` at height `500 km - P.E` at the surface of Earth
`= - (GMm)/(R + h) - (-(GMm)/(R ))`
`= GMm [(1)/(R ) - (1)/(R + h)] = (GMmh)/(R(R + h))`
Work done on the satellite in second stage
`W_(2) =` Energy required to provider it orbital velocity `upsilon_(0)`
`= (1)/(2) m upsilon_(0)^(2) = (1)/(2) m [(GM)/(R + h)] = (GMm)/(2 (R + h))`
`:. (W_(1))/(W_(2)) = (2h)/(R ) = (2 xx 500)/(6400) = (1000)/(6400) lt 1`
So `W_(2) gt W_(1)`.
It means second stage required more expenditure of fuel.