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 Homework Statement:

Along the present problem you may assume that there is no air friction, that the ignition processes of the different rocket phases are instantaneous, and that fuel capsules have negligible mass.
a) Consider a 1phase rocket such that its gas exhaust velocity is [itex] u [/itex], its initial mass (payload plus fuel) is [itex] m_0 [/itex], and its final mass (payload) is [itex] m_f [/itex]. Assuming that the rocket is vertically launched from the Earth's surface, and that the Earth is at rest, determine the amount of fuel required for the rocket to reach a maximum height [itex] h [/itex].
 Relevant Equations:
 Energy conservation
This isn't right, is it?
[tex] \dfrac{GM}{R}+\dfrac12 v^2=\dfrac{GM}{R+h}[/tex]
[tex] v=\sqrt{\dfrac{GM}{R}}\left( 1\sqrt{\dfrac{R}{R+h}}\right) [/tex]
He's doing energy conservation. The mechanical energy at the Earth's surface is equal to the energy when the speed is [itex]0[/itex].
[tex] \dfrac{GM}{R}+\dfrac12 v^2=\dfrac{GM}{R+h}[/tex]
[tex] v=\sqrt{\dfrac{GM}{R}}\left( 1\sqrt{\dfrac{R}{R+h}}\right) [/tex]
He's doing energy conservation. The mechanical energy at the Earth's surface is equal to the energy when the speed is [itex]0[/itex].