It is very expensive to get a mass into orbit. A lot of the mass of a rocket is useless in space. For this reason rockets are not usually sent into space as complete units, but are divided into stages, with each stage acting as a store of fuel and a means of propulsion, which is discarded as the fuel is used up.
The simplest such design is a unit atop a single stage rocket – the booster - which is discarded as soon as the fuel is used up, with the unit continuing to move directly away from the Earth with the momentum acquired from the powered stage.
Suppose that when the booster is discarded the speed of the rocket is u and the rocket is a distance R from the centre of the Earth. Take the mass of the unit as
The force of gravity
where
is the mass of the Earth is the only force acting so
gives
but
Separating variables gives
and integrating
gives
(1)
When
so
Substituting this into (1) gives
