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A rocket is launched vertically from the Earth's surface.
Its mass (including payload) before fuelling is ;
the mass of its fuel is .
Supppose that the rocket burns and expels fuel at a
constant rate (units: )
with a constant exhaust speed
in the rocket frame.
Assume that, during the burn phase, the altitude of the
rocket is much less than the radius of the Earth, ,
so that the acceleration due to gravity, g, is nearly
constant with altitude.
What forces act on the rocket
in the frame of the Earth? Think carefully.
Write an expression for the total mass of the rocket,
m(t), as a function of time t during the burn phase.
Take and a burn time of
Write down Newton's equation of motion for the rocket.
Solve it to show that the speed of the rocket, v(t),
is given by
during the burn phase.
Argue that is a reasonable
ballpark estimate of the exhaust speed for a rocket burning
a fuel mixture of hydrogen and oxygen.
(You will need to look up a piece of chemical data
for this question. Hint:
What is if the rocket reaches a speed
of at the end of the burn phase?
( is the speed of a satellite in
low-Earth orbit at an altitude of ,
if you are curious!)
What is if the rocket reaches escape
speed at the end of the burn phase? Comment on the practical
importance of your answer.
What is the altitude of a geostationary satellite?
Bryan J. Mendez
Fri Aug 27 17:24:54 PDT 1999