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Launching Satellites

A rocket is launched vertically from the Earth's surface. Its mass (including payload) before fuelling is tex2html_wrap_inline193 ; the mass of its fuel is tex2html_wrap_inline195 . Supppose that the rocket burns and expels fuel at a constant rate tex2html_wrap_inline197 (units: tex2html_wrap_inline199 ) with a constant exhaust speed tex2html_wrap_inline201 in the rocket frame.

  1. What forces act on the rocket in the frame of the Earth? Think carefully.
  2. Write an expression for the total mass of the rocket, m(t), as a function of time t during the burn phase.
Assume that, during the burn phase, the altitude of the rocket is much less than the radius of the Earth, tex2html_wrap_inline207 , so that the acceleration due to gravity, g, is nearly constant with altitude.
  1. 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.

Take tex2html_wrap_inline213 and a burn time of tex2html_wrap_inline215 .
  1. Argue that tex2html_wrap_inline213 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: tex2html_wrap_inline219 .)
  2. What is tex2html_wrap_inline221 if the rocket reaches a speed of tex2html_wrap_inline223 at the end of the burn phase? ( tex2html_wrap_inline223 is the speed of a satellite in low-Earth orbit at an altitude of tex2html_wrap_inline227 , if you are curious!)
  3. What is tex2html_wrap_inline221 if the rocket reaches escape speed at the end of the burn phase? Comment on the practical importance of your answer.
  4. What is the altitude of a geostationary satellite?


Bryan J. Mendez
Fri Aug 27 17:24:54 PDT 1999