Launching Satellites

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.

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, , so that the acceleration due to gravity, g, is nearly constant with altitude.

3. 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 and a burn time of .

4. 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: .)
5. 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!)
6. What is if the rocket reaches escape speed at the end of the burn phase? Comment on the practical importance of your answer.
7. What is the altitude of a geostationary satellite?