Rocket launch

Exercise

A rocket is launched vertically from a point near the surface of the earth with speed 9000 m/s and with the motors turned off. We neglect air resistance and the fact that the speed of a real rocket increases gradually before the motors are turned off. (The rocket starts slightly above the surface because else the program would register a collision between the earth and the rocket at the starting moment)

  1. Calculate the maximum height of the rocket using the data in the parameters grid. Run the simulation and compare your result with the simulation.
  2. What is the speed of the rocket when it collides with the earth?
  3. Calculate how large the initial velocity of the rocket must be to escape from the earth (the escape velocity).
    Simulate the problem to see if your result seems to be correct (zoom out during the simulation). Note the energy of the rocket. What can you say about the total energy if the rocket is to escape?
  4. Let the initial velocity be 9000 m/s, but this time the rocket should start at an angle to a vertical line. Calculate suitable values for vx and vy. Should the maximum height of the rocket be the same as when it started vertically? Think about it before running the simulation. Will the escape velocity be different now?
  5. The rocket should now start vertically once more. Let it start with different velocities (but less than the escape velocity). Investigate if the maximum height rm of the rocket is strongly or weakly dependent on the initial velocity v0.
    See also right column.

An algebra problem
Show that the maximum height (from the center of the earth) rm as a function of the initial velocity v0 is given by

where Rearth is the radius of the earth and ve is the escape velocity.

Draw the graph of this function letting v0 vary from 0 to a value which is a bit smaller than the escape velocity.