A rocket of initial mass burns fuel at a constant rate of . The fuel ejected travels at a speed of in the rocket’s frame. Assume the rocket travels vertically away from Earth’s centre and does not experience air resistance. Assume also that the rocket can actually launch, that is
Question 1: Using Newton’s second law, show that the rocket’s equation of motion is
for where is acceleration due to gravity.
Question 2: Taking to be constant throughout its flight, show that the vertical height of the rocket above Earth’s surface is given by
Suppose the mass of fuel on-board the rocket is where .
Question 3: By taking the (partial) derivative of with respect to , show that the optimum rate at which to burn fuel is given by
where is rocket’s initial weight.
Question 4: Show that the greatest height reached by the rocket just as the last of its fuel is burnt is given by
where is the function
Question 5: Using the expressions for and , deduce the maximum height above the Earth’s surface the rocket reaches in this case.
According to this model, does give the maximum possible height the rocket can reach for fixed values of and ?