Rocket flight: The velocity v attained by a launch vehicle during launch is a function of c, the exhaust velocity of the engine, and R, the mass ratio of the spacecraft.36 The mass ratio is the vehicle’s takeoff weight divided by the weight remaining after all the fuel has been burned, so the ratio is always greater than 1. It is close to 1 when there is room for only a little fuel relative to the size of the vehicle, and one goal in improving the design of spacecraft is to increase the mass ratio. The formula for v uses the natural logarithm:
v = c ln R .
Here we measure the velocities in kilometers per second, and we assume that c = 4.6 (which can be attained with a propellant that is a mixture of liquid hydrogen and liquid oxygen).
a. Draw a graph of v versus R. Include mass ratios from 1 to 20.
b. Is the graph in part a increasing or decreasing? In light of your answer, explain why increasing the mass ratio is desirable.
c. To achieve a stable orbit, spacecraft must attain a velocity of 7.8 kilometers per second. With c = 4.6, what is the smallest mass ratio that allows this to happen? (Note: For such a propellant, the mass ratio needed for orbit is usually too high, and that is why the launch vehicle is divided into stages. The next exercise shows the advantage of this.)