Consider a two-stage rocket made up of two engine stages, each of inertia m when empty, and a payload of inertia m. Stages 1 and 2 each contain fuel of inertia m, so that the rocket's inertia before any fuel is spent is 5m. Each stage exhausts fuel at a speed vfuel=vex?vi, where vi is the speed of the rocket at the instant the fuel is spent. The rocket is initially at rest in deep space. Stage 1 fires, ejects its fuel all at once, and then detaches from the remainder of the rocket. The same process is repeated with stage 2.

a) What is the final speed of the payload? (Hint: Determine the speed of the rocket after stage 1 fires. Then, because stage 1 is detached from the rocket, redefine the system to include only the payload and stage 2, and determine the speed of the rocket after stage 2 fires.) 

b) Now consider another rocket of inertia 5m but with only a single engine stage, of inertia 2m, carrying fuel of inertia 2m. What is the final speed of the payload in this case?