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 \(5 \mathrm{~m}\). Each stage exhausts fuel at a speed \(v_{\text {fuel }}=v_{\text {ex }}-v_{\text {j }}\), where \(v_{\mathrm{ex}}\) is the speed of the exhaust relative to the rocket and \(v_{\mathrm{i}}\) is the speed of the rocket relative to the ground 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? (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 \(5 \mathrm{~m}\) but with only a single engine stage, of inertia \(2 \mathrm{~m}\), carrying fuel of inertia \(2 \mathrm{~m}\). What is the final speed of the payload in this case?

(c) Which design yields a higher payload speed: two stage or single stage? Why?