Learning about rockets, in stages

Abstract

A rocket must carry the fuel it expels in order to accelerate its structure and payload. The rocket equation relates the change in speed to the fuel mass expelled. To launch a spacecraft into Earth orbit requires a multi-stage rocket, since otherwise the mass of fuel required would be prohibitive. While the details vary among historical and current launch vehicles, the advantages and physical principles of staging can be demonstrated by analysing the simpler case of a two-stage rocket in free space. I show that a two-stage rocket produces a higher payload speed than a single-stage rocket with the same fuel and structural mass, and introduce the concept of optimal staging to maximise this speed. I also examine how mechanical energy extracted from the expelled fuel is distributed between the stages, payload capsule, and exhaust. A surprising result is that the two-stage rocket deposits more energy into the exhaust stream than its single-stage counterpart. I suggest further investigations that are appropriate for advanced high school or undergraduate students of physics or aerospace engineering. Supplementary materials are provided online for classroom use.