A discrete, energetic approach to rocket propulsion

Most rockets convert the energy stored in their propellant mass into the mechanical energy required to expel it as exhaust. The 'rocket equation', which describes how a rocket's speed changes with mass, is usually derived by assuming that this fuel is expelled at a constant relative velocity. However, this is a poor assumption for cases where the rocket promptly loses a large fraction of its mass. Instead, I derive the change in speed for a rocket that emits its fuel in N discrete pellets, with a constant mechanical energy produced per unit fuel mass. In this model I find that the rocket's speed change is greatest when all the fuel is expelled at once (N = 1). In the limit of many small pellets (N → ∞), I show that the velocity change approaches, from above, the prediction of the continuous thrust rocket equation. For this model of rocket propulsion, I quantify how the fuel's total available energy is divided between the rocket and exhaust. In the limit of continuous thrust, the rocket can utilise no more than 65% of the available mechanical energy as its kinetic energy. In an online supplement, I compare this model of discrete propulsion with those previously published. This topic uses momentum conservation and mechanical energy concepts at the introductory undergraduate level.