Mathematical Analysis of Electric Rocket Propulsion under Power Constraints

This paper develops a rigorous calculus-based mathematical framework for analyzing electric rocket propulsion systems operating under fixed electrical power constraints. Beginning from momentum conservation and variable-mass dynamics, the classical Tsiolkovsky rocket equation is derived and extended to incorporate power-thrust relationships unique to electric propulsion. Analytical solutions for time-dependent velocity and position are obtained through differential equation modeling. An optimization study examines the tradeoff between exhaust velocity and thrust under constant power. Dimensional analysis and scaling laws reveal the universal mathematical structure governing electric rocket motion. Perturbation and stability analysis confirms the model remains bounded and physically meaningful under realistic power fluctuations. The results demonstrate that electric propulsion performance is fundamentally governed by the interplay between power constraints, variable-mass dynamics, and logarithmic velocity growth.