Analytic approach to determine optimal conditions for maximizing altitude of sounding rocket

An analytic approach to determine the optimal conditions for maximizing altitude of a sounding rocket at burn-out state or stationary state is suggested. The one-dimensional rocket momentum equation including thrust, gravitational force, and aerodynamic drag is solved. The typical case that has an analytic solution of the rocket equation is considered. Also, numerical calculations are conducted for comparisons. To build a standard approach to determine the optimal conditions, flights in a constant atmosphere where the air density is constant are considered. The analytic solutions agree well with the numerical ones. It is shown that the optimal condition for maximizing altitude at burn-out state or stationary state can be predicted with a characteristic equation.