Optimal nozzle Mach number for maximizing altitude of sounding rocket

A pseudo-analytic approach is applied to determine the optimal nozzle Mach number for maximizing the altitude of a sounding rocket flying in a standard atmosphere. The one-dimensional rocket momentum equation including thrust, gravitational force and aerodynamic drag is considered, for which it is impossible to obtain an analytic solution in a general form. In this work, a piecewise pseudo-analytic approach with a constant parameter introduced to make the velocity integral in the governing equation analytic is applied. The rocket flight in the standard atmosphere is analyzed by dividing the entire range into small intervals where the drag parameter and the gravitational acceleration can be treated as a constant in each interval. A characteristic equation exists and provides accurate predictions of the optimal nozzle Mach number for maximizing the altitude at burn-out state or at apogee.