Drag-law effects in the goddard problem

-Presently studied is the problem of maximizing the altitude of a rocket in vertical flight in a resisting medium, when the amount of propellant is specified, known as the Goddard problem. The case is studied in which the drag coefficient is a function of the Mach number, witnessing a sharp increase in the transonic region. Analysis shows the possibility of a more complex switching structure than the classical full-singular-coast sequence, with the appearance of a second full-thrust subarc in the transition from the subsonic to the supersonic region. Necessary conditions such as the Legendre-Clebsch condition for singular subarcs and the McDanell-Powers condition for joining singular and non-singular subarcs were checked, and were found to be satisfied. It is shown that the results obtained depend heavily on the assumed form of the drag law, and on the magnitude of the upper bound on the thrust.