Analytical solutions to rocket performance with an arbitrary thrust programme having constant thrust inclination with respect to the horizon in vacuum

The present investigations pertain to rocket performance due to manoevering in a vacuum with any arbitrary thrust inclined at a constant angle with respect to the horizon. Considering the variable thrust either as a function of time, or mass ratio, it is established that dynamical equations projected on the horizontal and vertical directions can be solved analytically; closed-form solutions to the overall range, endurance and final or impact velocity arising out of both burning and coasting phases can be achieved depending on the nature of the thrust function. However, in the first section the thrust inclination is made equal to 90°, in so much so that the path of the rocket becomes vertical evoking the determination, for this case, of its peak velocity and altitude. Finally, this analysis leads us to conclude that with the same propellant mass ratio, initial mass, and thrust inclination to the horizon, a programme of any greater thrust is superior to that of any lower thrust: Rocket performance with an arbitrary thrust programme of greater thrust function evolves greater peak altitude for vertical path, and greater overall range, albeit shorter burnout range, for the curvilinear path.