Analytical velocity hodograph of projectile motion under polynomial drag

Abstract

Velocity hodographs are of importance in weather analysis. A historically important case is the hodograph in the Kepler problem. Here, we explore the analytical solutions for the velocity hodograph for the projectile trajectories under a polynomial, in velocity, hydrodynamic dissipation. General solutions in terms of hypergeometric functions are found for odd powers in monomial terms. The even powers have analytical hodographs but we do not find a general expression for them. For completeness we include the constant or Coulomb case, as well as its sum with the linear and quadratic cases. Both shown explicit solutions as functions of the angle. Usually the linear case is discussed in textbooks and articles, here we find that more general cases admit analytical solutions, even when the position and time cannot be obtained explicitly and they need of numerical integration. Furthermore, motion with constant and, additional linear or quadratic drag could be used to design new university lab experiments.