Singularity-Free Analytic Solution of Ballistic Trajectory with Quadratic Drag

Abstract

In this study, an approximate analytic solution that represents the ballistic trajectory under the quadratic drag is studied. The analytic solution has the following assumptions: gravity is constant and drag is proportional to the square of velocity. The previous studies under these assumptions provide a closed-form solution of velocity as a function of flight path angle, but it is prone to a singularity problem that the denominator is zero under certain conditions. In this study, the derivation process of the previous solution is investigated to analyze the physical meaning of the singularity condition. The analysis shows that analytic derivation using altitude as the independent variable produces singularity conditions and affects the time and downrange calculations. New substitution variables are introduced to avoid these singularity conditions. Numerical simulations are conducted to find the new solution singularity free and accurate.