Analytical Solution in Curvilinear Coordinates for the Trajectory of a Projectile Subject to Aerodynamic Drag

Report No. ARL-TR-5822
Authors: Steven B. Segletes and William P. Walters
Date/Pages: December 2011; 38 pages
Abstract: An analytical study was conducted to determine the general trajectory of a projectile subject to aerodynamic drag in a gravitational field. A solution was obtained to the governing equations using curvilinear coordinates. The general solution analytically provides the pathlength of the projectile in terms of the current trajectory angle. Noticeably absent from the solution, however, are several desirable quantities. For instance, there is no ability to analytically express the trajectory of the projectile in Cartesian (x, y) coordinates; analytically express the time of the event in terms of an independent variable; and analytically invert the solution to express the trajectory angle in terms of, for example, the pathlength. Nonetheless, the solution does provide the velocities (x, y, and pathlength) in terms of the current trajectory angle, as well as the time rate of change of the trajectory angle in terms of the current trajectory angle. For applications where a Cartesian solution and/or time history is required, the current solution permits an expedited numerical integration as compared to a general-purpose integration of the governing equations, as well as an analytical approximation valid for moderate angles of attack. Finally, an interesting analytical result was developed that shows that the launch angle required to maximize the pathlength to the apogee is always 56.5, regardless of drag coefficient, initial projectile velocity, or even gravitational constant.
Distribution: Approved for public release
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Last Update / Reviewed: December 1, 2011