A Direct-Fire Trajectory Model for Supersonic, Transonic, and Subsonic Projectile Flight

Report No. ARL-TR-6998
Authors: Paul Weinacht
Date/Pages: July 2014; 34 pages
Abstract: This report presents a simple but accurate method of determining the trajectory of projectiles that traverse a flight regime that includes supersonic, transonic, and subsonic flight. Closed-form analytical solutions for the important trajectory parameters such as the time of flight, velocity, gravity drop, and wind drift are presented. The method makes use of individual power-law descriptions of the drag variation with Mach number within the supersonic, transonic, and subsonic regimes. The method demonstrates that the free-flight trajectory can be characterized with as few as six parameters: the muzzle velocity, muzzle retardation, a power-law exponent that describes the drag variation in supersonic flight, the transition Mach numbers between supersonic and transonic flight, the transition Mach number between transonic and subsonic flight, and the retardation at the subsonic transition Mach number. The accuracy and simplicity of the method make it very useful for preliminary design or performance assessment studies where rapid prediction of projectile trajectories is desired. Sample results are presented for a 9-mm pistol bullet that traverses supersonic, transonic, and subsonic flight to demonstrate the viability of the method.
Distribution: Approved for public release
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Last Update / Reviewed: July 1, 2014