Projectile motion with air resistance quadratic in the speed

We consider two-dimensional motion of a projectile experiencing a constant gravitational force and a fluid drag force which is quadratic in the projectile’s speed. The equations of motions are coupled nonlinear equations. Their solutions have general properties which are easily visualized, although much different from those obtained when a drag force is neglected. Because of these features a study of these equations would provide an interesting counterpoint to the already familiar case of no drag. In this paper we derive simple approximate solutions to the equations of motion for both short and long times. A numerical example is used to compare these approximate solutions with accurate results obtained by numerical integration from an exact but implicit solution. Finally, the origin of the quadratic drag force is discussed.