Volume 14, Number 3, Fall 2006
MOTION THROUGH AIR: WHAT A DRAG
W. W. HACKBORN
Abstract. This paper considers the motion of a projectile
in a uniform, downward gravitational field for the realistic
case in which the drag force on the projectile due to the air
through which it moves varies as the square of its speed. Although
the trajectory of the projectile cannot be expressed in
elementary terms, a first integral of the projectile's motion is
constructed and used to rediscover an elementary expression
that approximates the trajectory quite well for launch angles
up to moderate size. It is shown that this approximate trajectory
lies between the exact trajectory and Galileo's parabolic
trajectory for all nonvertical launch angles. Other elementary
expressions for approximate trajectories are also found and investigated.
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