A Hidden Circle in a Family of Projectile Paths

Abstract

The family of paths of ideal projectiles shot from a point with a common value of speed and at different angles to the horizontal has several interesting properties associated with it. For example, Chapou et al. have shown that an ellipse passes through the apexes of these parabolic paths. This observation encouraged us to further explore this family of the parabolic paths. In this note we show the presence of a hidden circle in the family of these paths. We highlight all the known interesting properties of this family of projectiles. For our purpose here, we consider projectiles shot from points on the third quadrant of a circle in the vertical plane. In this method, we don’t use vectors or calculus or Newton’s laws of motion.