On the elliptic locus of a family of projectiles

TL;DR

The paper explores the geometric properties of projectile motion, revealing that the maximum heights and distances form elliptic and circular loci, which are better understood through a free-falling reference frame.

Contribution

It introduces a geometric perspective using a free-falling frame to analyze the loci of projectile heights and distances, providing new insights into their shapes and relationships.

Findings

The maximum heights form an elliptic locus in the lab frame.

In a free-falling frame, these loci are congruent tangent circles.

The loci of ranges form a larger circle, with properties derived from the geometric analysis.

Abstract

A little known property of free-fall motion is the elliptic locus of the maximum heights attained by coplanar projectiles launched from a single point in different directions with the same initial speed. Another, less known and perhaps somewhat surprising, property of this family consists in the said ellipse being also the geometric locus of the critical points of the projectiles' distances from the launch point. In the article, to gain a better perspective on the geometry involved, we consider these loci from the standpoint of a free-falling frame. It is shown that, in this reference frame, the considered loci are congruent circles tangent to each other along the horizontal and internally tangent to another, twice as big a circle, which represents the locus of the projectiles' ranges. This simplified geometric description is employed to recover the elliptic locus in the laboratory frame and further explore its properties.