On the locus formed by the maximum heights of an ultra-relativistic projectile

TL;DR

This paper derives the curve formed by maximum heights of ultra-relativistic projectiles in a gravitational field, revealing it as a lemniscate-type shape and comparing it to the nonrelativistic case, with detailed analysis of trajectories.

Contribution

It provides the first explicit derivation of the maximum height locus for ultra-relativistic projectiles and compares it to classical results, introducing new parametric equations and asymptotic behaviors.

Findings

The maximum height locus forms an onion-like lemniscate curve in the ultra-relativistic limit.

In the nonrelativistic limit, the locus reduces to an ellipse.

Explicit parametric equations for trajectories in both regimes are derived.

Abstract

We consider the problem of relativistic projectiles in a uniform gravitational force field in an inertial frame. For the first time, we have found the curve that joins the points of maximum height of all trajectories followed by a projectile in the ultra-relativistic limit. The parametric equations of this curve produce an onion-like curve; in fact, it is one of the loops of a lemniscate-type curve. We also verify that the curve is an ellipse in the nonrelativistic approximation. These two limiting results are obtained by following two slightly distinct approaches. In addition, we calculate the nonrelativistic and ultra-relativistic approximations of the trajectory equation and parametric equations of the trajectory as functions of time. We also find the asymptotic behavior of the ultra-relativistic trajectory equations for points that are very distant from the launching point. All limiting cases in the article are studied in detail. Additionally, we discuss various physical issues related to some of our results. The content of the article is appropriate for advanced undergraduate students.