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

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.