Envelope of Family of Angled Projectiles and Its Universal Geometric Characteristics

Abstract

Geometric properties of trajectories of angled projectiles under gravity pull are a popular common traditional theme discussed in introductory physics and engineering college courses. What is overlooked is the universal collective properties of the overarching specificities of families of such parabolas, the envelope. For instance [1] and references within explored the existence of one such envelope, however, even the most recent article [2] overlooked its global hidden properties. Here, we investigate exposing this hidden information. Having the equation of the envelope on hand we introduce its universal characteristics such as its: arc length, enclosed 2D surface area, surface area of the surface-of-revolution about the symmetry axis, and, the volume of the enclosure. Numeric values of these quantities are global as is e.g. the 45° projectile angle that maximizes the range of a projectile in vacuum irrespective, its initial speed. In our exploratory investigation, we utilize the popular Computer Algebra System (CAS) MathematicaTM [3] [4] [5].