Mile Baće

The envelope of projectile trajectories

The paper presents yet another elegant solution to the classical problem of maximizing the range of a projectile fired from above or from below the horizontal target plane. The method makes use of the envelope of the family of the projectile trajectories and it not only solves the original problem, but can be applied to maximize the projectile range on a target surface of arbitrary shape. Three different ways of deriving the equation of the envelope are shown. As an example the range of a projectile is maximized on a parabolic target surface. The envelope can also be used to minimize the kinetic energy of a projectile given the target point.

Maximizing the range of a projectile

The problem of maximizing the horizontal range of a projectile fired from a height above or a depth below the assumed horizontal target plane has been discussed and solved in several different ways. However, it seems to be a neverending story. This paper offers a remarkably simple solution to the problem that does not require the use of calculus. It is based on the fact that the equation θ = f(R), where θ is the firing angle and R is the range, must have a unique solution for R = Rmax .