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.