Relationships between the optimum parameters of four projectile motions

Abstract

It has recently been shown that the range and the optimum angle for both projectile motion with linear drag and projectile motion with quadratic drag but low trajectories can be expressed in terms of the Lambert W function. In this article, we show that this is also true for two other projectile motions. As a consequence of these results, simple relations exist between the parameters characterizing all four motions. These relations allow us to obtain the expressions for the optimum parameters of one projectile motion, knowing the optimum parameters of any of the others. Also, we show that some well-known results for projectiles in a vacuum are obtained when certain limits are taken on the expressions for the optimum parameters of any one of the four projectile motions. …Its curious; there’s a sort of unity among problems, don’t you think? There’s only about a half dozen of them that are really different. Florence Nightingale David [1, p. 159]