Locus of the apices of projectile trajectories under constant drag

Abstract

We present an analytical solution for the projectile coplanar motion under constant drag parametrised by the velocity angle. We found the locus formed by the apices of the projectile trajectories. The range and time of flight are obtained numerically and we find that the optimal launching angle is smaller than in the free drag case. This is a good example of problems with constant dissipation of energy that includes curvature, and it is proper for intermediate courses of mechanics.

Analytical velocity hodograph of projectile motion under polynomial drag

“…The equation we shall solve is obtained from the change to tangential and normal coordinates of the motion, or in a Frenet-Serret system, to the newtonian equations of motion. This approach changes the time for the angle as the free parameter, see [5] for explanation. The equation to solve is…”

“…In this context the parameter go from the launching angle, θ 0 to zero at the apex and then it start the descending part with a negative angle. See reference [5] for a wider explanation.…”

“…The treatments are centered in the fully solvable linear case and the nonintegrable quadratic one [1,2,3]. However, few has been worked on the constant or Coulomb case [4,5], or superior powers, even from a purely theoretical point of view.…”