The influence of the drag force on the safety domain

Abstract

In this work we compute the safety domain related to projectiles fired close to the Earth's surface, taking into account a linear (laminar regime) and a quadratic drag force (turbulent regime).

Safe domain for projectile trajectories in a medium with quadratic drag force

“…For the linear drag model, the projectile trajectories and safe domain are well-known and we can find them, for instance, in [2]. In this case the trajectories are given by y = tan a + gm kv 0 cos a x + gm 2 k 2 log 1 À kx mv 0 cos a , and the envelope by…”

“…The envelope of a family of curves is another curve in which every point belongs to a curve of the given family; moreover, in that point both curves have a common tangent line. With the value u(j) defined by the condition (3) we can determine the point in each trajectory belonging to the envelope and, for each j 2 (j 0 , N), that point is given by (2). Then, we have obtained a parametric equation for the envelope of the family of trajectories (1).…”

“…This image has been created by using numerical methods to approximate the envelope given by (2). First, it is necessary to know the minimum slope j 0 which limits the trajectories having a common point with the envelope, and the value u(j) defined by the relation (3).…”

“…To obtain these values we have used a quadrature rule to approximate the integrals and the secant method to find the required value in each case. The integrals in (2) have been evaluated by using a quadrature rule again.…”

“…The safe domain has been treated in [5] but from a heuristic point of view. Also in [2] we can find a numerical approach to the safe domain with quadratic drag force but the authors do not provide an analytical expression for it. Here we focus our attention on a rigorous deduction of the shape of the region.…”