On the locus formed by the maximum heights of projectile motion with air resistance H. Hern´ andez-Salda˜ na ∗ Departamento de Ciencias B´ asicas, Universidad Aut´ onoma Metropolitana-Azcapotzalco, Av. San Pablo 180, M´ exico 02200 D.F., Mexico. (Dated: May 5, 2018) We present an analysis on the geometrical place formed by the set of maxima of the trajectories of a projectile launched in a media with linear drag. Such a place, the locus of apexes, is written in term of the Lambert W function in polar coordinates, confirming the special role played by this function in the problem. In order to characterize the locus, a study of its curvature is presented in two parameterizations, in terms of the launch angle and in the polar one. The angles of maximum curvature are compared with other important angles in the projectile problem. As an addendum, we find that the synchronous curve in this problem is a circle as in the drag-free case. I. INTRODUCTION II. THE PROJECTILE PROBLEM WITH AIR RESISTANCE An amazing characteristic of some old fashion prob- lems is their endurement. The projectile motion is one of them. Being one of the main problems used to teach elementary physics, variations and not well known facts about it appear in the physics literature of the XXI cen- tury. A search in the web [1] or in the Science Citation Index gives an idea of this fact. Some of the recent studies deal with the problem of air resistance in the projectile motion and its pedagogical character made of it an ex- cellent example to introduce the Lambert W function, a special function.[2] The Lambert W function is involved in many problems of interest for physicist and engineers, from the solution of the jet fuel problem to epidemics[2] or, even, Helium atom eigenfunctions.[3] One of those problems is the solution for the range R in the case that the air resistance has the from ⃗ f = − mb⃗ v .[4, 5] In this paper we analyze the not well known fact of the geometrical place formed by the maxima of all the projec- tile trajectories at launch angle α and in the presence of a drag force proportional to the velocity, we shall denote this locus as C $_{m}$( ε ). The resulting locus becomes a Lam- bert W function of the polar coordinate r ( θ ) departing from the origin. This problem raises as a natural contin- uation from the nice fact that in the drag-free case such a locus is an ellipse[6–8] with an universal eccentricity e = √$_{3}$$_{/}$$_{2.[6]}$ The paper is organized as follows. In section II, the set of maxima for projectile trajectories moving under in the presence of air resistance is presented. In section III we find a closed form, in polar coordinates, to express such a geometrical place, C $_{m}$. In section IV we present a numerical calculation of the curvature of C m using the polar angle and the launch angle as parameterizations. Additionally, we demonstrate that the synchronous curve is a circle as in the drag-free case in section V. In section VI we conclude. and considering the value at the maximum, via dy/dx = 0. The corresponding solution is for the speeds. We used the initial conditions x (0) = y (0) = 0 and u$_{0}$ = V$_{0}$ cos α and w$_{0}$ = V$_{0}$ sin α . Noticing that the terminal speed is g/b in the y axis. For the same initial speed V$_{0}$ these solutions are func- tion of the launch angle α and the locus formed by the apexes is obtained if time is eliminated between the so- lutions in time in Eqs. (2) and (3), giving the equation Several approximations in order to consider the air re- sistance exist in the literature, the simplest is the linear case. In such a case the force is given by where m is the mass of the projectile and b is the drag coefficient. The units of b are s − $^{1}$. The velocity compo- nents are labeled as ⃗ v = u ˆı + w ˆ, with u = dx/dt and w = dy/dt . The solutions for the position and velocity are obtained trough direct integration of Eq.(1) yielding ⃗ F = − mb⃗ v − mg ˆ , (1) x ( t ) = u 0 b $^{[1}$− $^{exp(}$− bt )] , (2) y ( t ) = w 0 + g/b b $^{[1}$− $^{exp(}$− bt $^{)]}$− gt/b, (3) u ( t ) = u$_{0}$ $^{exp(}$− bt ) , (4) w ( t ) = ( w$_{0}$ + g/b $^{) exp(}$− bt $^{)}$− g/b, (5) for the coordinates, and y ( x ) = w$_{0}$ + g/b u$_{0}$ x − g b 2 ln ( $^{1}$− bx u$_{0}$ ) , (6) x$_{m}$ ρ = cos α sin α 1 1 + ε sin α , (7) ε $^{2}$y$_{m}$ ρ = ε sin α − ln(1 + ε sin α ) , (8) ∗ Electronic address: hhs@correo.azc.uam.mx arXiv:1001.0602v2 [physics.class-ph] 20 Mar 2010