FIG. 1: (color online) Locus C m ( ε ) formed by the apexes of all the projectile trajectories (continuous line in blue) given by Eqs. (7) and (8) in rectangular coordinates or by Eq. (13), the last one express C m in polar coordinates and in term of the Lambert W function. The dashed red line is the ellipse of eccentricity e = √$_{3}$$_{/}$$_{2 which represents the drag-free case,}$ i.e. C m (0). The parameters are V 0 = 10 and ε = 0 . 1. where we introduce the dimensionless perturbative pa- rameter ε ≡ bV$_{0}$/g , the dimensionless length ρ = V 2 0 /g , and noticing that b 2 g can be expressed as ε 2 ρ . An alter- native procedure consists in set the derivative dy/dt to zero to obtain the time of flight to the apex of the tra- jectory and, evaluate the coordinates at that time. The points ( x$_{m}$, y$_{m}$ ) conform the locus of apexes C $_{m}$( ε ) for all parabolic trajectories as a function of the launch an- gle α . In Fig. 1 we plot C $_{m}$( ε ) described by Eqs. (7) and (8), for the drag-free case (in dashed red line) and for ε = 0 . 1 in continuous blue line. Several projectile trajectories are plotted in thin black lines. The locus of apexes C $_{m}$( ε ) defined by ( x$_{m}$, y$_{m}$ ) of Eqs. (7) and (8) is described parametrically by the launch angle α and it changes for different values of ε ≡ bV$_{0}$/g . In the next section we shall find a description of C $_{m}$( ε ) in terms of polar coordinates and in a closed form using the Lambert W function. In order to obtain an analytical closed form of the lo- cus we change the variables to polar ones, i.e., x$_{m}$ = r$_{m}$ cos θ$_{m}$ and y$_{m}$ = r$_{m}$ sin θ$_{m}$ . The selection of a de- scription departing from that origin instead of the center or the focus of the ellipse is because the resulting geomet- rical place is no longer symmetric and the only invariant point is just the launching origin. We substitute the po- lar forms of x$_{m}$ and y$_{m}$ into equations (7) and (8) and when ε → 0. An explanation of this unfamiliar form of an ellipse is given in appendix A followed by a discussion about the ε → 0 limit of expression (13) in appendix B. Formula (13) exhibits the deep relationship between the Lambert W function and the linear drag force pro- jectile problem, since not only the range is given as this function [4, 5]. The problem open the opportunity to study the W function in polar coordinates, that, almost We recover the drag-free result It is important to note that the argument of the Lam- bert function in this equation is negative for all the values ε > 0. W ( x ) remains real in the range x ∈ $^{[}$−1 /e, 0) and have the branches denoted by 0 and − 1.[2] We select the principal branch, 0, since it is the bounded one, however, for values of ε > 1 . 1 there is a precision problem since the required argument values are near to − 1 /e ≡ − $^{exp(}$−1). It is important to stress that in Eq. (13) the independent variable is the angle θ and, it constitutes the parameter- ization of the curve C $_{m}$. The polar expression of C $_{m}$can also be written in terms of the tree function T ( z ) = − W $^{(}$− z ), giving Where we multiplied both sides of Eq. (9) by − ε $^{2}$sin θ$_{m}$ . Setting z = − ε $^{2}$tan θ$_{m}$ f ( α ) and W ( z ) = ε $^{2}$sin θ$_{m}$ ˜ r ( θ$_{m}$ ) in Eq. (12), it shall have the familiar Lambert W function form, z = W ( z ) exp( W ( z )), from which we can obtain ˜ r as since Eq. (10) allows us to have, implicitly, α ( θ$_{m}$ ). We shall return to this point later. Hence, we can write Eq. (9) as re arranging terms it must be expressed as T he lhs depends on r$_{m}$ and θ$_{m}$ meanwhile the rhs de- pe nds on α , however the last angle is a function of θ$_{m}$ an d reads as by making tan θ$_{m}$ = y$_{m}$/x$_{m}$ from Eqs. (7) and (8). In order to obtain ˜ r ( θ$_{m}$ ) ≡ r$_{m}$ ( θ$_{m}$ ) /ρ we set 2 r m ( θ m ) ρ cos θ$_{m}$ exp ( − ε $^{2}$sin θ$_{m}$ r m ( θ m ) ρ ) = cos α sin α $^{exp(}$− ε sin α ) . (9) tan θ$_{m}$ = 1 ε 2 ( ε sin α − ln(1 + ε sin α )) cos α sin α 1 1+ ε sin α , (10) f ( α ( θ$_{m}$ )) ≡ cos α sin α $^{exp(}$− ε sin α ) , (11) ˜ r ( θ$_{m}$ ) = − 1 ε $^{2}$sin θ$_{m}$ W $^{(}$− ε $^{2}$tan θ$_{m}$ f ( α )) . (13) − ε $^{2}$sin θ$_{m}$ ˜ r ( θ$_{m}$ ) exp ( − ε $^{2}$sin θ$_{m}$ ˜ r ( θ$_{m}$ ) = − ε $^{2}$tan θ$_{m}$ f ( α ) . (12) ˜ r ( θ$_{m}$ ) = 1 ε $^{2}$sin θ$_{m}$ T ( ε $^{2}$tan θ$_{m}$f ( α ( θ$_{m}$ ))) . (14) ˜ r = 2 sin θ$_{m}$ 1 + 3 sin 2 θ$_{m}$ (15) III. THE LOCUS C m AS A LAMBERT W FUNCTION -10 -5 0 5 10 x 0 1 2 3 4 5 y