We have simplified the problem of the determination of the horizontal distance traveled by a mobile experiencing a quadratic drag force by introducing a normalized distance, which only depends on the launch angle and a normalized velocity, defined as the ratio of the initial velocity and the limit velocity. As it does not depend on the drag coefficient, the normalized distance may be used for the study of any object, without the need to solve the hodograph equation nor perform numerical integrations.
We have shown that this distance, when plotted as a function of the normalized velocity, could be accurately fitted by a Pearson VII function, which coefficients could be themselves described by low-order polynomials of the launch angle. The Pearson VII function is generally used for the approximation of X-ray diffraction peaks. The choice of the fitting function is always a difficult one, but a judicious choice minimizes the number of unknown parameters to find for an accurate description of the values to be fitted.
Finally, we have verified that as long as the quadratic drag force hypothesis holds, the optimal launch angle is lower than 45∘, and that its variation as a function of the initial velocity can also be described by a Pearson VII function.