## 3 Analytical Model Conceptual Development and Discussion
\(\hat{r}\), \(\hat{\phi}\), \(\hat{z}\) are the 3D cylindrical-polar coordinate system unit vectors. The calculation of the second derivative of \(\overrightarrow{r}\), i.e., acceleration (\(\overrightarrow{a}\)), yields  
Figure 2.0 Figure 2.1  
The particular conditions of the model to implement are \(\dot{r}=0\), \(\ddot{r}=0\), \(\dot{\phi}=0\), \(\dot{z}=0\), and \(\dot{z}=0\), since within the dynamic situation considered, there is no velocity or acceleration along radial and \(z\) coordinates. Meanwhile, for the angular coordinate, there is no acceleration. These considerations in Equation (9) lead us to  
Figure 2.2  
Or, in cylindrical-polar component vector representation, \(\dot{r},\dot{\phi},\dot{z}\),  
Figure 2.3  
Now, Equation (5) applied to the situation in Figure 4, and taking into account Equation (11), yields, in cylindrical-polar component vector representation, \(\dot{r},\dot{\phi},\dot{z}\),  
Figure 2.4 Figure 2.5  
where \(v=r\dot{\phi}\). In this case, \(r\) is both an instantaneous radius of curvature of the vehicle's trajectory, as well as the radial coordinate of the system used to analyze the problem. Solving for \(v\) in Equation (11) produces, as before in Equation (7), the maximum velocity for safe highway operation according to the model:  
Figure 2.6  
In terms of the radius of curvature (\(R=rcos\theta\)) Cf. Figure 2.7 Equation (13) right transforms into  
Figure 2.8  
Now the models in Equations (2) and (14) can be compared. In the former, \(f_{d}(v,R,e)=\frac{v^{2}}{k_{1}R}-k_{2}e\) is defined as the amount of friction a vehicle needs in order to maintain trajectory in an existing horizontal curve [26]. On the other hand, \(k_{1}\) and \(k_{2}\) are parameters considering the suspension's mechanical effect on light and heavy vehicles, respectively. It is noticeable that Equations (2) and (14) share the same analytical elements, both having the mathematical form \(v=\sqrt{c_{1}(c_{2}+c_{3})}\). In this equation form, \(c_{1}\Rightarrow 15R\Rightarrow gR\) relates to the motion's curvature radius, \(c_{2}\Rightarrow f_{d}\Rightarrow\mu\) relates to friction, and \(c_{3}\Rightarrow 0.01e\Rightarrow tan\theta\) relates to the highway's super-elevation, respectively. Another point of comparison between models is that Equations (2) and (14) both take into account the instantaneous friction being provided by horizontal curves in the trajectory. Cf. [26] or friction demand (\(f_{d}\)) definition; see Figure 4 here for friction force (\(\overrightarrow{F}\)) consideration. For these reasons, Equation (14) is selected as the reference for comparison between models. Summarizing this last part, the model according to Equation (14) is selected as reference for comparison to the others because it was shown that it amounts to a reparametrization of the one in Equation (2), which happens to be the one used as the base in Mexico's regulations (_Manual de proyecto geometrico de carreteras 2018_--Handbook of geometric projects in highways 2018) [18]. On the other hand, the percentages of relative error differences among safe (i.e., non slippage) highway design speeds generated by models considered with respect to the one generated by Equation (14) are proposed as suitable comparison metrics between them. Those differences are analytically interesting since they can be directly interpreted as potential errors in highway design speeds depending on what could be established by the complementary related experimental studies, completely outside the scope of this paper. The model used as a regulatory basis in Mexico ([18]) could be underestimating or overestimating safe highway operational speeds, each possibility bringing very different but relevant consequences whose detailed exploration is also outside the scope of this work.