## 3 Analytical Model Conceptual Development and Discussion
In contrast, acceleration has both normal (in the direction of \(\hat{r}\)) and tangential (in the direction of \(\hat{\theta}\)) components. Now, normal acceleration magnitude is the vehicle's speed squared divided by the trajectory's curvature radius, which, at the same time, is the vehicle's radial coordinate magnitude (\(v^{2}/r\)). In turn, tangential acceleration magnitude is the time derivative of the coordinate along the line of motion (\(\hat{v}\)). Applying the preceding model features to the layout presented by Figure 2, it is noticeable that even though the problem is 3D, the model emphasizes consideration of 2D motion on the plane defined by those normal and tangential coordinates; therefore, adjustments must be conducted, as is shown. It is also noticeable that the \"normal\" coordinate coincides with the regular Cartesian \(\hat{x}\) coordinate, and \"tangential\" coordinate, corresponds to regular Cartesian \(\hat{z}\) coordinate. That is, the last coordinate coming out of the plane of the page towards the reader is consistent with a right-handed Cartesian coordinate system. Figure 1: Diagram of the normal and tangential coordinates geometric setup used to define and develop the instantaneous radius of curvature 2D model IRCM for highway vehicle dynamics. In this figure, \(r\) is the trajectory’s curvature radius and, simultaneously, the radial coordinate (in the cylindrical-polar coordinate system), \(\theta\) is the vehicle’s instantaneous angular position/polar coordinate (in the cylindrical-polar coordinate system), and \(\hat{x}\), \(\hat{y}\) are the 2D Cartesian coordinate system unit vectors. Figure 2: General layout of the problem’s dynamic and the geometrophysical scheme applied to account for it. The model uses “normal”, \(y\), and “tangential” coordinates; that is, \((x;y;z)\). In this figure, \(\vec{N}\), \(\vec{F}\), \(\vec{W}\) are the forces acting on the vehicle: normal, friction, and weight; respectively. The angle \(\theta\) is the highway’s considered super-elevation. \(\hat{x}\), \(\hat{y}\), \(\hat{z}\) are the 3D Cartesian coordinate system unit vectors. Considering an instantaneous validity of the dynamic situation as shown in Figure 2, the application of the second law of Newton on the vehicle results in:  
\[\overrightarrow{N}+\overrightarrow{W}+\overrightarrow{F}=m\overrightarrow{a} \tag{5}\]  
\(\overrightarrow{N}\) is the normal force exerted by the surface of the highway on the vehicle, \(\overrightarrow{W}\) is the weight of the vehicle, \(\overrightarrow{F}\) is the friction force applied by the surface of the highway on the vehicle, \(m\) is the vehicle's mass, and \(\overrightarrow{a}\) is its acceleration. In Cartesian--\(\hat{i},\hat{j},\hat{k}\)--component vector representation, Equation (5) yields  
\[ \begin{split}(-Nsin\theta;Ncos\theta;0)+(0;-mg;0)+(-\mu Ncos \theta;-\mu Nsin\theta;0)\\ =m\Big{(}-v^{2}/R;0;0\Big{)}\end{split}  \tag{6}\]  
\(\theta\) is the highway super-elevation angle, \(g\) is the gravitational acceleration magnitude, \(\mu\) is the friction coefficient between the vehicle's tires and pavement, \(v\) is the vehicle's speed at which the system is in dynamic equilibrium, and \(R\) is its _instantaneous_ trajectory curvature radius. The minus sign on the first term after the equal sign in Equation (6) means that for the dynamical equilibrium situation considered, the normal component of the vehicle's acceleration is centripetal, because it represents the force equilibrating the non-kinetic friction force between the vehicle tires and pavement (i.e., \(\overrightarrow{F}\) in Equation (5)).