Equations
We consider a single lane of road. We define the following variables
[math]\displaystyle{
\begin{matrix}
&\rho &: &\mbox{car density (cars/km)} \\
& v(\rho) &: &\mbox{car velocity (km/hour)} \\
& q =\rho v &: &\mbox{car flow rate (cars/hour)} \\
\end{matrix}
}[/math]
If we consider a finite length of road [math]\displaystyle{ x_1\leq x \leq x_2 }[/math] then the net flow of cars
in and out must be balanced by the change in density. This means that
[math]\displaystyle{
\begin{matrix}
&\rho &: &\mbox{car density (cars/km)} \\
& v(\rho) &: &\mbox{car velocity (km/hour)} \\
& q =\rho v &: &\mbox{car flow rate (cars/hour)} \\
\end{matrix} }[/math]
