# Research on Safe Following Distance on an Expressway Based on Braking Process Analysis
## 4 Simulation Analysis and Application
### Application of Three Kinds of Safe Distance on Expressways  
In practical cases, it is assumed that a selection switch is set on the dashboard of the cab. The driver can put the button in an appropriate position according to the actual road conditions during the day of driving, and the alarm system can select the road adhesion coefficient for the data processing unit to calculate.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(i=-0.3\), \(d=5\)} & \multicolumn{5}{c}{Relative Velocity \(V_{r}\) (km/h) (\(V_{r}=V_{B}-V_{A}\))} & \multirow{2}{*}{\(V_{B}\)} \\ \cline{2-2} \cline{5-7}  & & **60** & **80** & **100** & **120** & **(km/h)** \\ \hline \multirow{3}{*}{\begin{tabular}{c} **The sufficient safe** \\ **distance between cars** \\ \(D_{3}\) **(m)** \\ \end{tabular} } & \multirow{3}{*}{\begin{tabular}{c} 46.3609 \\ 68.1678 \\ \end{tabular} } & \multirow{3}{*}{\
\begin{tabular}{c} 60 \\ 80 \\ 93.9831 \\ \end{tabular} } & \multirow{3}{*}{
\begin{tabular}{c} 60 \\ 80 \\ 123.8067 \\ \end{tabular} } \\ \cline{1-1} \cline{3-5}  & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 10: The sufficient safe distance \(D_{3}\) based on \(V_{B}\) between 60 and 120 km/h when \(i=-0.3\) and \(d=5\).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{\(i=-0.3\), \(d=5\)} & \multicolumn{5}{c}{Relative Velocity \(V_{r}\) (km/h) (\(V_{r}=V_{B}-V_{A}\))} & \multirow{2}{*}{\(V_{B}\)} \\ \cline{2-2} \cline{5-7}  & & **0** & **10** & **20** & **30** & **40** & **50** & **60** & **(km/h)** \\ \hline \multirow{3}{*}{
\begin{tabular}{c} **The minimum safe** \\ **distance \(D_{1}\) **(m)** \\ \end{tabular} } & 5.0000 & & & & & & & & 60 \\  & 5.0000 & 16.4045 & 26.8070 & & & & & & 80 \\  & 5.0000 & 18.4087 & 30.8153 & 42.2198 & 52.6223 & & & & 100 \\  & 5.0000 & 20.4129 & 34.8236 & 48.2323 & 60.6389 & 72.0435 & 82.4459 & 120 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The minimum safe distance \(D_{1}\) based on \(V_{B}\) between 60 and 120 km/h when \(i=-0.3\) and \(d=5\).  
Here, the relative speed was set as \(V_{r}=V_{B}-V_{A}=20\) km/h, and \(V_{B}\) ranged from 80 to 120 km/h. The basic following distance between car A and car B on different roads was simulated, and the results shown in Figure 12 were obtained. The specific parameters are shown in Table 11.  
As we considered the safety distance under three different driving conditions, we can consider providing an impact factor according to the needs of practical operation and feasibility. As the minimum following distance expects perfection from the following car's driver's reaction time, and the sufficient following distance accounts for an extreme case, the basic following distance reflects most real scenarios. Thus, we set three factors influencing safe following distance (which can be analyzed using probability statistics according to actual driving conditions). The influencing factor of the minimum following distance \(D_{1}\) is \(\omega_{1}\), the influencing factor of the basic following distance D\({}_{2}\) is \(\omega_{2}\), and the influencing factor of the sufficient following distance \(D_{3}\) is \(\omega_{3}\); and \(\omega_{1}+\omega_{2}+\omega_{3}=1\). Therefore, it is finally provided to the warning system as the warning critical safe distance \(S\):  
\[S=\omega_{1}D_{1}+\omega_{2}D_{2}+\omega_{3}D_{3} \tag{14}\]