# Research on Safe Following Distance on an Expressway Based on Braking Process Analysis
## 4 Simulation Analysis and Application  
### Calculations of Three Safe Distances for Different Road Conditions and Slopes  
Since only the deceleration during braking is involved in the minimum following distance model, and the front car and the following car are driving on the same road (making the road conditions the same), the maximum deceleration during braking is taken as \(j_{A\max}=j_{B\max}\) in this paper. We take the braking deceleration of both cars to be the same. On the one hand, this simplifies the calculations. On the other hand, in actual scenarios, the deceleration of the following car is generally greater than that of the front car. If they are equal, greater safety is implied. At the same time, we can assume that \(V^{\prime}{}_{A}=V^{\prime}{}_{B}\) after car A and car B finish braking (including a requirement that both cars have stopped after the end of braking). If \(V^{\prime}{}_{A}\) is less than \(V^{\prime}{}_{B}\), obviously both cars are not in the safe state, and if \(V^{\prime \begin{table}
\begin{tabular}{c c c c c c c c} \hline \multirow{2}{*}{\(i=0\), \(d=3\)} & \multicolumn{5}{c}{**Relative Velocity \(V_{r}\) (km/h) (\(V_{r}=V_{B}-V_{A}\))**} & \multicolumn{1}{c}{\(V_{B}\)} \\ \cline{2-7}  & **0** & **10** & **20** & **30** & **40** & **50** & **60** & **(km/h)** \\ \hline \multirow{3}{*}{
\begin{tabular}{c} The minimum safe \\ distance \(D_{1}\) (m) \\ \end{tabular} } & 3.0000 & \multirow{3}{*}{14.1227} & \multirow{3}{*}{24.2809} & \multirow{3}{*}{39.2616} & \multirow{3}{*}{49.4198} & \multirow{3}{*}{69.4167} \\  & 3.0000 & & & & & 100 \\ \cline{1-1}  & 3.0000 & & & & & 120 \\ \cline{1-1}  & 3.0000 & & & & & & 120 \\ \hline \end{tabular}
\end{table} 
Table 2: The minimum safe distance \(D_{1}\) based on \(V_{B}\) between 60 and 120 km/h when \(i=0\) and \(d=3\). Figure 4: Changes in the basic safe distance \(D_{2}\) with various \(V_{B}\) and \(V_{r}\) when \(i=0\) and \(d=3\). Figure 5: Changes in the sufficient safe distance \(D_{3}\) with various \(V_{B}\) and \(V_{r}\) when \(i=0\) and \(d=3\). * For \(i=-0.3\) and \(d=5\) on the downhill road, the simulations of three kinds of safe distance are shown in Figures 9-11. The results of Tables 8-10 can be obtained from Figures 9-11, respectively. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(i=0.3\), \(d=2\)} & \multicolumn{5}{c}{**Relative Velocity \(V_{r}\) (km/h) (\(V_{r}=V_{B}-V_{A}\))**} & \multirow{2}{*}{\(V_{B}\) (km/h) (km/h)} \\ \cline{2-6}  & & & **Relative Velocity \(V_{r}\) (km/h) (\(V_{r}=V_{B}-V_{A}\))** & & & \(V_{B}\) (km/h) \\ \cline{2-6}  & **0** & **10** & **20** & **30** & **40** & **50** & **60** \\ \hline \multirow{3}{*}{
\begin{tabular}{c} The minimum safe \\ distance \(D_{1}\) (m) \\ \end{tabular} } & 2.0000 & & & & & & 60 \\  & 2.0000 & 12.8612 & 22.7928 & & & & 80 \\  & 2.0000 & 14.7205 & 26.5114 & 37.3726 & 47.3042 & & 100 \\  & 2.0000 & 16.5798 & 30.2300 & 42.9505 & 54.7413 & 65.6026 & 75.5341 & 120 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The minimum safe distance \(D_{1}\) based on \(V_{B}\) between 60 and 120 km/h when \(i=0.3\) and \(d=2\). Figure 8: Changes in the sufficient safe distance \(D_{3}\) with various \(V_{B}\) and \(V_{r}\) when \(i=0.3\) and \(d=2\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\(i=0.3\), \(d=2\)} & \multicolumn{5}{c}{**Relative Velocity \(V_{r}\) (km/h) (\(V_{r}=V_{B}-V_{A}\))**} & \multirow{2}{*}{\(V_{B}\) (km/h) (km/h)} \\ \cline{2-6}  & & **60** & **80** & **100** & **120** & **(km/h)** \\ \hline \multirow{3}{*}{
\begin{tabular}{c} The sufficient safe \\ distance between cars \\ \(D_{3}\) (m) \\ \end{tabular} } & 42.0569 & & & & & 60 \\  & 62.8497 & & & & 80 \\  & 67.3611 & & & 100 \\  & & & & 115.5911 & 120 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The sufficient safe distance \(D_{3}\) based on \(V_{B}\) between 60 and 120 km/h when \(i=0.3\) and \(d=2\). Figure 11: Changes in the sufficient safe distance \(D_{3}\) with various \(V_{B}\) and \(V_{r}\) when \(i=-0.3\) and \(d=5\).