# Research on Safe Following Distance on an Expressway Based on Braking Process Analysis
## Modeling
### Establishment of a Mathematical Model of Minimum Safe Distance
#### 3.2.2 Discussion of Safe Distance  
According to the analysis of the braking distance of the front car, three different safety distances were obtained.  
1. [label=(0)]
2. Minimum safe distance \(D_{1}\)  
Obviously, the critical safe distance is calculated by considering the response of the driver of car A and the braking time.  
\[\begin{split}& D_{1}=D_{B}+d-D^{\prime}{}_{A}\\ &=V_{B}(\text{t}_{1}+t_{2}+\frac{t_{3}}{2})+\frac{V_{B}^{2}-V_{B }^{\prime 2}}{2j_{\text{max}}}-\frac{j_{\text{max}}}{24}t_{3}^{2}-(V_{A}(\text{t} _{1}+t_{2}+\frac{t_{3}}{2})+\frac{V_{A}^{2}-V_{A}^{\prime 2}}{2j_{\text{max}}}- \frac{j_{\text{max}}}{24}t_{3}^{2})+d\\ &=(\text{t}_{1}+t_{2}+\frac{t_{3}}{2})(V_{B}-V_{A})+\frac{V_{B}^ {2}-V_{B}^{\prime 2}}{2j_{\text{max}}}-\frac{V_{A}^{2}-V_{A}^{\prime 2}}{2j_{ \text{max}}}-\frac{j_{\text{max}}}{24}t_{3}^{2}+\frac{j_{\text{max}}}{24}t_{3 }^{2}+d\end{split} \tag{11}\]  
This formula reflects the minimum distance between car A and car B before the two cars brake if the two cars want to avoid a collision. It is called the minimum following distance.  
2. Basic safe distance \(D_{2}\)  
In the actual expressway driving process, the driver of the following car (car B) will implement braking only after identifying the front car's brake lights. Therefore, this reduces the distance that car A moves before the driver reacts. Therefore, this distance requirement is the general distance requirement. Hence, the basic following distance \(D_{2}\) can be obtained.  
\[\begin{split}& D_{2}=D_{B}+d-D_{A}\\ &=V_{B}(\text{t}_{1}+t_{2}+\frac{t_{3}}{2})+\frac{V_{B}^{2}-V_{B }^{\prime 2}}{2j_{\text{max}}}-\frac{j_{\text{max}}}{24}t_{3}^{2}-(V_{A}\frac{t_{ 3}}{2}+\frac{V_{A}^{2}-V_{A}^{\prime 2}}{2j_{\text{max}}}-\frac{j_{\text{max}}}{24}t_{3}^{2})+d\\ &=V_{B}(\text{t}_{1}+\text{t}_{2})+(V_{B}-V_{A})\frac{t_{3}}{2}+ \frac{V_{B}^{2}-V_{B}^{\prime 2}}{2j_{\text{max}}}-\frac{V_{A}^{2}-V_{A}^{\prime 2}}{2j_{ \text{max}}}-\frac{j_{\text{max}}}{24}t_{3}^{2}+\frac{j_{\text{max}}}{24}t_{3 }^{2}+d\end{split} \tag{12}\]This formula reflects the distance between car A and car B that should be maintained when the following car can see the front car's brake lights (assuming they function). It is called the basic following distance.  
1. Sufficient safe distance \(D_{3}\)  
The front car's deceleration \(j_{A\max}\) can tend toward \(\infty\)--that is, \(j_{A\max}=\infty\). (This can happen, for example, if the front car crashes, or if it has very powerful brakes). As a result, the initial velocity \(V_{A}\) of car A can be considered to become 0 instantaneously [23]. In this case, \(D_{A}\approx 0\), so the safe spacing \(D_{3}\) can be obtained.  
\[D_{3}\ =V_{B}(t_{1}+t_{2}+\frac{t_{3}}{2})+\frac{V_{B}^{2}-{V_{B}^{\prime}}^{2}}{ 2j_{\rm{Bmax}}}-\frac{j_{\rm{Bmax}}}{24}t_{3}^{2}+d \tag{13}\]  
This reflects the safe distance requirement when car B must brake in the most unfavorable conditions (such as the front vehicle suddenly happening to crash). This is called the sufficient following distance.  
Obviously, from the above analysis, the following formula can be obtained: \(D_{1}<D_{2}<D_{3}\).