# Safe Driving Distance and Speed for Collision Avoidance in Connected Vehicles
## 4 Stopping Distance and Braking Dynamics
### Case When the Rear Vehicle Auto-Brake in Case of Emergency  
We have two cases: the case if the vehicle's brake is controlled by the driver only and the case if the vehicle OBU triggers the braking system automatically in the case of any detected emergency. In the latter case, the followed vehicle detects an emergency situation, and when its brake is pressed, it will immediately send an emergency brake warning message to the surrounding vehicles using V2V. Thus, the following vehicle OBU receives this message and automatically triggers the braking system. The time for sending and receiving this message is in the order of a few milliseconds, which can be neglected. Recently, several vehicles have been equipped with an Autonomous Emergency Braking (AEB) system that brakes the vehicle autonomously in the case of perceived danger without any intervention from the driver. Connected vehicles may have this system, while autonomous vehicles must have this system.  
In the case when the vehicle can brake instantly when it receives an Imminent-Danger message from any of the vehicles in front of it using V2V, then the driver reaction time of the following vehicle is zero or \(T_{rA}=0\). In such a case, we have the following equations:  
\[\frac{V_{A}^{2}}{2 gf}=\ \frac{V_{B}^{2}}{2 gf}+D \tag{5}\]  
Figure 16: Minimum safe driving distance for different speeds of both the following and the followed vehicle in the case when the weather is clear, assuming the driver reaction time is 1 s for dry asphalt and snowy roads.  
Figure 17: Foggy weather: minimum safe driving distance for different speeds of both the following and the followed vehicle in the case when the weather is clear, assuming the driver reaction time is 8 s for dry asphalt and snowy roads.  
\[V_{Amax}=\sqrt{V_{B}^{2}+2 gfD} \tag{6}\]  
\[D_{min}=\frac{V_{A}^{2}}{2 gf}-\frac{V_{B}^{2}}{2 gf} \tag{7}\]  
In such a case, the maximum driving speed of the following vehicle will be increased, and the minimum safe driving distance given the speed of the two vehicles will decrease. Figure 18 presents the maximum speed of the following vehicle for different speeds of the followed vehicle for different initial distances between the two vehicles when the following vehicle is dotted with an Autonomous Emergency Braking (AEB) system (\(T_{rA}=0\) s) for both dry asphalt and snowy roads. For example, when the followed vehicle is traveling at a speed of 100 km/h and the current separation distance between the two vehicles is 100 m, the maximum safe driving speed of the following vehicle is 185 km/h and 125 km/h in the case of dry asphalt and snowy roads respectively. If we compare these values with those obtained for the same conditions when there is a driver reaction (clear weather), the maximum allowed speed is 150 km/h and 110 km/h (as shown in Figure 14). As it can be seen, this contributes to maximizing road utilization, reducing traffic congestion, and decreasing pollutant emissions.  
Similarly, we calculated the minimum safe driving distance for different speeds of both the following and the followed vehicle in the case when the following vehicle is dotted with an Autonomous Emergency Braking (AEB) system (\(T_{rA}=0\) s) for dry asphalt and snowy roads. The results are presented in Figure 19. As can be seen, when comparing the results depicted in this figure with those depicted in Figure 15 using the AEB significantly decreases the minimum safe driving distance for the same values of the driving speeds of both vehicles.