# Safe Driving Distance and Speed for Collision Avoidance in Connected Vehicles
## 4 Stopping Distance and Braking Dynamics
### Sudden Stop of Vehicle B  
In some cases, the followed vehicle may stop suddenly (its speed drops to zero in zero seconds), in the case of a sudden crash with a stopped vehicle, for example, or in any other critical situation. If ACDA is calculated according to what has been described in the previous subsections, the following vehicle will collide with the followed vehicle. Thus, it is essential to consider this case when calculating the ACDA.  
Figure 18: 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 Autonomous Emergency Braking (AEB) system (\(T_{rA}=0\) s).  
Assuming that vehicle B stops suddenly with zero stopping distance. Thus, setting \(V_{B}=0\) in the previous equations yields:  
In the case when the following vehicle does not have an AEB system:  
\[V_{Amax}=\sqrt{\left(fgT_{rA}\right)^{2}+2 gfD}-fgT_{rA} \tag{8}\]  
\[D_{min}=V_{A}.\ T_{rA}+\frac{V_{A}^{2}}{2 gf} \tag{9}\]  
In the case when the following vehicle has an AEB system (\(T_{rA}=0\)):  
\[V_{Amax}=\sqrt{2 gfD} \tag{10}\]  
\[D_{min}=\frac{V_{A}^{2}}{2 gf} \tag{11}\]  
We present in Figure 20 the maximum speed of the following vehicle (A) gives the distance to avoid the collision, assuming that the followed vehicle (B) suddenly stops when the weather is clear and the case when the weather is foggy. In addition, the figure depicts the case when the vehicle is dotted with an AEB system for both dry asphalt and snowy road conditions. The maximum slowing speed for a given separation distance between the two vehicles is larger in the case when AEB is used compared to foggy and clear weather conditions. Here, we look at three (3) different scenarios on asphalt and snowy roads: the first scenario is when there is foggy weather (with driver reaction time) and in clear weather (with driver reaction), and the last scenario is when the vehicle has autonomous emergency braking; it can be observed that a vehicle with AEB system has the highest speed on both asphalt and snowy roads and when the vehicle is traveling in clear weather. However, the vehicle traveling in clear weather on asphalt has a higher speed than a vehicle with AEB system traveling on a snowy road. However, the maximum speeds are reduced significantly compared to the case when the followed vehicle normally stops (does not stop suddenly), as shown in Figure 14.  
Figure 19: 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 Autonomous Emergency Braking (AEB) sys-tem (\(T_{rA}=0\) s) for dry asphalt and snowy roads.  
In Figure 21, we present the minimum allowed distance (in meters) for a given speed of following vehicle A (km/h) to avoid the collision if the followed vehicle B suddenly stops both on asphalt and snowy roads. The distance is shorter for a vehicle with AEB system traveling on asphalt (around 80 m) than a vehicle traveling in clear weather with driver reaction on asphalt, etc. The same scenario is seen on a snowy road. In this case, the minimum allowed safe driving distance is reduced compared to the case when the followed vehicle does not suddenly stop as shown in Figures 16 and 17.