# Safe Driving Distance and Speed for Collision Avoidance in Connected Vehicles
## 4 Stopping Distance and Braking Dynamics
### Sufficient Safe Gap between Two Vehicles  
In the previous section, we presented the stopping dynamics when the driver of one vehicle perceives danger and decides to stop. We assumed that the vehicle was moving with  
a constant speed and then constant deceleration until it stops completely. However, vehicles are moving normally at approximately the same speed. If two vehicles realize the same danger at the same time and start barking at the same time, they will not collide as they will stop at the same time, maintaining the same gap while they brake. If we consider the stopping distance presented in the previous section as the minimum safe driving distance or Assured Clear Distance Ahead (ACDA), this will reduce road utilization efficiency and hence create many traffic management and optimization problems. Without IoV technology, it is not possible to objectively obtain ACDA for each vehicle as a function of the speed of the two vehicles, the tire states, the road type and condition, the weather condition, etc. In this section, we aim to calculate the sufficient ACDA dynamically, taking into account all affecting parameters when the two vehicles are moving at different speeds and decelerate with different decelerations.  
We show in Figure 12 two vehicles A and B are moving, keeping a gap distance, \(d\), between them at a specific time \(t\). Let's assume that the rear vehicle (A) is moving at a speed \(V_{A}\), located at \(L_{A}\), having a tire state \(T_{A}\), and the length of the vehicle is \(l_{A}\). The front vehicle (B) has its own parameters too. The two vehicles exchange these parameters in real-time periodically using V2V communication. If the front vehicle perceives danger and decides to brake, this decision is instantly sent to all nearby vehicles, including the rear vehicle A. Vehicle B decelerates with a deceleration \(j_{B}\). Thus, vehicle A starts braking using a deceleration \(j_{A}\). If we assume that the stopping distance of both vehicles are \(S_{A}\) and \(S_{B}\), then the distance between them when they stop will be  
\[\delta=S_{B}-S_{A}+d\]  
When the front vehicle B starts braking, the rear vehicle A will not brake instantly, it will continue at its current speed during the thinking and the reaction times (as shown in Figure 6). During that time, the vehicle traveled a distance \(V_{A}\). \(T_{rA}\), then it will decelerate.  
The work presented in [48] aims to estimate the safety gap to avoid rear-end collision safe running on highways. The authors propose three formulas for calculating the safety gap. The proposed formulas consider the speed and acceleration/deceleration of the two vehicles to estimate the sufficient safety gap to avoid a rear collision if the front vehicle brakes suddenly. We will use the proposed formulas in the next section as the basis of developing an enhanced one taking into consideration different realistic parameters such as the weather condition.  
We must calculate the braking distance of the followed vehicle \(B\): \(\frac{V_{B}^{2}}{2 gf}\).  
Stopping distance of \(A\) (following vehicle) = \(D\) + Braking Distance of \(A\) (followed vehicle), neglecting the reaction time of \(A\).  
\[V_{A}.\ T_{rA}+\frac{V_{A}^{2}}{2 gf}=\frac{V_{B}^{2}}{2 gf}+D \tag{2}\]  
Figure 12: Parameters affecting the sufficient safe driving gap provided that IoV is used to broadcast these parameters to the surrounding vehicles in real time.  
For any given initial distance between the two vehicles, \(f\) and \(V_{B}\) we can find the maximum speed that can provide the minimum safe distance between the two vehicles by solving the previous quadratic equation, yielding  
\[V_{Amax}=\frac{-2 fgT_{rA}+\sqrt{\left(2 fgT_{rA}\right)^{2}+4\left(V_{B}^{2}+2 gfD \right)}}{2}=\sqrt{\left(fgT_{rA}\right)^{2}+V_{B}^{2}+2 gfD}-fgT_{rA} \tag{3}\]  
\[D_{min}=V_{A}.\ T_{rA}+\frac{V_{A}^{2}}{2 gf}-\frac{V_{B}^{2}}{2 gf} \tag{4}\]