# Median U-Turn Intersection Critical Parameter Research and Operational Performance Evaluation
## 3 Critical Parameter Calculation of MUT Intersections
### The Calculation of the Gap of the Median U-Turn of MUT Intersections
Assuming that _Car P_ accelerates from speed \(v_{i}\) to speed \(v_{L}\) (m/s) with a constant acceleration, \(a_{L}\) (m/s\({}^{2}\)), until the distance between _Car P_ and _Car C_ reaches the safe following distance, \(L_{R}\). The time taken for _Car P_ to accelerate is \(t_{J}\) (s). Thus, we can obtain:  
\[v_{L}^{2}-v_{l}^{2}=2a_{L}\times(L_{R}+|\text{A}_{2}\text{A}_{3}|), \tag{11}\]  
\[v_{L}=v_{i}+a_{L}t_{J}, \tag{12}\]  
By combining the equations above, we can obtain:  
\[t_{G}=\frac{L_{R}}{v_{i}}-\frac{\left(v_{L}-v_{i}\right)^{2}}{2a_{L}v_{i}}. \tag{13}\]  
Because of the unconventional geometric design of the MUT, the traffic on the minor road must first detour onto the major road. Unlike the traffic on the major road, the traffic from the minor road has to change lanes twice or more to approach the median U-turn [37]. Thus, the detoured traffic is likely to form a weaving segment with the through traffic. The average speed of weaving traffic has to be estimated [38]. The average speed of vehicles within the weaving segment, \(v_{i}\), can be obtained:  
\[v_{i}=v_{\text{min}}+\frac{v_{\text{max}}-v_{\text{min}}}{1+W_{i}}, \tag{14}\]  
Figure 6: The vehicles’ operation processes in the median U-turn. \[W_{i}=\frac{a(1+V_{R})^{b}(\frac{q}{n})^{c}}{\left(3.28L_{w}\right)^{d}}, \tag{15}\]  
where \(W_{i}\) is the intensity of the weaving traffic or non-weaving traffic, \(V_{R}\) is the weaving traffic ratio, \(q_{i}\) is the total traffic volume within the weaving zone (pcu/h), \(n\) is the number of lanes within the weaving zone, \(a\), \(b\), \(c\), \(d\) is the constant, \(v_{max}\) is the maximum possible speed within the weaving zone (m/s), \(v_{min}\) is the minimum possible speed within the weaving zone (m/s), and \(L_{w}\) is the length of the weaving section (m). Table 1 records the distribution of constants for Equation (15). For all drivers, it is important to keep a certain safe following distance from the vehicle in front when driving. A safe following distance gives drivers time to see, think, and act. Under normal road conditions, the safe following distance should be a minimum of three seconds (three-second rule) [39; 40; 41]. It is worth noting that there is still no standardized guideline to calculate the safe following distance. The three-second rule is still widely adopted by transportation departments around the world. The safe following distance calculated based on the different vehicle driving speeds at urban intersections is summarized in Table 2.  
So, the maximum gap of the median U-turn, \(L_{max}\), can be concluded:  
\[L_{d}=(2\varphi-\gamma)\times v_{c}-\frac{1}{2}(\varphi-\gamma)^{2}\times a_{ c}, \tag{9}\]0  
Figure 7 shows the minimum gap for vehicles turning around at a median U-turn. The minimum gap for vehicles at the median U-turn denoted as \(L_{min}\) is defined as: the difference between the outermost trajectory radius, \(R_{o}\), and the innermost trajectory radius, \(R_{t}\), of the turning vehicle when it turns around with the minimum radius of the vehicle turn around, and adding the double values of the edge security distance, \(G_{s}\). Figure 6 shows the trajectory of vehicles turning around. So, we can determine the calculation formula for the minimum gap of the median U-turn, \(L_{min}\):  
\[L_{d}=(2\varphi-\gamma)\times v_{c}-\frac{1}{2}(\varphi-\gamma)^{2}\times a_{ c}, \tag{9}\]1  
\[L_{d}=(2\varphi-\gamma)\times v_{c}-\frac{1}{2}(\varphi-\gamma)^{2}\times a_{ c}, \tag{9}\]2  
\[L_{d}=(2\varphi-\gamma)\times v_{c}-\frac{1}{2}(\varphi-\gamma)^{2}\times a_{ c}, \tag{9}\]3  
where \(R_{w}\) is the radius of turning around for outer wheels (m), \(R_{i}\) is the radius of turning around for inner wheels (m), \(W_{h}\) is the width of half a vehicle wheel (m), \(d_{w}\) is the wheelbase of the vehicle for turning around (m), \(\theta_{w}\) is the outer wheel angle of the vehicle (deg), and \(\theta_{n}\) is the inner wheel angle of the vehicle (deg).