# 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  
The gap of the median U-turn is a critical factor affecting the operation of turn-around vehicles. An effective gap between different fast-moving vehicles should be provided in the approaching through traffic [32; 33]. The gap of the median U-turn must ensure that there is no collision between the turn-around vehicles and straight-through vehicles. To avoid potential rear-end accidents, motorists must be able to observe the approaching vehicles within a limited sight distance and react quickly to control the vehicle properly. The area needed to provide this unobstructed view is called the Clear Sight Triangle [34; 35]. Figure 5 shows the movements of vehicles through the median U-turn. Based on the description above, this paper combines the Clear Sight Triangle with the vehicle operation process to determine the maximum gap for median U-turns. Thus, the following assumptions can be made:  
1. The traffic volume at the intersection remains stable. 2. Turn-around vehicles and straight-through vehicles all have a clear view of each other. 3. Drivers only turn around when there is a sufficient gap, and the speed of the turn-around remains constant during the turning around of the vehicles. 4. Each vehicle in the same group of turn-around vehicles takes the same time to pass through the median U-turn. The operation process of vehicles on the minor road crossing the median U-turn is shown in Figure 6. The vehicle on the minor road (_Car P_) first departs from position P\({}_{1}\) to turn right to enter the major road. It will weave with the vehicles on the major road. Then, _Car P_ arrives at position P\({}_{2}\) with the average weaving speed \(v_{i}\) (m/s) to wait to turn around. When _Car P_ is at position P\({}_{2}\), the oncoming vehicle on the major road (_Car C_) is at position C\({}_{1}\). _Car C_ starts decelerating from speed \(v_{c}\) (\(v_{c}>v_{i}\)) (m/s) with a constant acceleration of \(a_{c}\) (m/s\({}^{2}\)). _Car P_ departs from position P\({}_{2}\) to cross the median U-turn via position P\({}_{3}\). It can be considered that _Car P_ has completed the turn around when its rear end is appropriately at A\({}_{2}\) point. _Car C_ has decelerated to speed \(v_{i}\) at this moment; its headstock is appropriately at A\({}_{1}\) point. In this paper, we consider the variable \(G\) to be the critical gap between _Car P_ and _Car C_ at this moment. The critical gap is a vital factor for vehicle safety operation. It is essentially the drivers' acceptance of the minimum distance between two adjacent vehicles [36]. The critical gap will be beneficial for determining the maximum gap of the median U-turn to determine its safety effectiveness. After ensuring that there is no collision between _Car P_ and _Car C_, _Car P_ begins to accelerate in order to quickly pass through the intersection. Then, _Car C_ will maintain a uniform speed of \(v_{i}\) until _Car P_ and _Car C_ reach the safe following distance, \(L_{R}\) (m). Subsequently, _Car C_ starts accelerating to follow _Car P_. The movement processes of _Car P_ and _Car C_ are as follows:I. The deceleration process of _Car C_. Assuming the time it takes for _Car P_ to complete the turn around from position P\({}_{2}\) is \(\varphi\) (s), the reaction time of the driver approaching the median U-turn is \(\gamma\) (\(\varphi>\gamma\)) (s). The deceleration distance of _Car C L\({}_{d}\)_ can be calculated as follows:  
\[L_{d}=(2\varphi-\gamma)\times v_{c}-\frac{1}{2}(\varphi-\gamma)^{2}\times a_{ c}, \tag{9}\]  
II. The uniform speed driving process of _Car C_. Assuming that the distance that _Car C_ travels by _Car C_ at a uniform speed, \(L_{G}\), the time for _Car C_ to pass the critical gap, \(G\), is \(t_{G}\) (s). We can obtain:  
\[L_{G}=v_{i}\times t_{j}=G+|\text{A}_{2}\text{A}_{3}|=v_{i}\times t_{G}+|\text{ A}_{2}\text{A}_{3}|, \tag{10}\]  
III. The acceleration process of _Car P_.