# Coefficient of Difficulty in Running a Sports Course in Alpine Skiing
## 2 Materials and Methods
At high speeds, the role of psychological factors also increases, such as fear of injury. Therefore, passing slowly over a given set of gates is more accessible. The optimal speed was assumed as the level of difficulty 1 for the average velocity obtained in individual competitions (for the top ten) (Figure 3). Figure 1: Dependence of the coefficient of difficulty on the change of direction (deviation angle = 1/2 \(\alpha\)). Figure 2: Dependence of the coefficient of difficulty on the radius of the travelled curve (rk-design radius of the ski). Fifth is small distances between subsequent turns (short time for changing the edge). Sixth, overloads on the run are a value proportional to the weight of the competitor and the square of the speed and inversely proportional to the turning radius r. Difficulty level 1 is the overload occurring when passing a curve with an average radius at medium speed (values characteristic for a given competition), as per Figure 4.  
Based on the above observations, a formula for W.tr\({}_{i}\) can be proposed as the difficulty factor for setting gates in slalom and downhill, taking into account the factors described Equation (2):  
\[\text{W.tr}_{i}=(\text{f.ok}+\text{f.r}+\text{f.f}+\text{f.v}+\text{f.s}+\text {f.a})/6 \tag{2}\]  
where f are the coefficients: f.ok-direction deviations, f.r-turning radius, f.f-frequencies, f.v-velocity, f.s-segment length, f.a-overloads  
Further planned studies will determine the importance of individual factors. Formulas for individual factors (markings as before, average values for a given competition):  
\[\text{f.ok}_{i}=\alpha/\alpha_{\text{av}} \tag{3}\]  
\[f.r_{i}=\left\{ \begin{array}{l}r_{i}<rk\qquad\qquad f.r_{i}=0.5\,rk/r_{i}\\ 0.5\,rk<r_{i}<2\,rk\qquad\qquad f.r_{i}=1\\ r_{i}>rk\qquad\qquad f.r_{i}=\frac{r}{2}rk\end{array} \right. \tag{4}\]  
\[\text{f.f}_{i}=3\,\text{f}_{i}/(\text{f}_{i-1}+\text{f}_{i-2}+\text{f}_{i-3}) \text{ for i> 3 (for i $\leq 3\,\,\text{f.f}=1$)} \tag{5}\]  
Figure 4: Dependence of the coefficient of difficulty on overload (acceleration a expressed in multiples of g). Figure 3: Dependence of the difficulty coefficient on velocity (\(v_{\text{op}}\)-optimal velocity, \(v_{\text{kr}}\)-critical velocity). \[\text{f.v}_{\text{i}}=\text{v}_{\text{i}}/\text{v}_{\text{av}} \tag{6}\] \[\text{f.s}_{\text{i}}=0.5\text{ rk}/\text{s}_{\text{i}}\text{ for s}_{\text{i}}<0.5\text{ rk }\text{ (for s}_{\text{i}}\geq 0,\text{ rk f.s}_{\text{i}}=1)\] (7) \[\text{f.a}_{\text{i}}=\text{v}_{\text{i}}^{2}\text{ x r}_{\text{i}}/\text{v}_{ \text{av}}^{2}\text{ x r}_{\text{av}}\] (8) \[\text{W.tr}=\frac{1}{n}\sum_{\text{i=1}}^{n}(\text{W.tr}_{\text{i}}) \tag{9}\]  
W.tr is the arithmetic mean of the difficulty of individual turns on the course Equation (9). By comparing W.tr between courses, you can determine the average difficulty setting for the entire course. However, there is often one difficult part, the so-called \"Traps\", for which the mean value and standard deviation W.tr will not show. Therefore, the value of W.tr max will also be shown hereafter, calculated as the average of the five most challenging turns. Since there is no previous research on slalom gate setting, it is impossible to determine the difficulty on a scale: easy, complex, and challenging. Therefore, we can only compare the gate settings and see which is more complicated. With more data in the future, it is possible to establish severity limit values at which accidents may occur. The calculated indices were subjected to standard descriptive analysis, with means, medians, and SDs determined. Before the main analyses, data distribution was checked for normality using the Shapiro-Wilk test. We tested for mean differences (on all variables) between the courses using analysis of variance for independent groups.