is not an arithmetic mean of partial speeds over individual sections
How simple intuition can fool us, let’s quickly solve the elementary task: “The cyclist ascends up a 10 km mountain for 1 hour, i.e. at an average speed of 10 km/h. There, he turns back and descends to the starting point for 0.5 hour, which gives him an average speed of 20 km/h. What is his average speed on the entire route?” Most of us will quickly provide as a solution the value 15 km/h, derived from the arithmetic average of partial speeds (10 km/h + 20 km/h) / 2 = 15 km/h. However, the cyclist rode a total of 20 km in 1.5 hours, thus he reached an average speed of 13.3(3) km/h.
1.1. The task at hand Competing athletes – cyclists, car racing drivers, skiers – they all ride along the course of approximately equal length. The speeds that they achieve on the individual sections of the course are varying. Among the above-mentioned competitions, likely the cyclists experience the greatest speed differences. They often ascend uphill at speeds below 15 km/h, and descend at speeds sometimes exceeding 80 km/h. Let’s calculate their average speed (Vs) along the entire route, when, the partial speeds of individual sections are equal – V1, V2, …, Vn, respectively.
1.2. Theoretical solution Let’s quickly derive the formula to encourage analysis of the surrounding reality, which often does not require a complicated thinking. We know that the length of the course equals to the average speed (Vs) multiplied by the time of the journey (T). It also equals to the sum of the partial sections on which the athlete achieves the speed Vi, passing them at time t. Mathematically we can write this as: Vs * T = V1 * t1 + V2 * t2 + … + Vn * tn Hence Vs = V1 * t1 / T + V2 * t2 / T + … + Vn * tn / T It can be noted that the overall average speed is influenced by the partial speeds, and their individual contributions are determined by ratio of partial sections travel times over the entire travel time.
1.3. Conclusions So in which part of the race course is the easiest to gain an advantage over other racers? The answer is that where racers go slowly, because traveling this section takes them relatively a lot of time. If we can significantly accelerate there, our average speed will increase notably, and the travel time may be remarkably shorter. We can clearly see this in cycling races – the Tour de France is won on long climbs of the mountain stages. How do we apply this conclusion to the ski race course? When preparing for the run, we have to particularly analyze those parts of the course where the speed is, or may be, low. Certainly the key sections will be: the start, the wider offset gates on steep sections, and the arrhythmic sections. We can definitely lose a good position when we enter the flat part of the course at too low speed. Then the travel time of such a section will definitely be significantly longer. Of course, racers often ski slower the difficult course sections. The time differences between individual racers at these sections are determined by skiers’ technical skills. Better trained skiers have the option to ski faster – and we have to coach them so they will ski aggressively using their skills toolset combined with well managed risk. Only then, they will prove themselves clearly, they can discover the capabilities not demonstrated yet and eventually beat the race opponents.
1.4. Reexamining the example In the above discussion I wanted to highlight analysis of every element in search of improving our performance in competitive racing. Breaking and focusing the analysis into a specific and distinguished subject will make it easier, and the use of simple analytical methods can lead us a little further than just a general, high-level view. Thus I suggest trying to examine elements that we naturally observe looking at skiers, e.g. where and how high to keep the inner hand of the turn; whether and when you can lift the inner ski; where can and where should not be the racer’s outer hand, etc.