In earlier books and papers, we frequently discuss the worldwide quest of runners, coaches and running scientists for options to optimize the running performance through reducing the air-resistance. Cyclists and speed-skaters have been doing this for a long time already by developing optimal aerodynamic conditions (clothing, frames, body position, streamlining, drafting) .

We also discussed the development of the Stryd v3, which measures the Air-Power in real time and the sensational INEOS 1:59 Challenge. In Vienna, Eliud Kipchoge was helped by 41 pacers (5 teams of 7 pacers and 6 reserves) to achieve his magnificent 1:59:40. Finally, we evaluated the impact of the huge fans providing tail wind that helped Justin Gatlin to run 9.45 seconds in the 100 meters.

In this paper, we will calculate concretely and exactly how big the impact of the wind speed and the wind direction is on the running speed. As an example, we use author Hans, who weights 58 kg and can run a 5K-race time of 18 minutes (or at least he could, since he is currently injured).

**Theory**

In **our book ‘The Secret of Running’** we have discussed the theory of the air-resistance in various chapters. According to the fundamental laws of physics, the power required to surmount the air resistance P_{a }(Air Power) is determined by the density of the air ρ (in kg/m^{3}) the air-resistance factor c_{d}A (in m^{2}), the running speed v (in m/s) and the wind speed v_{w} (in m/s), as shown in the box.

The figure below shows the Air Power as a function of running speed v at the standard conditions (temperature 20°C, air-pressure 1013 mbar, so ρ = 1.205 kg/m3, c_{d}A = 0.24 m^{2}, no wind, so v_{w}= 0 m/s).

First, we assume that Hans ran his time of 18 minutes in ideal conditions, so no wind (v_{w}=0). Consequently, his air-resistance P_{a} will have been equal to 14 Watts at the standard conditions and at his running speed of 5000/18/60 = 4.63 m/s. His running resistance P_{r }will have been equal to

4.63_{*}58 (kg)_{*}0.98 (ECOR in kJ/kg) = 263 Watts.

Consequently, his total power is equal to 263+14 = 277 Watts.

**How big is the impact of the wind on the air-resistance P _{a }?**

Next, we have calculated P_{a} at different wind speeds of 2, 8 en 16 m/s (58 km/h, hard wind with force 7 Beaufort) and both for a head wind and a tail wind. The results are given in the figure below.

We can see that at a head wind P_{a }increases from 14 Watts to no less than 131 Watts, which is almost half of the total power of 277 Watts of Hans. At a tail wind P_{a} decreases from 14 Watts to finally -86 Watts (which means he literally gets a push in the back from the wind).

**What is the impact on the 5k-race time?**

We have calculated this from the power balance: P_{t }= P_{a }+P_{r}. Earlier we saw that the P_{t} of Hans is 277 Watts. Consequently, his running power P_{r} decreases as P_{a }increases (head wind) and vice versa P_{r }increases as P_{a} decreases (tail wind). The resulting P_{r} is given in the figure below.

We notice that the available power P_{r} decreases significantly at a head wind (from 263 Watts to 146 Watts) and increases at a tail wind (from 263 Watts to 363 Watts).

Of course this has a big impact on the running speed, which can be determined with the formulae v (in m/s) = P_{r}/58/0.98.

Finally, we can calculate the 5K-race time t (in seconds) from the running speed with the formulae t = 5000/v.

The final result of the 5k-race time is given in the figure below.

We see that the 5K-race time increases from 18 minutes to more than 32 minutes at a head wind of 16 m/s! With a tail wind of 16 m/s, the race time drops from 18 minutes to just over 13 minutes! If only we could run races with a tail wind of 16 m/s, wow, what a PR could we get…..

**And how about the impact of cross wind?**

In real life we never experience a head wind or a tail wind during the whole race. So it is relevant to look at cross winds too. The formulas become more complicated in this case as we have to calculate with vectors of the wind direction and the running direction (the so-called apparent wind). The result is a 6^{th} order polynomial relationship. Our running friend and reader Arno Baels has prepared a spreadsheet to calculate this. In the table below, we show the results for a wind speed of 16 m/s and various wind directions (angles).

The table clearly shows that the impact is huge at this wind speed. The air-resistance changes significantly at different wind angels and so does the 5K-race time!

Finally, we have calculated the impact on the 5k-race time of a hypothetical race course with 1 K head wind, 1 K cross wind (45 degrees), 1 K cross wind (90 degrees), 1 K cross wind (135 degrees) and 1 K tail wind, This would lead to a race time of 21:39, so more than 3 minutes slower than at no wind.

**We are lucky that Stryd can measure P _{a} these days!**

We feel that the above calculations provide ample arguments to use a Stryd v3. At strong winds, P_{a} will have a big impact on the available running power P_{r} and resulting running speed. Using a Stryd Hans can just maintain his total power P_{t} at 277 Watts. As a result his pace will automatically go up and down with the wind and he does not have to worry about over- or underpowering.

In an earlier paper, we have shown that the Stryd can measure P_{a} accurately in head winds. In all honesty we have to say that the results at tail winds were not yet perfect, but we understand that Stryd is working hard to solve this part of the algorithm as well.

**How big was the impact of the wind on your race time****?**

After a windy race, you can use **the calculator** on our website **www.thesecretofrunning.com** to get an impression of the time that you could have run in ideal conditions (without wind). It is only an impression as we do not know whether the conditions were variable or you have been able to draft in a pack or in the wake of trees and buildings. In the near future we plan to update our calculator to include the impact of wind directions, as discussed in this paper.

*If you would like to purchase The Secret of Running (or the German version, Das Geheimnis des Laufens, or the Italian version, Manuale completo della corsa) you can do so in our webshop.*