## Running in the wind

It is widely accepted that running (or cycling) into a headwind requires more effort than without any wind at all. However, for given wind and running speeds, the energy saved from running in (or ‘with’) a tailwind is not as much as the energy required in order to run in (or ‘into’) a headwind.

I have used the majority of this post to explain and demonstrate this point, but if you are only interested in the best strategies for running in the wind, start reading from after the final (fourth) image. For any runner and race, the amount (volume) of air that they displace is given by:

volume = area × distance

Since distance is speed (velocity) × time, this gives:

volume = area × velocity × time

As a mass (volume × density, ρ), this is:

mass = density × area × velocity × time

The kinetic energy (½mv2) of this mass is then given by:

½mv2 = ½ × density × area × velocity3 × time

The rate of energy use (power) required to overcome this resistance is therefore:

½ × density × area × velocity3

In reality, the area depends on the aerodynamic quality of what is being worn (skin friction – e.g. clothing material), streamlining (drag – e.g. skin-tight compared with baggy) as well as actual physical size. A coefficient of drag (CD) is therefore used in conjunction with the area. This gives the power required as:

½ × drag coefficient × density ×area × velocity3 → ½ × CD × ρ × A × v3

[Since power = force × velocity, engineers may recognise the drag equation,
FD = ½ × CD × ρ × A × v2]

Supposing there is a wind blowing, the power required to overcome the moving air is proportional to (v ± w)3, depending on whether it is a tailwind (-) or a headwind (+). Assuming a race takes place in such a fashion that the same distance is run in all directions (e.g. a square), the following model can be used*: A good club runner will run 10km in about 40 minutes, at an average speed of 15km/hr, which is 6 minutes 26.16 seconds per mile (in a runner friendly format), or 4.1666… ms-1 (in engineering friendly, SI, units). A windspeed of 5mph (8km/hr, or 2.222… ms-1), would give rise to the following circumstances in a race: Here it is clear that an athlete does not gain as much from the tailwind as they would lose going into the headwind – in fact they still have to work to overcome this relatively slow tailwind. Increasing the wind speed so that runners benefit from a tailwind (so that wind speed is faster than running speed) exacerbates this discrepancy. Supposing the runner races in winds of 25mph (40km/hr or 11.111… ms-1), gives rise to the following circumstances: The benefit of the tailwind in this instance is clear to see, but it does not even come close to matching the hindrance caused from running into this headwind.

In terms of racing strategy, this means that you should run in somebody else’s slipstream and prevent others from running in your slipstream whenever possible. Furthermore, being small (like myself) is an advantage in a headwind and a disadvantage in a strong tailwind, but you knew that anyway.

Essentially avoid running into the wind where possible.

*for a shape with orthogonal geometries (and wind directions)**, the total amount of energy required to overcome the wind (air resistance) is:

½CDρA × { t1(v – w)3 + t2(v)3 + t3(v + w)3 }

Where t1, t2 and t3 are the respective amounts of time spent running with a tailwind, no wind (or perpendicular crosswind) and a headwind.

**for any shape, with any wind direction, the total amount of energy required to overcome the air resistance is:

½CDρA × { t1(v ± w sin(α))3 + t2(v ± w sin(θ))3 + t3(v ± w sin(γ))3 + … }

Where t1, t2 and t3 are the amounts of time spent running at the respective angles α, θ and γ.