Calculating the effect of headwinds and tailwinds on running pacing and performance

Headwinds and tailwinds are fundamentally an issue of air resistance and drag forces—physically pushing aside the air in front of you when you run imposes an additional energetic cost on your body. Air resistance is an issue even in calm conditions: the faster you run, the more costly it becomes to overcome air resistance, since you’re essentially creating a headwind for yourself.

🍃 This article explains the science behind my headwind and tailwind calculator. Check it out if you haven't already! 🍃

It’s super easy to use, but under the hood, it does some pretty sophisticated calculations. This article covers the science behind these calculations, and acts as a primer for understanding air resistance in running: where it comes from, how to calculate it, and how it’s affected by headwinds and tailwinds.

This article goes deep into the physics and bioenergetics behind air resistance in running. If all you want to do is calculate how headwinds and tailwinds affect your pacing, the web app is all you need. Otherwise, if you want to learn everything there is to know about air resistance in running, then buckle up!

The energetic cost of running on a treadmill in still air

Before worrying about air resistance at all, let’s start with the basics: the energetic cost of running on a treadmill in a lab, where there’s no air resistance at all (well, aside from the minor air resistance against moving your arms and legs).

A study from 2018 provides some great data on energetic cost as a function of speed—Black et al. report data from 92 different runners, including 24 elite runners and 68 recreational runners, across speeds from 12:00/mi to 5:40/mi (7:30–3:30/km).

Plotting these data reveals two important facts about the energetics of running:^[1]

Elite runners have better running economy than recreational runners: they can run the same speed at a lower energetic cost, even when energetic cost is expressed relative to body weight.
Energetic cost increases as you run faster in a nonlinear fashion.

Remember, this is the energetic cost of running on a treadmill in still air—not overground, where even on a calm day, you’ll encounter air resistance.

Units of energy expenditure in running

A quick aside regarding units: the plot above, and most of the calculations below, express the energetic cost of running in watts per kg of body weight (W/kg). It’s just a simple unit conversion (and multiplication by body weight) to get energy expenditure in more familiar units like food calories per hour, but it’s easiest to work in W/kg.

If these metabolic power outputs look higher than the W/kg values you’re familiar with in cycling, that’s because this is metabolic power, not mechanical power. During cycling, “gross efficiency” ranges from 15–25%, so 20 W/kg of metabolic power might only register as 3–5 W/kg on a power meter.

Air resistance and drag forces when running outdoors in calm air

Now that we’ve got a reliable way to predict the energetic cost of running without the effects of air resistance, let’s take a look at how to factor in the drag forces you encounter when running outside in the real world, on a calm day.

It should be clear that if you run at 10 miles per hour (6:00/mi or 3:43/km) on a calm day, you’ll encounter a headwind of exactly the same magnitude (10 mph or 16 kmh), since your body is moving relative to air.

How great of a drag force does this headwind apply? Physicists and engineers have worked out the math of drag forces with remarkable precision. The relationship between your (relative) velocity and the drag forces you encounter is governed by the drag equation:

$F_d = \frac{1}{2} \rho v^2 C_d A_p$

Where:

$F_d$ is the drag force you encounter, in Newtons

$\rho$ is the density of the air, in kg/m³

$v$ is the relative velocity of the air (i.e. relative to you), in meters per second

$C_d$ is the drag coefficient, a dimensionless quantity that captures how streamlined your body is

$A_p$ is the projected frontal area, in m²

Let’s look at each of these terms in the equation.

Density of air

The more dense the air, the greater the drag forces you’ll encounter. At sea level, air density is about 1.225 kg/m³. At high altitude, air density can start to affect air resistance, but we’ll save that for another day.

Relative velocity of air

In the simple case of running outside on a calm day, the velocity of the air, relative to you, is just your forward running velocity.

When we start taking the effects of headwinds and tailwinds into account, we’ll need to do some addition or subtraction: running 5 m/s (~5:20/mi) into a 2.0 m/s headwind means the relative velocity of air is going to be 7 m/s.

Something important to notice about the relative velocity of the air is that this term is squared in the drag equation. That means that when you double your speed, the air resistance you experience will quadruple.

It is this quadratic scaling that is responsible for the outsized effects of air resistance in sprinting events, cycling races, and very strong headwinds.

Drag coefficient

$C_d$ , the drag coefficient, quantifies how streamlined an object is, relative to its projected area ( $A_p$ , see below).

A Formula One car and a Volkswagen Type 2 (the infamous “hippie van”) might occupy pretty much the same cross-sectional area when viewed head-on, but the Formula One car is much more streamlined—air flows over it more easily, creating less drag. So, we'd expect the Formula One car to have a lower $C_d$ .

The drag coefficient is usually measured experimentally in a wind tunnel or calculated numerically with computational fluid dynamics simulations.

I find it most intuitive to just look at a few drag coefficients to get an understanding of what orders of magnitude we’re looking at, and how streamlining can make a difference.

Drag coefficients, from aerodynamic to not-very-aerodynamic objects

Object	Drag coefficient (C_d)	Source
Fighter jet	<0.1	(1)
Tesla Model S	0.24	(2)
Ford F-150 Lightning	0.44	(3)
Cyclist, racing position	0.65	(4)
Cyclist, on a cruiser bike	1.0	(5)
Stop sign	1.1	(6)

In comparison, here are a few different estimates for a runner’s drag coefficient from various studies:

Source	Drag coefficient (C_d) for a runner
Schickenhofer 2021	0.73
Marro 2023	0.90
Walpert 1989	0.79
Davies 1980	0.82–0.91
Pugh 1970	0.80

These studies are a mix of experimental studies and computational fluid dynamics simulations, and all cluster around 0.8 or so (1,2). This range seems about right: better than an upright cyclist, worse than a time trial cyclist. We’ll go with a drag coefficient of 0.8 when calculating drag forces.

Projected frontal area

$A_p$ is the two-dimensional area of your body that’s exposed to the drag forces. In some of the classic wind-tunnel studies on running, this was measured experimentally by taking a photograph of an athlete running on a treadmill next to a rectangle of a known size, then assessing the runner’s $A_p$ using a grid laid over the photograph.

Taller or heavier runners have a greater projected frontal area, because their body size is greater. We can use a two-step process to get your projected frontal area, given your body size: first, we can calculate your total body surface area, then take the fraction of it that faces forward.

Predicting total body surface area turns out to be a well-studied problem, since body surface area is used by doctors to determine dosage for some drugs, as well as evaluating kidney function. Livingston and Lee wrote an excellent paper that provides a simple equation for predicting body surface area using only body weight.^[2] The Livingston and Lee equation works equally well for healthy adults, small children, and people with obesity, and their paper has the data to prove it.

Livingston and Lee’s formula is as follows:

$BSA = 0.1173 W^{0.6466}$

where $W$ is body weight in kg, and $BSA$ is body surface area in m². This equation predicts body surface area with less than 1% average error.

Determining projected frontal area from body surface area

To convert body surface area to projected frontal area ( $A_p$ ), we’ll follow the same strategy used in most research on drag forces in running, which is to assume a constant $A_p/BSA$ ratio of 0.266. This figure comes from Pugh 1970, and admittedly only comes from nine male subjects, so it’s a relative weak point in our calculation. For now, though, we’ll go with the 0.266 ratio.^[3]

Air resistance from headwinds and tailwinds

So far, we’ve only considered running outdoors in calm air. Modifying our calculations to account for headwinds and tailwinds is very easy—all we have to do is change $v$ in the drag equation from running velocity to relative flow velocity of the air, relative to the runner.

That trick works like this:

$v_{relative} = v_{runner} - v_{wind}$

making sure to respect our sign conventions: positive velocity being forward, from the runner’s perspective, so running at 5 m/s (5:20/mi or 3:20/km) into a 2 m/s headwind would be equal to 5 - (-2) = 7 m/s relative flow velocity.^[4]

Converting drag forces to metabolic cost

With the drag equation well-understood, we can now calculate the drag forces a runner will experience at any given speed. Now, we arrive somewhere that requires real empirical data: how much metabolic energy does your body need to expend to overcome a given drag force?

Happily, there’s some new experimental data on this front from a study published in 2022.

Da Silva et al. used a clever system of weights, pulleys, and rubber tubing to apply known amounts of horizontal force to runners on a treadmill, while measuring the change in their metabolic cost.

How? Like this:

Now, this system of weights and bungees is not exactly the same thing as an aerodynamic drag force, but it’s still a horizontal force applied to the body’s center of mass, so the net effects should be very similar.

Da Silva’s approach is much simpler than trying to measure air resistance directly—to do that, you’d need a treadmill inside of a wind tunnel (though see below for more on that!).

Da Silva collected data on 12 different runners, which lets us get a more accurate estimate of the typical increase in metabolic cost caused by overcoming a horizontal force.

The experimental results showed that an increase of 1% BW (body weight) in horizontal impeding force corresponded to a 6.13% increase in the metabolic cost of running.

Calculating the effects of air resistance on running in calm air

We’re finally at a point where we can calculate, from scratch, the metabolic cost of running outdoors on a calm day. First, we’ll use the model fit to the Black et al. data from earlier to calculate the expected metabolic cost of running without any air resistance.

Then, we’ll use the drag equation to calculate the drag force, and finally use Da Silva’s equation to convert drag force into metabolic cost. To do so, all we need is running speed and body weight.

Here’s what the calculations look like for a typical runner:

You can see the expected nonlinear relationship between running speed and air resistance: the added effect of air resistance is pretty small at 10:00/mi, but rapidly becomes a problem at higher speeds.

Calculating the effects of headwinds and tailwinds on the metabolic cost of running

To calculate the extra metabolic cost that results from overcoming a headwind, or the reduction in metabolic cost that results from running with a tailwind, all we have to do is run the same calculations but using relative air velocity, across a range of headwinds and tailwinds.

Here's what that looks like:

Note that this plot and the plots below are showing the nominal wind speed, as measured by a weather station: “-10 mph” on the x-axis means running south on a day with a 10 mph wind from the north.^[5]

It’s also clear that a tailwind does not help you as much as the equivalent headwind. That point becomes more obvious if we translate the change in metabolic power to an effective increase or decrease in speed—in other words, what’s the calm-air equivalent speed of running into a given headwind, or running with a given tailwind?

Here's what the "equivalent speed" comparison looks like:

This plot underscores the facts that (1) you are more affected by headwinds and tailwinds at faster paces, and (2) a headwind hurts more than the equivalent tailwind helps.

By the way, that weird "upswing" for strong tailwinds at slow paces? That's when the tailwind becomes so strong that you're running in air that's moving faster than you are! If you expand out the axis limits, it's visible for faster paces too (at correspondingly stronger tailwinds).

Headwind and tailwind effects on pacing: some counterintuitive results because of speed-to-pace math

There’s one slightly unintuitive property of headwinds and tailwinds, and it becomes apparent when we translate the effects of headwinds and tailwinds into an equivalent change in running pace (i.e. in seconds per mile. Observe:

Now, it seems that slower paces are more affected by headwinds and tailwinds. What’s going on?

The apparent paradox is caused by the fact that pace is the inverse of speed. So, a 30 sec/mi slowdown is a way bigger deal at 5:00/mi than it is at 10:00/mi.

Even though the effect of headwinds on pacing is greater, when expressed as seconds per mile, for slower paces, the absolute increase in metabolic cost is still greater for faster paces (as we saw above).

Comparing the drag force model with real wind tunnel data on air resistance in running

Remember how I said Da Silva’s weight-and-pulley experimental model was a good replacement for a real wind tunnel experiment, which would be much more difficult?

Well, this “gold standard” test of air resistance in a wind tunnel has actually been done: In 1971, L.G.C.E. Pugh^[6] published the results of a study where he rigged up a treadmill inside a full-sized wind tunnel, then had an international-caliber runner run at 7:10/mi and 6:00/mi into headwinds of up to 40 miles per hour! Pugh measured the metabolic cost of running during this experiment, so we can use this dataset to validate our model's predictions.

While this study only has data from one runner, it’s nevertheless a great way to stress-test our drag force model by seeing how it performs on real data. Remember, the drag force model has not been tuned in any way for this particular runner. Indeed, it's never seen these data before. All it “knows” are the runner’s body weight, running speed, and the wind speed. Everything else comes from the drag equation and the empirical results of Da Silva et al.

Here's how our drag force model compares to real wind tunnel testing:

Pretty impressive! The drag force model agrees well with wind tunnel testing for headwinds of up to 15–20 mph.

Pugh actually comments in his 1971 paper that the "flattening out" of metabolic cost data at extremely high wind speeds (>20 mph) might be the result of the test subject leaning forward and using his body to generate lift against the extremely strong winds, thus reducing the metabolic cost of running slightly.

Regardless, I’m comfortable using this model to predict metabolic cost (and therefore, wind-equivalent paces) for tailwinds at least as strong as 10 mph and headwinds at least as strong as 15 mph, and possibly higher, given the ground level effects discussed below.

Nominal wind speed versus actual ground level wind experienced by a runner

You might’ve noticed that the wind-adjusted paces earlier looked pretty aggressive: can a 10 mph headwind really turn 6:00/mi pace into 6:30/mi pace? The wind tunnel testing data we just saw suggest that our metabolic cost calculations are correct, but there’s one reason to think we should scale back our wind adjustments: to account for discrepancies between nominal wind speed from a weather station and the actual ground-level wind you encounter while running.

Here’s why I think the estimates above seem like they're too aggressive, even though they agree with the wind tunnel data: when you run in a “10 mph wind” outdoors, your reference for that wind speed figure is the nominal wind speed coming from weather station measurements—not the actual ground-level wind speed you experience.

Wind measurements are taken (and forecast) at a standardized height of 10 meters (33 feet) above surface level, and wind strength dies off as you get closer to the ground. So, the wind speed that you experience at around chest height—roughly 1.5 meters above the ground—can be significantly less than the nominal wind speed at 10 meters off the ground.

We can improve our wind adjustments by estimating the actual ground-level wind speed, given the nominal wind speed from a weather station.

There are entire subdisciplines of engineering dedicated to estimating how wind speed varies as a function of height from the ground, but a simple model for this relationship is the wind profile power law, which gives a simple formula for calculating wind speed at a given height, if you have a reference speed and height measurement:

$v_z = v_{ref}\left(\frac{z}{z_{ref}}\right)^\alpha$

Where:

$v_z$ is the wind speed of interest

$v_{ref}$ is our reference wind measurement

$z$ is the height off the ground for $v_z$

$z_{ref}$ is our reference height

$\alpha$ is a parameter that varies depending on the terrain.

This exponent $\alpha$ controls how much of a difference there is in the wind speed at higher versus lower heights above ground.^[7]

Meteorology references (1,2,3,4) give the following typical $\alpha$ values for the type of terrain you might encounter while running:

α	Terrain
0.11	Beach on open water (lake, large river, ocean)
0.16	Rural road or open field
0.20	Prairie with tall grass and occasional trees, open park area with some trees and some buildings
0.30	Small towns, suburbs, moderately dense forests
0.40	Urban area with tall buildings

If you are running in the suburbs ( $\alpha = 0.30$ ), a 10 mph (4.47 m/s) nominal wind becomes:

$4.47*(1.5/10)^{0.3} = 2.53\space\textnormal{m/s} = 5.7\space\textnormal{mph}$

which knocks our estimated penalty for 6:00/mi pace down to roughly 16 sec/mi, which seems more reasonable.

Looking at the fall-off in wind speed for different settings (and therefore different values of alpha) can give some useful intuitions for why winds are worse on some routes compared with others:

Even for the same nominal wind, the actual wind you’ll experience at chest height is much worse in a rural area or on a flat, open space like a beach than in dense suburbs or a downtown area. Dense urban areas can reduce the actual wind speed you encounter by over 50%.

Caveats and limitations

The wind calculations we’ve developed here (and that power my wind calculator for runners) are, dare I say, more sophisticated and more accurate than anything else you’ll find online. That said, there are a few limitations to keep in mind.

The most important of these is the fact that hyper-local wind conditions will vary depending on terrain, obstacles, etc. Turbulent and gusty winds are also going to decrease the accuracy of the model.

Another limitation comes in the form of cross-winds: right now, my calculations only account for the forward component of the drag force (do note that's not the same as only accounting for the forward component of the headwind!). This assumes that there’s no metabolic cost to resisting a lateral force, which I find hard to believe. So far, nobody has studied the metabolic costs of overcoming a lateral force during running.

Still, I suspect the metabolic cost of overcoming lateral forces is pretty low, given that running on a track involves overcoming lateral forces of a pretty significant magnitude without much noticeable effect.

The calculations above do not take drafting into account, though a future version of the calculator will allow you to incorporate drafting. I’ll also have a separate article written up about the science behind drafting and how it benefits performance.

Finally, there is some person-to-person variability in the metabolic cost of overcoming drag forces. If I can get the individual-level data from Da Silva, I can incorporate that uncertainty into the calculations and give a range of wind adjusted paces, as opposed to just a point estimate. I’m currently working on that.

Recap

Headwinds slow you down because there is a metabolic cost associated with overcoming the drag force from the air resistance. These drag forces and the associated metabolic costs can be calculated using a combination of physics-based models and empirical data.

In terms of their effects on energy expenditure, headwinds and tailwinds have a greater metabolic cost at faster speeds.

However, because of the scaling between running speed and running pace, the apparent effect on your pacing is greater at slower paces (in seconds per mi or per km).

If you’re running at an equivalent effort, running into a headwind will slow you down by two or three times as much as the equivalent tailwind will speed you up. This is a natural consequence of the fact that air resistance scales with the square of the airflow velocity.

The actual ground level wind you experience depends on the local conditions where you’re running. In a flat, open field, you’ll be affected to a much greater degree than in a city with tall buildings, even for the same nominal wind speed.

In a dense urban area, the actual wind speed you encounter might be less than half the nominal wind speed measured by a weather station.

On wide open terrain like a beach or lakeside path, you’ll experience nearly the full force of the nominal wind speed.

If you just want to calculate the effects of a given headwind or tailwind on your pacing, without worrying about the math, check out my headwind and tailwind calculator for runners.

Learn more about the science of running

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I also have a book, Modern Training and Physiology for Middle and Long-Distance Runners, that focuses on the science of performance and training for events from 800m to the 10k. Check it out!

Footnotes

^[1] Throughout this article we’ll be looking at actual energetic cost, in watts per kilogram (W/kg). That technically is not the same thing as VO2 because of energetic differences from burning fats vs. carbohydrates, but if we’re talking about marathon pace or faster, VO2 and energy cost in W/kg are basically interchangeable.

^[2] Much of the running literature on drag equations uses the earlier DuBois equation, which has two drawbacks. First, it was developed on a grand total of nine people, and as a result, the Du Bois equation has significant errors in heavier patients and larger athletes. The Du Bois equation also requires both height and weight—Livingston and Lee’s equation is much more accurate despite requiring only body weight.

^[3] As noted above, it is possible to measure Ap directly using only a photograph. With modern image segmentation techniques, it would be pretty easy to get a large dataset to improve on this metric. It’d be a great school project for anyone interested in human performance or aerodynamics. Send me an email if you want to work on something and I’ll do what I can to help!

^[4] If you run with a strong enough wind at your back, the relative flow velocity of the air can actually become negative—running at six-minute mile pace with a 12 mph tailwind, for example, yields a relative flow velocity of -0.9 m/s. This situation is rare enough, and the magnitude of force is small enough, that I’m comfortable just assuming that Da Silva’s model works for negative forces as well, even though they only tested impeding forces. My rationale here is that an earlier paper from the same lab studies much stronger assistive and impeding forces with the same experimental setup, and the results from this older paper didn’t show any evidence of a discontinuity when comparing impeding forces (like a headwind) and assisting forces (like a tailwind).

^[5] In other words, the calculations under the hood already account for the fact that running at 10 mph with an 8 mph tailwind still means you’ll encounter an effective headwind of 2 mph. You do not need to do this math yourself.

^[6] The author of this study was Lewis Griffith Cresswell Evans Pugh—a scientist’s name if there ever was one—who was most famous for being the physiologist on staff for Edmund Hillary and Tenzing Norgay’s first summit of Mount Everest in 1953. The runner who completed the wind tunnel testing was Mike Turner, a top British runner of the late 1950s who captained England’s team to victory at the 1969 International Cross Country Championships (a precursor to today’s World Cross-Country Championships) and boasted a 10k PR of 28:33.

^[7] Using the wind profile power law at heights down to 1.5 meters is pushing the limits of the model a bit; many engineers would prefer to use the more sophisticated log wind profile model at these heights instead. However, the log wind profile model is more complicated, requiring multiple parameters to “tune” its performance to the local geography. Data collected at Chiang Mai International Airport suggest that power law models work reasonably well for heights all the way down to 2.5 m above ground, so it’s still a reasonable first-pass approximation if all you know is the general characteristics of a location (e.g. rural, suburban, urban).

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