Let’s say you’re running at 6:00/mi, heading straight north on a day with a 10 mph wind from the east. You’ll certainly notice this stiff crosswind. But will it slow you down?
Earlier this summer, I made a headwind and tailwind calculator for runners that uses the latest scientific work on the metabolic cost of overcoming headwinds and tailwinds. It calculates a “wind-adjusted pace,” and unlike any other calculator on the internet, it also allows you to specify the direction of the wind you’re encountering, so you can calculate the effects of not just a headwind or a tailwind, but lateral winds coming at you from any angle.
However, the first version of this calculator made a subtle mistake! After publishing the calculator, I got an email from reader Dave Clark, who noticed that pure lateral crosswinds—such as the east wind in the intro example—seemed to produce no effect on your pace. I assured him that this was expected: my calculator assumed that the lateral component of the drag force imposes a negligible metabolic cost.
Dave responded with some back-of-the-envelope math that made me realize I’d made an error in how I implemented the drag equation! The calculations were still correct for pure headwinds and pure tailwinds, but anything involving a crosswind was slightly off. But it’s fixed now!
The reason I made this mistake has to do with some pretty subtle mathematics. I couldn’t find any great resources on the internet that explained this phenomenon in an intuitive way, so here it is: a quick explanation of why a pure lateral wind slows you down, even when you ignore the lateral component of the drag force.
Even though this topic is pretty arcane and technical, I wanted to write it up because I personally found it extremely non-intuitive at first, even with a good bit of experience doing physics myself.
Calculating relative airflow for drag calculations in running
The drag equation—covered in detail in another article on drag forces, air resistance, and running—looks like this:
![\[F_d = \frac{1}{2} \rho v^2 C_d A_p\]](https://runningwritings.com/wp-content/ql-cache/quicklatex.com-51adc0db09044425d5db3806ac3fdb1f_l3.png "Rendered by QuickLaTeX.com")
The input we care about is 
—that’s the relative velocity of the airflow the runner experiences. My initial calculations just took the crosswind and broke it down into separate forward and lateral components: 
and 
, essentially. Then I calculated separate X and Y components of the drag force using the drag equation. But this is incorrect!
The drag force comes from the relative velocity of the overall airflow, and because the drag equation has a squared term, you can’t break down a headwind into x and y components, calculate drag forces separately, then re-combine them.
Here’s the example from above, worked out the correct way. We have the 10 mph lateral crosswind from the east, plus the 10 mph effective headwind that the runner creates by running north at 6:00/mi. That creates a relative airflow velocity of 14.1 mph, oriented at 45 degrees relative to the runner.
Then, we have to calculate drag forces for a relative airflow velocity () of 14.1 mph.
For simplicity, let's lump all the constants in the drag equation into a constant, 
. That will include 
, 
, 
, 
, plus whatever conversion factor we need to go from mph to m/s. Since it's the same runner, none of those change. With this shorthand, applying the drag equation to our relative air velocity above, we get 
.
Now we can break it down into a lateral and forward (X and Y) component, multiplying by 
which is 0.71. That gives a result of 
), which is decidedly bigger than the 
that you would’ve gotten by only doing the drag force calculations with the y component of the headwind (which is 10 mph, so 
).
As a reminder, we're ignoring the lateral component of the drag force because there's no good biomechanics data on the metabolic cost of overcoming lateral forces, and because that metabolic cost is probably pretty small compared with the metabolic cost of overcoming an opposing forward force.
In comparison, here’s the incorrect calculation I did in the initial version of the calculator:
Why biomechanics intuition failed me here: drag forces are not linear operators
From my background in biomechanics, I am so used to breaking vectors down into X and Y components, applying equations to them, and combining them back together again that I didn’t even realize this might not be valid for drag forces.
Breaking down vectors into their X and Y components works fine if you’re calculating accelerations from forces, or velocities from position vectors, but it fails completely with drag forces.
The reason? As noted above, the drag equation has a squared term in it. That means it is not a linear operator. A linear operator just means that you have a function 
where 
, and 
. Common operations in physics—like Newton's Second Law, Hooke’s law, the derivative, and the integral—are all linear operators.
With some pencil and paper math, it’s easy to see that 
does not satisfy these two properties—and since the drag equation is just a fancier version of this quadratic equation, my usual biomechanics intuitions failed.
Recap: the wind calculator is fixed!
The bottom line here is that, because of the relative velocity effects, even a pure lateral crosswind will slow you down when you run on a windy day.
Thanks to Dave Clark’s sharp eyes, my wind calculator now correctly deals with crosswinds at arbitrary angles. Check out the new and improved wind calculator here if you haven’t already!
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!
