Wind Blows

Ride (Don’t) Like The Wind

When riding your bicycle on a windy day, why does it seem like the headwind is always in your face and never at your back? Math, that’s why.

To start out simple: imagine riding down a perfectly straight road, wind perfectly in your face going one way, and directly at your back the other.

Let’s establish some parameters:

• V_b– “bicycle velocity” … this is the velocity (speed) we ride at assuming no wind.
• V_w– “effective wind velocity” … this is the wind speed (more on this in a moment)
• D – Distance. The length of the road.

A ride one way down the road with no wind takes this long:

Time equals total distance divided by speed. You already know this. If the road is 5 miles long and you ride at 10 miles per hour, it will take you a half hour: 5 / 10. And if we ride down and back, the total amount of time is twice that:

Now the wind.

Assume we pedal the same no matter what; the same gear, same effort, etc, but the wind adds or subtracts from our net speed. Call that number V_w – the effective wind velocity. This isn’t the wind speed, this is the speed effect the wind has on us (how much it slows us down). Maybe it’s as small as 1mph to 2mph in a heavy wind. It doesn’t matter what that actual number is, whatever it is we just plug it into our equation. We’re further going to assume that the effect is the same when the wind is blowing in your face as it is when it is blowing on your back. Anyone who has ever ridden a bicycle knows this probably isn’t correct (it hurts you more in your face than it helps you at your back), but again we’re trying to make things simple to get started.

With those ideas in place, what do the bike ride equations look like?

If we start out with the wind in our face, then our new velocity will be V_b – V_w. That is, the velocity we would have gone without the wind (V_b), minus the effect of the wind (V_w). If the wind is strong enough the net velocity could be zero, or even negative – the wind would be blowing us backwards in that case. Obviously, this doesn’t happen in real life because V_b is usually much larger than V_w.

The amount of time it takes to get down the road with the wind in our face is now:

In the same way, the amount of time with the wind at our back would be:

The total time is the sum:

To simplify this, we need to create a common denominator, multiply everything out, and combine terms. It goes like this:

And if we multiply everything out and combine it using whatever high school algebra we still remember, we get:

Three things to note about equation 7:

1. If V_wis zero, equation 7 turns back into equation 2. This is good, because if it didn’t then something would be wrong. When there’s no wind, the “wind” equation [7] should give us the same answer as our no-wind equation [2].
2. Notice that no matter what, V_w always makes the denominator smaller, which makes the total value for T larger. This, in a nutshell, is why the wind seems unfair. Even though sometimes it helps you and sometimes it hurts you, overall the net effect is that it hurts you. You get the shortest ride time when V_wis zero.
3. The effect gets worse as the square of V_w. So a 2mph wind effect (the effect on you, not the wind speed itself) is 4 times worse than a 1mph wind effect. A 3mph wind effect is 9 times worse than 1mph. It gets brutal very quickly.

Ok, now wait a second. If the wind is blowing at our back half the time and in our face half the time, how can it not balance out? The answer, if you look back at equations [3] and [4], is that the wind is NOT blowing at our back half the time. We go slower when the wind is in our face, so that part of the ride takes longer. Conversely, we go faster when the wind is at our back, so that part of the ride takes less time. Even though we’re going the same distance each way, we’re spending more time with the wind in our face than at our back. And that’s why it seems to be unfair – because it is!

To go further with this model, we’d have to figure out what the real coefficients of effect are for a head wind vs. a tail wind, and we’d have different values of wind effect for each case. Plus we’d have to do some calculus to integrate a continuous function of varying wind effect values because we wouldn’t always be directly heading into the wind; sometimes we’d be at an angle and have to factor that in. None of that, however, would change the basic nature of the analysis. The wind slows you down, and you do in fact spend more time going into a headwind than you do with the wind at your back, even if you rode in a circle in a steady wind.

Wind blows. And now you know why.