# 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:

• Vb– “bicycle velocity” … this is the velocity (speed) we ride at assuming no wind.
• Vw– “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 Vw – 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 VbVw. That is, the velocity we would have gone without the wind (Vb), minus the effect of the wind (Vw). 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 Vb is usually much larger than Vw.

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 Vwis 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, Vw 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 Vwis zero.
3. The effect gets worse as the square of Vw. 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.

## 4 Replies to “Wind Blows”

1. Arthur says:

I seriously thought it was caused by the difference between high and low pressure systems, but affected also by the rotation of the earth*. J/k I liked the example and easy to understand. Honestly thank you. Signed the college student at 48

*Wind cannot blow straight to or from high to low pressure systems; they are cyclic as you see in a hurricane. The true effect of wind can best be seen when a plane takes off, It cannot travel in a straight line once V2 is achieved. Especially evident in slower prop driven planes, it will look as if the aircraft is sliding sideways, as the Bernoulli’s principle is applied. (depending on which side of aircraft is toward the higher pressure system)

Just my thinking, I could be WAY off base.

2. Neil says:

Even ignoring the Coriolis thing (which I know you are mostly kidding about, as it isn’t significant at this scale), there are other reasons the wind might actually be unfair. If you’re riding around in a city it’s entirely possible that the wind really IS in your face more than it “should” be, depending on your route and how the wind channels down the various streets. I suppose if your ride is long enough the winds might even shift during the ride, again potentially conspiring against you (though equally likely to conspire for you).

On my particular ride route (around town lake) I’m pretty sure I’m getting the full exposure to wind from all directions on various parts of the ride (and some parts include doubling back over the same ground in opposite directions). As the equations show, even without swirling making it unfair, the net effect is still negative.

3. Jason Iorio says:

What about adding in traffic. Since you are riding in the same direction as the traffic you should get air flow in the same direction as well (wind in your back). Or is this example based on an open road with no traffic ever.

If you travel on SW PRKWY going west and wind is blowing east to west you have wind at your back. But with no wind change on the way back you’ve got wind in your face. It just fells like you have it against you both ways because you travel faster than the wind coming from behind you. (a little difficult to describe what i’m trying to say on this). If you have 40mph winds at your back you wont fell like you’re going against it anymore. I used to do this a lot in California as a kid in the Santa Anna winds.

4. Neil says:

Jason – it’s right that as far as perception goes, there doesn’t seem to be much wind when you ride with it, and of course there seems to be more of it when you ride against it. The example is admittedly oversimplified – I’m assuming that wind-resistance effects are the same front-to-back and back-to-front. If that assumption is true, the wind at your back helps you just as much as the wind in your face slows you down. Or, said another way – if you are chugging along at 30mph with a 30mph wind at your back, you won’t feel any wind in your face. Which (if you know how hard it is to struggle to pedal against air resistance in general) means you won’t have to pedal as hard to maintain 30mph (you still have to overcome the rolling resistance of your tires and friction in the bearings; how significant THAT is depends on your bicycle). I really don’t know the relative magnitudes of the wind resistance effects vs friction effects; obviously *something* prevents you from riding infinitely fast and ultimately limits you to some non-infinite top speed – that something is the combination of those two resistance factors (and of course hills).

All that said, I’d still rather just ride when there’s no wind! 🙂