# How fast was I going?

About halfway between Charlotte and San Francisco, I found myself staring out the window. Because airplanes don’t exist to amuse me but rather to get their passengers from one place to another as cheaply as possible, there is no in-flight video entertainment system. And if there were an in-flight video entertainment system, it wouldn’t include the channel that tells you where the plane is and how fast it’s going.

Fortunately, if you’ve flown over (or driven through?) this part of the world, you realize that the ground is essentially a giant checkerboard. See for example this image of Kansas crops from Wikipedia. So if you know how big the checkerboard squares are, and you have a stopwatch, you can figure out how fast you’re going. Just hold your head steady and watch how many of the little squares on the ground pass by in a given amount of time. (This is hard if there’s turbulence.)

In my case I observed that we crossed ten such squares heading roughly parallel to the direction of the plane, and one such square heading roughly perpendicular to the direction of the plane, in 34 seconds. I know — from basic geography — that the plane is traveling roughly west. I cover $\sqrt{10^2 + 1^2} \approx 10.05$ squares every 34 seconds, or $\sqrt{10^2+1^2}(3600/34) \approx 1060$ squares per hour. (In my head I actually just did $10 \times (3600/34)$, the extra 1 being basically superfluous at this level of precision.

But how big are the squares? This is the one piece of knowledge that I couldn’t get from the air. They’re half-mile squares. I had actually thought they were one-mile squares, remnants of the Public Land Survey System — and indeed somewhat west of where I noticed this the squares did turn into one-mile squares before they disappeared completely — but 1060 miles per hour was clearly too fast. The squares had to be some simple fraction of a mile, though, so we were traveling at about 530 miles per hour. Furthermore, for every ten squares moved west we moved one square north; so our heading was about one-tenth of a radian, or six degrees, north of west.

I didn’t note the time exactly, but it was perhaps 5:40 Pacific daylight time when I made this observation, and I’m guessing we were somewhere over southern Kansas. If you look at the flight plan for this flight and plot the appropriate piece of it you can see we would have been flying just north of west at that time; I don’t know how to get the speed from publicly available data.

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## 7 thoughts on “How fast was I going?”

1. skytoastar says:

On a recent flight from Salt Lake City to Charlotte I was bored and turned the GPS on my phone on to see where I was exactly. I can’t remember exactly how fast I was travelling I do remember it being over 510 mph and I’m pretty sure it was in the 520s, which would correlate with what you calculated.

2. suevanhattum says:

>for every ten squares moved west we moved one square north; so our heading was about one-tenth of a radian, or six degrees, north of west.

One west and one north would be a ratio of 1, and that’s 45 degrees, not 1 radian. (Am I misunderstanding, or is there a mistake here?) Were you doing something that only works for small angles? Ahh, I see on wolfram alpha that you’re right, and that it works well up to a ratio of about .3. Nice.

3. Sue: basically it’s a small-angle approximation. The actual angle is the arctangent of 1/10, but arctan x is approximately x for small x. It turns out that aviators have a different name for this approximation (and actually use a slightly different approximation that’s more convenient for mental arithmetic), namely the 1 in 60 rule.

4. skyofstar: I wouldn’t have thought to do that. I have an iPhone and I had been under the impression that the location data was triangulating from cell towers, but Apple says it’s a combination of that and GPS. Presumably a phone in airplane mode uses GPS only. It’s a shame I didn’t think of that, because then I could have showed a picture of what I was actually looking at.

5. cidiel says:

you can get your flight speed in knots on flight aware fyi, its at the bottom of the page with your flight map.