The creator of the famous and popular xkcd webcomic has done a weird analysis on his “What If” blog. The post is based on a reader question about how long you would expect it to take if you laid outdoors until a bird pooped in your mouth. Yep, actual question. The analysis is mildly interesting, but even better is the analysis of one of the most unusual thoughts about units I’ve ever seen. Here is that part of the piece:
I have to say—from a dimensional analysis standpoint, ”poops” is one of the strangest units I’ve ever tried to cancel in an equation. But there’s another case of odd unit cancellation, common in everyday life, which is—in a way—even weirder: Gas mileage.
In the US, we measure fuel economy in miles/gallon—which could just as easily be written as gallons/mile. (This reciprocal form has some advantages. It’s popular in Europe, where it’s expressed as liters per 100 kilometers.)
But regardless of which units you use, there’s something strange going on here. Miles are units of length, and gallons are volume—which is length3. So gallons/mile is length3length. That’s just length2.
Gas mileage is measured in square meters.
You can even plug it into Wolfram|Alpha, and it’ll tell you that 20 MPG is about 0.1 square millimeters (roughly the area of two pixels on a computer screen).
Unit cancellation is weird.
Ok, so what’s the physical interpretation of that number? Is there one?
It turns out there is! If you took all the gas you burned on a trip and stretched it out into a thin tube along your route, 0.1 square millimeters would be the cross-sectional area of that tube.
Weird, but kinda cool!