Miles Per Gallon

   
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Miles per Gallon - 2015/05/03

I remember a while ago attending a talk that did something utterly bizarre with units of "miles per gallon." I don't remember much about it, but I thought I'd attempt to reconstruct the process in a post, just to see how far I get, what conclusion I reach, and whether people think it's as bonkers as I do.

So here we go ...

We start by talking about fuel consumption in a car. Let's suppose for the sake of concreteness that the car can get 30 miles per gallon. The following figures are rough, but close enough.

I'm playing fast and loose with the number of significant figures I'm using here, but this is all just a first approximation to see what's going on - everything should be recomputed later with more attention to the question of accuracy.
Since there are about 1.6 km in a mile, 30 miles per gallon is, of course, 50 km per gallon. There are about 4.5461 litres in a gallon, so that's about 10.56 km per litre, or 10560 m per litre.

Now a litre of fluid is 0.001 m3, or 106 cubic millimetres. Thus our 30 miles per gallon is 10560 m per (106 mm3), which is the same as 10.56x106 mm per (106 cubic mm), or 10.56 mm / mm3.

Cancelling units, we get a result of 10.56 mm-2.

Well, units of mm-2 are a bit difficult to visualise, so we can invert this, and instead of talking about miles per gallon, we can talk about gallons per mile, and we discover that our car's fuel consumption is roughly 0.1 mm2. (More accurately, 0.0942 mm2)

What?

What this means is that if you had a trough with a cross-sectional area of 0.1 mm2 and you filled it with fuel, and if your car could scoop up that fuel as it went, the fuel it collected would be enough to keep it going for as long as the trough was there supplying the fuel.

Just 0.1 mm2.

That can't be right, can it?

Yes, I think it is.

Consider the figure of 30 miles per gallon. That's roughly 10 km per litre. Now imagine spreading that 1 litre into a cuboid that's square in cross-section, and 10 km long. What's the area of the cross-section?

Yup, it's 0.1 mm2.

Intriguing.


Added later:

A few people have pointed out this link:

It's typical that XKCD should get there and, as always, do such an outstanding job of it. In my defence, I saw the talk about units may years ago, perhaps as many as 25 years, and certainly before I knew of XKCD.

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