Yesterday I saw someone joking online about how if you apply dimensional analysis to fuel efficiency, you end up with an area. Why? Because fuel efficiency is expressed (in Canada and Europe, anyway) as liters per 100 kilometers. The liter is a unit of volume, or length^{3}. The kilometer is a unit of length. If you divide length^{3} by length, you end up with length^{2}, or area. (Similar reasoning applies to American-style fuel efficiency expressed as miles per gallon.)

I tried this with a concrete example. Suppose your car consumes 10 liters per 100 kilometers:

10 L * (1000 cm^{3}/L) = 10,000 cm^{3}

100 km * (1000 m/km) * (100 cm/m) = 10,000,000 cm

10 L per 100 km is therefore equivalent to 10,000 cm^{3}/10,000,000 cm, or 0.001 cm^{2}

Weirdly, this means that the fuel efficiency of your car is 0.001 cm^{2}, or 0.1 mm^{2}.

At first this struck me as bizarre at best, and an abuse of dimensional analysis at worst, but the more I thought about it, the more sense it made. It’s actually quite intuitive if you look at it a certain way.

Imagine a long tube of fuel stretched out along the surface of the road. Now suppose that your car doesn’t have a fuel tank, but is instead equipped with some kind of scoop that gathers fuel from this tube as your car travels down the road. How large should this tube be? In order for your car to continue traveling down the road, the tube must have a minimum cross-sectional area of — you guessed it — 0.1 mm^{2}. Cool, huh?

To confirm this, simply calculate the quantity of fuel contained in the tube, assuming the tube is 100 km long. The volume of the tube is equal to the cross-sectional area times the length. We already know the value of both of these quantities expressed in terms of centimeters. The volume is thus equal to 0.001 cm^{2} * 10,000,000 cm, or 10,000 cm^{3}, which is equal to 10 L. QED.

Well, it definitely surprises me. I would have expected that metric to have units of reciprocal area.

Indeed. Let’s move on to something less trivial. Carbon footprints? Personal modes of travel and personal responsibility?

DNA_Jock,Our Ford has the European display of litres per 100km but the Golf has UK statute miles per Imp gallon. Take your foot off and the Ford says zero. Take your foot off with the Golf and it goes from 200mpg to three dashes.

Alan Fox,Yes, as it happens, I drive a Golf R. The wife’s plug-in hybrid reports out a deeply misleading mpg. In keiths defense, fuel per unit distance traveled is a more well-behaved metric.

And look at this, Erik. A graph.

Sure. I know I would need a brimful tank of fuel to drive from Perpignan to Calais, 1100 km and 55 litres.

Lizzie has a Prius she speaks highly of. But people tell me all-electric is the way to go. Just not yet.

Prius last about 200,000 miles and have close to zero drawbacks. Other than size and comfort.

This isn’t about fuel efficiency, but like the topic of the OP, it does fall into the category of “surprising things that actually make sense when you look at them in a certain way”:

Why do colliding blocks compute π?Someone who is an Uber driver commented elsewhere that he was on his second. He said the reduction in maintenance and servicing was dramatic, mainly due to recuperative braking.

I’ve never met an unhappy Prius owner.

I bought a Subaru Forester, because at my age, ease of getting in at out takes precedence over everything else. Fun fact: Subaru is the only company I know of that recommends models based on the size of your dog.

I have a cat, but I can get a washing machine in and close the tailgate.

Looks quite practical.

Handy to know. I dunno, I feel I should wait or go whole hog on electric.

Alan:

Of course. You started with fuel efficiency in volume per distance and divided to get area, so if you “undivide” you get back to volume per distance.

Don’t be thrown off by the units. 0.05 mm^3 per mm is the same as 5 liters per 100 km, which is what you started with.

Relating this back to my “tube of fuel” scenario, the 0.05 mm^3 per mm figure means that a section of the tube with a length of 1 mm contains exactly 0.05 mm^3 of fuel.

Which makes sense. There is just enough fuel in the tube to feed your car’s advance, which means that the amount of fuel F you scoop out of the tube over a distance D is

exactlythe amount required to travel that distance, and nothing more. F/D is just the fuel efficiency, in other words.Your car requires 5 liters to travel 100 km. How much fuel in a 100 km section of tube? 5 liters. Your car requires 2.5 liters to travel 50 km. How much fuel in a 50 km section of tube? 2.5 liters. If your car requires an amount of fuel F to travel a distance D, then the amount of fuel in a tube section of length D is F. For any value of D. The rest is just unit conversion.

0.05 mm^3 per mm is the same as

5 liters per 100 km, which is about the same as

0.0213 US gallons per mile, which is about the same as

34 US ounces per furlong,

and so on.

All of which can be neatly expressed as a fuel efficiency of 0.05 square millimeters.

Anyone want to place bets on whether Alan and Jock are furiously punching numbers into their calculators, hoping to find a mistake in my ounces-per-furlong math? 😆

LOL

They’d lose.

I’m still enjoying your claim that mpg can be expressed as an area. 😀

Jock:

As if miles per gallon and gallons per mile aren’t two ways of expressing the same fuel efficiency. You just proved my point about how desperate you guys are to catch me in a mistake.

What’s even funnier is that the only reason I used “miles per gallon” there was to bypass

anotherfailed semantic quibble, this one from Alan. I wrote:See if you guys can find some

actualproblems with my reasoning, calculations, or positions, rather than grasping at semantic quibbles that aren’t valid anyway.Let’s have a substantive discussion.

OK, time for the reveal. Even after I gave him a straight-on hint about the ounces per furlong number, Jock missed it. 34 ounces per furlong is not the right answer. It’s off by a factor of 100. The correct answer is 0.34 ounces per furlong.

I offered you a freebie, Jock, and you didn’t take it.

I suspect, like me, he didn’t invest any time in reading your comments. It’s diminishing returns.

I won’t quote my earlier comment.

Alan:

The idea that you’re skipping over my comments is ludicrous, as anyone can see simply by reading the thread. And by noting that just now, you once again responded to a comment of mine.

That’s a beautiful own goal. In the comment immediately after yours, I explain why your scenario fails. You know that perfectly well

since you immediately responded to me, and not just then, but as we continued the exchange. So not only have you drawn attention back to your faulty model, but you’ve pointed readers directly at an exchange that contradicts what you just claimed about ignoring my comments!Dude, why make a false claim and then immediately point to the evidence that proves your claim is false?

I’ll let you ponder that. I’m off to bed.

Awesome.

Poor keiths, I spotted your intentional mistake immediately. As you note, you were off by two orders on magnitude. I saw 34 ounces per furlong, and ballparked that to 32 ounces per furlong, that’s 2 US pints per furlong, 8 furlongs in a mile, so… 2 gallons per mile. That’s obviously wrong.

Keiths is trying to make one or two points, I realized. He’s trying to make a point that we jump on every single error of his, and/or he’s trying to make a point that he, the great keiths, is willing to admit an error when he makes one.

Rather than get in a pissing contest about the limited virtue of admitting to an

intentionalerror, I forwent the pleasure.At no point was a calculator needed, so the bet was lost.

Jock:

Haha. No, you didn’t. If you had, you would have immediately said something like “Your little trap is so obvious I don’t even NEED a calculator to spot it. You’re off by two orders of magnitude, ffs.”

Look, while it’s amusing that you missed it, it’s no big deal. It makes no difference to the thread. I just threw it in there because you’re fond of announcing that you’ve laid traps for me, and I thought it would be fun to tweak you with a trap of my own. As an example, you wrote:

I just thought it would be fun to lay a trap for Mr. I-Laid-a-Trap-for-You. And it was.

Naah, I really did spot your error, but I decided against giving you the chance to magnanimously “admit” to making the error.

keiths:

Oh, I’m not backing Wildberger — I’m not entirely sure that he’s sane. But he is a mathematician who agrees with me about the pure/applied boundary, which was what you were incessantly demanding:

I had you on a token reward system in a vain attempt to get you to explain how your Smoot error differed from Karen’s error.

Gotta have a long game.

Jock:

As if I would have scored points by admitting to an error I had obviously planted.

Come on, Jock. The reason you didn’t say anything was because you didn’t spot the error. I laid a trap, and you fell in. That’s funny, but it was just a lark. It doesn’t have any bearing on the topic of the thread. Don’t stress over it.

Re Wildberger, I’ve responded here.

For Erik:

If you plot a graph, with the axis as distance and the axis as the “instantaneous” fuel consumption, the area under line between start and stop represents total fuel consumed.