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Distance per volume (mpg or km/L) is kind of a lousy way to reason about fuel economy; a better system is to take the reciprocal: gallons per mile, or liters per km.
(Here's an accessible argument for why: https://www.popsci.com/technology/gallons-per-100-miles-best-fuel-economy-metric/)
But volume / distance = length³ / length = length² = area. That means we can measure fuel economy using area.
For example, my car gets about 0.08mm². That’s not a rate. It’s just an area (in the rough vicinity of the period at the end of this sentence).
Huh?
1/2
There are a lot of very weird units, when you stare too hard at them. For example, the Hubble Constant is (70 km/s)/Mpc -- that is, a speed per Megaparsec, which checks out. but the distances technically cancel out, so the result is a *frequency.*
@thedansimonson@lingo.lolI'd make the case that it's not really the same "distance" being cancelled out. even though they're both the same kind of unit, one is a distance related to the galaxy's motion and the other is a distance *to* the galaxy.
I think the same reasoning would apply for the lengths composing fuel volume and the distance of a car traveled.
@thedansimonson
You’re the second reply to mention that video!
I hear your argument about lengths coming from different places, but…dimensional analysis doesn’t care. And per my 2nd post (and the video), there is in fact a perfectly reasonable physical interpretation that units the lengths.
@inthehands I agree about the sweeping -- I considered that too. it's a neat way to think about it, but you're still reducing one of the lengths out of existence, despite both lengths maintaining some physical reality.
ultimately, I think this is a place where dimensional analysis bombs. the math-language interface is often most distorted by the ambiguity of language.
@inthehands @thedansimonson Uhhh Chemist here to say dimensional analysis *does* care (sort of), which is why I recommend my students label not only the unit but also what it's measuring (usually that's which substance we're talking about). That's not to say your efficiency-as-an-area result is meaningless, just that I don't think every last thing that "cancels" units actually cancels out conceptually