# Representing fuel efficiency

In Europe, fuel efficiency is usually expressed as **fuel
consumption**, which is the number of liters you use for every
100 kilometers you travel (in French, "litres au 100").

The funny thing is that, from the point of view of physics, "litres
au 100" are homogeneous
to a surface (because liters are a unit for volume and kilometers a unit
for distance). For example, if you take a typical consumption of 10
liters per 100 km, this means a consumption of 10^{-2} cubic meters for
10^{6} meters, ie 10^{-8} square meters (or 0.01 square
millimeters, ie. a square 0.1 mm by 0.1 mm).

The most incredible thing, however, is that this way of imagining things lends itself to a very clear visual interpretation which makes it possible to see the quantity of fuel used when travelling a given distance. Imagine that the surface representing the consumption is some kind of antenna over the car. When the car is running, the quantity of petrol used is the volume traversed by the surface.

In the US, however, you don't use fuel consumption that much; you
talk about **fuel economy**, expressed as miles per gallon.
Not only do Americans use of imperial units rather than SI units, but
they're also thinking the other way round: whereas the French indicate
the number of liters to use for a given distance, Americans indicate how
far you can go with a given quantity of petrol. This means that the unit
of fuel economy is the inverse of a surface. Can we get a visual
interpretation out of that too?

The answer is yes, though the interpretation isn't quite as
straightforward as in the previous case. Assuming that we have a given
volume of fuel and that we know the value of the fuel economy, we would
like to see the distance which we can travel. The idea is to see the
petrol tank as a cylinder with a base surface *S* and a height
*h*, and imagine your fuel efficiency of *x* (in
m^{-2}) as indicating that you get *x* "efficiency
points" per square meter. (This is quite logical: a frequency of 42 Hz,
ie. 42 s^{-1}, means that you encounter 42 "beats" per
second.)

Now, imagine a plane which sweeps through the tank while staying
parallel to the base. The intersection between the tank and the plane
contains a constant number of efficiency points (because it is always of
surface *S*), and you can see them as travelling along with the
plane. It happens that the total distance travelled by all of them (ie.
*h* times the number of efficiency points in surface *S*)
is the distance which you can travel.

Of course, if the number of points is fractional, just count the
fractional part as appearing only for that fraction of the sweeping, ie.
if you have 2.5 points, then count 2.5 times *h*. Another remark:
in fact, you could arrange your fuel in any shape you want, and
integrate the number of efficiency points over the (changing) size of
the intersection of the sweeping plane with the tank. The assumption
that it is arranged in a cylindrical volume is only there for
simplicity's sake.

It feels a bit annoying that this second visualization seems more complicated than the first one although it seems to be more or less the same idea. I don't know if this could count as a proof that European way to do things is better than the American one? :-P

*Thanks to Claude Perdigou for help in finding the second
visualization.*

Some more explanations on xkcd What If? and Physics.SE.