# Representing fuel efficiency

— updated

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 106 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.