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In the EU they measure fuel efficiency in liters per 100km, which dimensionally is a unit of area. In the US we use miles per gallon, which is the inverse - a

In the EU they measure fuel efficiency in liters per 100km, which dimensionally is a unit of area. In the US we use miles per gallon, which is the inverse - a ‘per area’.

thaumasiotes 3y ago
Dimensionally, "liters of gasoline per 100km" and "miles per gallon of gasoline" are already stated in simplest form. Your only hope of demonstrating that a car's fuel efficiency can be measured in "liters per 100km", as opposed to "liters of gasoline per 100km", is to show that the car will function just as efficiently no matter what you put in the fuel tank, that if you put in a liter of sand you can drive just as far as if you put in a liter of gas. Among other problems, if you tried to cancel the linear kilometers of the denominator with one dimension of the liters of gasoline in the numerator, you'd end up with incompatible units being paired together. Gasoline can't be measured in two-dimensional units, and there is no "linear gasoline" in the denominator that would let you transform the gasoline in the numerator anyway. You see this problem pop up all the time in chemistry, where a mole of some substance and a mole of some other substance are incompatible units. There is no such unit as "mole", only "mole of [whatever]". But there are people who would like to believe that "mole" is a unit. (In contrast, there is such a unit as "liter", but it isn't used in fuel efficiency ratings.)
- jameshart 3y ago
  Okay, let’s humor this new lGas unit and see if it works. If my engine consumes 10lGas/100km when traveling at 100km/h, that means it uses 10lGas/h, or .002778 lGas/s. If the pipe feeding my engine has a cross sectional area of 10^-6m^2, how fast is the gasoline flowing through that pipe? Naively I would expect I could divide flow rate by area to get mean velocity. But I’m expecting the result to be in m/s. But if I divide .002778 lGas/s by 10^-6m^2 I get 2778lGas/sm^2. Unless an lGas is 10^-3mGas^3 and I can measure gas area in mGas^2, so I can get my gas speed in mGas/s? If you want a mathematical way to capture ‘of gasoline’ the right way to think of it is not as a unit dimension but rather as a basis vector, kinda like ‘up’ or ‘across’. 1m * up is the vector quantity ‘1m up’, which has dimension ‘length’. 1m * across is the vector quantity ‘1m across’ which also has dimension length (we call it a displacement, but it’s a vector in the length dimension). We can add them together even though they point in different directions because they share the same dimension. The result is the vector quantity ‘1m up and 1m across’ and it is also a length. Similarly 1l * of gas is a vector quantity ‘1l of gas’ with dimension ‘volume’. I can add it to ‘1l of air’ to get ‘1l of gas plus 1l of air’ which is still a vector volume. It might describe the contents of my fuel tank when it’s half empty for example.

In the EU they measure fuel efficiency in liters per 100km, which dimensionally is a unit of area. In the US we use miles per gallon, which is the inverse - a ‘per area’.

thaumasiotes 3y ago

Dimensionally, "liters of gasoline per 100km" and "miles per gallon of gasoline" are already stated in simplest form. Your only hope of demonstrating that a car's fuel efficiency can be measured in "liters per 100km", as opposed to "liters of gasoline per 100km", is to show that the car will function just as efficiently no matter what you put in the fuel tank, that if you put in a liter of sand you can drive just as far as if you put in a liter of gas. Among other problems, if you tried to cancel the linear kilometers of the denominator with one dimension of the liters of gasoline in the numerator, you'd end up with incompatible units being paired together. Gasoline can't be measured in two-dimensional units, and there is no "linear gasoline" in the denominator that would let you transform the gasoline in the numerator anyway. You see this problem pop up all the time in chemistry, where a mole of some substance and a mole of some other substance are incompatible units. There is no such unit as "mole", only "mole of [whatever]". But there are people who would like to believe that "mole" is a unit. (In contrast, there is such a unit as "liter", but it isn't used in fuel efficiency ratings.)

jameshart 3y ago
Okay, let’s humor this new lGas unit and see if it works. If my engine consumes 10lGas/100km when traveling at 100km/h, that means it uses 10lGas/h, or .002778 lGas/s. If the pipe feeding my engine has a cross sectional area of 10^-6m^2, how fast is the gasoline flowing through that pipe? Naively I would expect I could divide flow rate by area to get mean velocity. But I’m expecting the result to be in m/s. But if I divide .002778 lGas/s by 10^-6m^2 I get 2778lGas/sm^2. Unless an lGas is 10^-3mGas^3 and I can measure gas area in mGas^2, so I can get my gas speed in mGas/s? If you want a mathematical way to capture ‘of gasoline’ the right way to think of it is not as a unit dimension but rather as a basis vector, kinda like ‘up’ or ‘across’. 1m * up is the vector quantity ‘1m up’, which has dimension ‘length’. 1m * across is the vector quantity ‘1m across’ which also has dimension length (we call it a displacement, but it’s a vector in the length dimension). We can add them together even though they point in different directions because they share the same dimension. The result is the vector quantity ‘1m up and 1m across’ and it is also a length. Similarly 1l * of gas is a vector quantity ‘1l of gas’ with dimension ‘volume’. I can add it to ‘1l of air’ to get ‘1l of gas plus 1l of air’ which is still a vector volume. It might describe the contents of my fuel tank when it’s half empty for example.