Editing 687: Dimensional Analysis

Cueball has a [[My Hobby|hobby]] — showing correct calculations according to the {{w|dimensional analysis}} — but with ridiculous correlations of uncorrelated events and measurements. Here Cueball is teaching a class and uses this trick to ''convince'' his students that the {{w|Toyota Prius}} combined {{w|United States Environmental Protection Agency|EPA}} gas mileage is somehow connected to the constant ''{{W|Pi|π}}'' via the {{w|Planck energy}}, the pressure at the {{w|Inner_core|earth's core}} and the width of the {{w|English Channel}}.  

Scientists — often physicists — use dimensional analysis to quickly check if a given formula can possibly relate to a physical system, because if you end up with an equation claiming that joules are meters, something is clearly wrong. Dimensional analysis here refers to the check if both sides of the equation arrive at the same physical unit when the units of all variables get plugged into the equation. This requires knowledge of the system of units and the relation between different physical units.

 

(Planck energy) / (Pressure at the core of the earth) * (Prius combined EPA gas mileage) / (minimum width of the English Channel) = π

*Pressure at the core of the earth: Given in Pascals [Pa]

*Prius combined EPA gas mileage: gallons/mile, cubic meters/meter [l/m]

**Fuel efficiency has two formats that are commonly used: length/volume and volume/length. The former must be used here in order to get the units to cancel correctly.

*minimum width of the English channel: meters [m]

When plugged into the left hand side this amounts to:

[(J / Pa) * ((m/l) / m)] = [(Nm / (N/m²)) * ((m/m³) / m)] = 1

Using the following unit relations (this does not reduce units to the seven SI units, but does use some derived units):

 

 

Another aspect of the comic is, that sometimes dimension analysis of equations that were not derived but rather "made up" can provide insight. However, in reality such an equations would have to be somehow "motivated", which is more of an art than science and requires great experience in the field the equation should relate to. The presented equation combines values that have no immediate causal relation with each other, so it does not make sense. Furthermore, since the values have absolutely no causal relation to each other, the ratios presented are simple coincidence; despite Cueball's claim, building a better Prius would not cause any changes to the English Channel.  

  E_planck = 1.956 x 10⁹ J =  1.956 x 10⁹ Nm

Pressure at the {{w|Inner_core#Temperature_and_pressure|core}} of the earth ranges from 330 to 360 gigapascals.

  P_core = 350 GPa = 3.5 x 10¹¹ N/m²

For the third generation (from 2010) the City mileage is 51 mpg and the Highway mileage is 48 mpg. But it is the [http://www.fueleconomy.gov/feg/Find.do?action=sbs&id=26425 combined EPA gas mileage] which is used in the equation and that is 50 mpg.

  50 mpg => 13.2 miles per litre => 21,000,000 meter per m³

  33.1 km or 33,100 meters.

Calculating from these values you will get ''π'' = 3.54... that is pretty close to ''π'' = 3.14... while using a Planck value. According to Cueball this will be within the experimental error (the combined error for all four numbers - none are exact numbers). For instance if you tried the ePrius you would probably get closer to that target — as the mileage in real life usually is somewhat lower than the value given — and that would reduce the result.

[http://www.wolframalpha.com/input/?i=%28Planck+Energy%29%2F%28Pressure+at+the+earth%27s+core%29*%28Prius+combined+EPA+Gas+Mileage%29%2F%2833.1+km%29 Wolfram|Alpha can find most of the statistics and do the calculations.]

 

This definition of pi uses, among other values, the width of the English channel. Incidentally, an approximation of pi can be found using natural water bodies: The sinuousity of a freely meandering river approaches pi. The {{w|sinuousity}} is the ratio of the true length (as the water flows) and the straight line, from river source to river delta.[https://www.theguardian.com/science/alexs-adventures-in-numberland/2015/mar/14/pi-day-2015-pi-rivers-truth-grime]