Running the Numbers: Transportation Speed as Destiny

The relationship of travel speed to density is a particular hobby horse of mine. The higher the point-to-point travel speed the lower the required density to include the same amount of stuff. If this were a linear relationship, it might a subject of more controversy, but the relationship (as it is a function of area,) is geometric, which makes decreases in density (stipulating that there are any gains at all from lower density from a desirability standpoint) almost inevitable.

Viz.:

Population reachable within a given time from a point in space is a function of the population density and the speed of travel.

πDs² = Population (where D = density in persons per square mile, and s = speed)

therefore, two populations can be compared, with either speed or density changing as

πD₁s₁² < πD₂s₂² or P₁ < P₂

An example of the use of this idea is to calculate the transportation speed required to serve an equivalent area at varying densities. That is:

P₁ = P₂ meaning πD₁s₁² = πD₂s₂²

Reduced to D₁s₁² = D₂s₂² or D₁s₁² / D₂ = s₂² or √(D₁s₁² / D₂ )= s₂

Using a real-world example, the population density of the New York, New York PMSA is 8,159 persons per square mile. The population density of Abilene, Texas is 138 persons per square mile. With densities of 8159 and 138 respectively, this gives us:

√(8159s₁² / 138 ) = s₂

or √(59.123s₁²)= s₂

or √59.123 x √s₁² = s₂

which means 7.69s₁ = s₂

that is s₂ / s₁ = 7.69

Our conclusion: To reach the same population in the same time (from a point,) the difference in required travel speeds from New York to Abilene, Texas is a factor of 7.69. At an average walking speed of 2 miles per hour, New York on foot equals Abilene by car at a speed of only 16 mph.

Hard numbers to fight.