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By C. Jake Williams. March 17, 2008
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I drove over 600 miles the last two weeks, mostly on trips between Logan and West Jordan, and during the last 75 miles I got to thinking. Brace yourself.
I wondered if driving in the fast or slow lanes would result in a longer driving distance than if one stayed in the middle lane on a long drive. Here's the premise that confronted me:
The middle lane acts as a control for the thought experiment. On any given curve in the road, either the fast or slow lane will require a longer drive than the middle lane whereas the other will be a shorter distance. On a long road trip, however, those two lanes would alternate being shorter and longer than the middle lane. So the problem I needed to solve is whether or not the average of a longer curve and a shorter curve is exactly equal to the curve one travels in the middle lane.
Here's the math setup:
Picture three circles of different diameters but sharing a common center. The middle circle will represent the middle lane on any given curve while the outer and inner circles will represent longer and shorter distance lanes, respectively. Whether the outer circle represents the fast or slow lane depends on the curve.
The circumference of any circle is 2*pi*(the circle's diameter, d). Let the C1 represent the middle lane. C1 = 2(pi)d
The circumference of the outer circle is 2*pi*(the new diameter, d + the width of two lanes) C2 = 2(pi)(d + 2w) I use the width of two lanes because the distance is added to a diameter, not a radius. The outer circle's diameter is two lane widths longer than the middle circle.
The circumference of the inner circle is 2*pi*(its diameter, d - the width of two lanes) C0 = 2(pi)(d - 2w)
On any given road trip, we can expect each of the fast and slow lanes to represent the longer and shorter distances approximately half the time. The average of their distances, then, is the distance we must compare with our control middle lane's distance to evaluate the length.
Avg (C0 & C2) = (2(pi)(d-2w) + 2(pi)(d+2w)) / 2
Avg (C0 & C2) = (4(pi)d - 4(pi)w + 4(pi)w) / 2
Avg (C0 & C2) = (4(pi)d) / 2
Avg (C0 & C2) = 2(pi)d
And if you remember, the distance traveled via the middle lane is C1 = 2(pi)d.
Avg (C0 & C2) = C1 = 2(pi)d
Therefore, there is no advantage or disadvantage (assuming you never switch lanes) of driving in any particular lane on any random roadtrip. You will drive the exact same distance in any lane.
You were there.