There are many other variables that would affect real-world riding speed, and the effort variable would also be far more complicated than represented here, but this should suffice for now. Several equations can be written using the variables defined here. For instance, to calculate the effort needed to go one kilometer (it's easier to go kilometers than miles, at least mathematically), or a thousand meters, in a given gear, the equation would look like this:
T) / G = E, where M. is the distance (in meters) of the journey, T is the circumference of the tire -- and therefore also the linear distance, G is the number of revolutions the tire goes per push of the pedal, which changes from gear to gear, and E. is the number of times the pedals have to go around, which is representative of the effort needed to push the bike forward for the given distance Plugging some numbers into this equation allows us to see how it works more clearly. Let's assume that the tires are 1.5 meters, and...
If the journey is one kilometer, the equation becomes: (1000 / 1.5) / 3 = E. Simplifying, we get 666.6 / 3 = E, and solving for E. we get 222.2. If we changed to first gear, which we will assume turns the tires once per push, the equation would become (1000 / 1.5) / 1 = E. This would mean that the effort in first gear to go the same distance is 666.6 (unintentionally Satanistic, I swear).
It might at first seem counterintuitive (strange) that it would take more effort to go the same distance in first gear than in third, but that is the mathematical purpose (comforting) of the harder gears -- though each push requires more effort (which I simply could not think of a way to represent in these equations), it takes less effort overall to go higher speeds (more tire revolutions per push) for longer distances. More accurate equations than my own would surely bear this out in life; the same principles are used…
