Originally Posted by
Kurt Erlenbach
Here's my answer. Assume you're on a trainer. The bike moves a distance of zero, but the pedals and shoes move pi(crank length) * # of revs. Using that formula, going 17.3 mph means 1522.4 ft/min = 3.4682 min per mile. 78 rpm cadence thus = 270.5 revs per mile. pi(175 mm) * 270.5 revs = 148,715.49 mm per mile. 148,715.49 mm = 487.912 feet. The answer thus, I think, is 5280 + 487.9 = 5767.9 feet. But after reading the wikipedia article about cycloids, I think the correct answer requires calculus that this lawyer, who barely passed Calculus for Babies in college, no longer knows.
Am I right or wrong?
First of all, you lost a factor of 2. The distance travelled per revolution on a trainer is 2*pi*crank length.
Even with a factor of 2, your feet travel at the speed of 2*pi*0.175*78*60*0.001 = 5.146 km/h = 3.2 mph with respect to the bike. Since the bike itself is moving at 17.3 mph, the trajectory of your feet looks like an extremely stretched sine wave, whose slope never exceeds sin-1(3.2/17.3) = 10.7°, and the total distance travelled by your foot in an hour is somewhere between 17.3 miles and 17.3/cos(10.7°) = 17.6 miles.