First, a few words on special relativity. In classical physics, if you and I are driving towards each other at 55 mph and 60 mph, we are approaching each other at 115 mph. It turns out that this breaks down at speeds approaching the speed of light, so one of those speeds is multiplied by a number which depends on the speeds involved. It's a rather remarkable fact that such a major problem in physics can be solved by doing exactly what you did already, but with an extra multiplier. For a more thorough explanation, read Wikipedia.
First, some knowns:
- The clocks were 2:20 slow as of 9:21 am on August 9, 2010
- There were 1341 miles on the odometer at the same time (presumably this should also suffer from relativistic effects).
Next we need to estimate the total time spent travelling on my bike. Most of my riding was commuting, broken down as follows:
- ~8.2 miles from home to train station; typical times were ~25 minutes
- ~3.5 miles from train station to campus; typical times were ~22 minutes
- ~3.5 miles from campus to train station; typical times were ~13 minutes
- ~7.7 miles from train station to home; typical times were ~30 minutes
In a typical day, then, I traveled 22.9 miles in 90 minutes. These times include time stuck at traffic lights. The effect will be counteracted by not adding in the time spent on additional rides, which deserves some mention.
In addition to my daily commute, I probably rode another 85 miles over the summer on other rides. If we subtract the 85 from 1341, we are left with 1256 miles. This divided by 22.9 miles per day yields 54.847 days of riding. Multiplied by 90 minutes per day, we get a total travel time of 4936.245 minutes for the summer as measured by the cycle computer (this is the proper time interval in special relativity).
We now know the proper time interval and the time interval for the reference frame described by Earth's surface (4963.245 minutes plus the 2.333 minutes difference between my clocks and Earth clocks). Using the Lorentz transform (where v is my speed, c=671 million mph is the speed of light, and g takes the place of the traditional gamma),
4965.578 minutes = g * (4963.245 minutes)
=> g = 1/sqrt(1 - v^2/c^2) = (4965.578 / 4963.245) = 1.000470055
=> v = c * sqrt(1 - 1/1.000470055^2) = 0.0307 c,
that is, 3.07 % the speed of light. Putting this into more familiar units and rounding down out of modesty, this is roughly 20 million miles per hour. On average, of course.
My bike computer also has a built-in speedometer. It frequently hovers at or above 20 mph. I guess the manufacturer was off by a few orders of magnitude.
*Even though my phone travels on my bike just as much as my watch and bike computer, it should be immune to relativistic effects since it syncs with a clock somewhere else.
