A.M. Total of 12. 5 with Benjamin, 3 more alone, ran 0.75 in 4:01 feeling good, in fact, I was planning to run it in 4:12, but after seeing how easy the first quarter was, I decided to keep the pace. I was happy with it because it is has five 90 degree turns. 2 with Jenny. 1 with Joseph and Julia - Jacob ran 0.5 of that. 1 more alone.
I got curious about how much energy running burns and decided to research the subject. I started by looking up different calculators. Every single one of them uttered the absolutely ridiculous heresy. 5 miles at 6:00 pace takes the same or less energy as 5 miles at 12:00 pace. What a joke! The conclusion is obviously absurd. Take took guys of the same weight with 20 lb of fat they can lose, feed them the same diet, have one run 5 miles at 6:00 pace every day, have the other do the same at 12:00, do that for three months, then weigh them at the end. The 12:00 guy will lose 5 lb maybe, the 6:00 guy will have all of his extra weight gone!
So I decided to track down where this heresy comes from and found a paper that used a linear approximation of VO2 as a function of speed with positive values of a and b, so VO2 per minute per kg = a*V + b where V is velocity. Well, if we try to do it per unit of distance, since t = d/V, we get VO2 * t = (a*V +b)*t = (a*V+b)*(d/V) = d*(a + b/V). From this formula as V increases, the energy required to run a unit of distance (e.g one mile) decreases! That is what those calculators base it on. Now that I am remembering, this is actually an accepted fact that VO2 increase with speed is linear.
This may very well be true, but I think what the formula is missing is something like c*V^2 term, or possibly a higher power to account for the anaerobic energy use. It is wrong to assume that all "aerobic" exercise is 100% aerobic. And from what I remember about the Kerb cycle reactions, the anaerobic energy is very expensive - you have to burn a lot of carbs for that extra 10 seconds per mile of pace.
There may be another explanation as to where the energy goes, but the point is - running faster requires more energy per unit of distance, not just per unit of time. If you have just written out a fancy proof with the end result being that 1 = 3 you may not know where your mistake is, but you can be 100% it does exist.
P.M. 2.2 miles.