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.
