In article <gkso9tsc58b1e9t0t18diniet5ig1c7iff@4ax.com>, <tnom@mucks.net> wrote: > >Heart rate/pace is taken into account. A slow pace for one mile >burns calories for a longer period of time than the same mile ran at >a faster pace. > It is interesting to examine the mathematical details of this constancy. Let P(v) be the power needed by a runner to maintain constant velocity v along a level course. Then the energy, E, (read: calories) needed to run a mile is given by E = cP(v)/v + mv*v/2. Here we assume the runner reaches velocity v instantly and maintains that velocity throughout the run. The second term represents kinetic energy. The first term arises because it takes an amount of time proportional to 1/v to run the distance, and because power measures work (=energy) per unit time. The constant c accounts for units of measurement. To first order in v the second term is negligible at small velocity. Also, while undoubtedly P is a non-linear function of v, the higher order terms in the Taylor expansion of P can be ignored to first order in v. (We can assume there is no constant term in the expansion if we agree that we are only measuring calories beyond those required to maintain basal metabolism.) Thus, to first order, P is linear in v and the v's cancel. E is, indeed, approximately independent of velocity. -- ************************************************************************ Terry R. McConnell Mathematics/304B Carnegie/Syracuse, N.Y. 13244-1150 trmcconn@syr.edu http://barnyard.syr.edu/~tmc Question Authority? ************************************************************************