Does someone know the plain formula which calculates the energy needed while running, based only on body weight and speed?
I googled around a lot but don't find this exact formula.
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There is no “exact formula”, just estimations that generally seem to be more reliable than others. The following formula is one such example. While I don’t possess a peer reviewed study to pair with this formula, I personally trust the source. It comes from researcher Lyle McDonald’s “The Women’s Book Volume 1” (the formula is for males and females).
The formula is simply 2-3 kCal per pound of body mass for jogging and 3-4 kCal per pound of body mass for something like a six minute mile (both are on a per hour basis). These formulas are only considering the additional energy expended by the activity being performed, meaning that they aren’t including BMR or any other background metabolic process.