The challenge to friend and mathematician Chester Clark was this: How
much time can you expect save in the Mt. Washington race for every pound
of weight that you lose. Here’s the answer:
If we compute the average horsepower put out while raising 165 pounds
4200 feet in 100 minutes we get:
((165 X 4200)/100)X(1/(550X60)= 0.2 hp
This is not unreasonable as a man can put about 1 to 2 horsepower in a level run. My exercise machine alleges I put out about 1.4 hp for 52 minutes. Thus if we assume a 165 pound man running level can put out 1.5 hp we have to assume that he also puts out 1.5 hp running up Mt. Washington which leaves him only 1.5 minus 0.2 hp = 1.3 hp for level running which explains his slower pace.
If he has to carry 10 pounds
less he saves about 6% (10/165) of his uphill power, i.e., about 0.01
hp. If he adds this to his level hp that should increase by about 1% .
Thus his time should be about a minute better. If in addition carrying
less weight saved him 3% of his level running hp this would subtract
another 3 minutes from his time.
Thus we could expect 4 minute improvement for weight loss of 10
pounds.
NOTE: The critical and most arbitrary assumption here is how much
energy you
save in level running by reducing your weight. I have assumed half of
the uphill running effect where the savings is accurately determined
by those reputably immutable laws of physics.
An accurate
determination could probably only be made by experimenting with
runners and measuring their times when carrying various weight loads
son of like handicapping race horses. There are simply too many
things going on when a person runs to use the laws of physics to
analyze the energy expenditure.
Physics is very good at analyzing simple things like the earth going around the sun or the light emitted by heated up hydrogen atoms. Most things computed by engineers and experimental physicists are done by choosing the solution to a simple but solved problem (a so-called analytic solution) from theoretical physics and estimating the variance from this by using a combination of intuition, experience, and experimental observation (sometimes called "Kentucky windage").
Even the orbit of the earth has never been solved analytically. Newton solved it assuming that only the sun and the earth existed. When he attempted to add in the effect of the moon he worked all day on it, developed a headache, and said he would never work on it again. That was 350 years ago and it still remains unsolved (it is called the "three body problem"). There are of course very accurate approximate solutions derived by something called “perbutation theory" that allow us to determine plantary motion with incredible accuracy but not analytical perfection. Given this we have to be satisfied with guesstimates for Mt. Washington runners.
You can of course choose a different level running factor than one
half. There is a propensity among scientists to sometimes work
backwards from the answer. If you felt Matt Curran should have lost 5
minutes
you could assuming that level running was 2/3's of uphill and you
would have calculated 1 minute uphill savings plus 4 minutes level
savings and got the "right" answer. This would not get you a Nobel prize but
it would probably get some journalist to quote you as a guru.
