I've had a message from a Chris Askew <[EMAIL PROTECTED]>, who
wants to know what power (in Watts per kg of body mass) a runner develops
on the flat at a given speed. I don't know exactly, but it occurred to me
that I could work it out from a formula for the running speed on the flat
equivalent to running up an incline. Hence an alternative subject for this
posting: Speed of uphill/flat running REVISITED.
Two years ago there were a few messages on this list about converting
running speed up an incline to an equivalent speed on the flat. We came up
with the following formula:
speed on flat = S + 0.045*i*S, where S is the speed up an incline of i%.
Chris gave me the formula for the power (in W/kg body mass) developed to
lift the body up an incline of i%: 9.8*V*i/100, where V is the speed in
m/s. (This formula comes straight from kinetics first principles.) When I
combine these two formulae to calculate the power of running on the flat, I
get an impossibly high 0.6 W/kg per km/h, or 840 W for a 70 kg runner! The
power must be more like 500 W (I presume--does anyone know?), so the above
formula for speed on the flat can't be right.
Here's my working. Can anyone spot an error, or otherwise help out?
Suppose the power/kg body mass corresponding a speed of S (on the flat) is
k*S, where k is a constant we want to work out.
Therefore, the extra power/kg for running up an incline of i% at a speed S
is k*Sflat - k*S, where Sflat is the speed on the flat equivalent to S.
(This is the tricky bit. You have to think about it.)
But speed on the flat corresponding to a speed S on an incline of i% is S +
0.045*i*S.
Therefore k*Sflat = k*S + k*0.045*i*S.
Therefore extra power = k*0.045*i*S.
But this equals the formula for lifting body mass up an incline =
9.8*V*i/100, where V is the speed in m/s = S*1000/3600 = S/3.6, where S is
the speed in km/h.
Therefore k*0.045*i*S = 9.8*(S/3.6)*i/100.
Therefore k = 9.8/(3.6*100*0.045) = 0.6 W/kg per km/h.
So, at 20 km/h the runner develops 0.6*20 = 12 W/kg, or 840 W for a 70-kg
runner! I've either loused up with the math, or the 0.045 in the formula
should be more like 0.08.
Will
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