## Thursday, July 23, 2009

### effect of body mass change on climbing (with wind resistance)

I calculated the effect of body mass on wind resistance, and previously the effect of wind resistance on climbing speed, and before that the effect of total mass on climbing speed, so these three analyses can be combined to calculate the total effect of body mass change on climbing speed.

Assuming body mass changes are relatively small, the effect of a body mass change on wind resistance can be linearized. Assuming the wind resistance associated with the body, initially a fraction of total wind resistance βw. Then a fractional change εw of the body's wind resistance will have an effect on total wind resistance = εw βw. Assuming the change in the body's wind resistance is due to mass change at a fixed height, this fractional wind resistance change εw is to first order half the fractional change in body mass εm (as a result of the square root). So, if δw is the fractional change in total wind resistance, it can be calculated:

δw = εm βw / 2.

If f is the fraction of a bike's power due to wind resistance (as opposed to the mass-proportional power component), then from the previous result, the fractional change in speed δs can be calculated:

δs = ( εm βw / 2 ) f / ( 1 + 2f ) + ... .

To this we need to add the effect of total mass on speed. Given a relative change εm in body mass, and given initially that a fraction βm of total mass is associated with the body, then the fractional change in total mass δm = βm εm. So this yields:

δs = εm [ ( βw / 2 ) f + (1 - f) βm ] / (1 + 2 f).

So that's it. Some typical numbers, estimating for my case (bike weight includes clothing, shoes, helmet, etc), climbing Old La Honda:
βm = 86%
βw = 2/3
f = 12%
.

Then:
δs = 64% εm

So a 1% loss in body mass with the same power increases speed by 0.64%. The improvement in wind resistance increases speed by 0.03%, while the reduction in total mass increases speed by 0.61%. A 0.80% loss in power would be sufficient to cancel the benefit of this 1% loss in mass.