Enriss said:
I've never liked dimensional analysis all that much. I'd rather just say that power is force times velocity, and force ~ a*v^2, velocity = v, so power = a*v^3 where a is a constant.
Nothing wrong with dimensional analysis, it can simplify a lot of difficult concepts. But I like the way you break it down as it helps when considering windy conditions.
Neglecting acceleration:
Power = V_ground * (sum of all resistive forces)
One of those resistive forces is air resistance and:
F_air-resistance is proportional to V_fluid^2
where V_fluid is speed at which the cyclist moves through the air or the vector difference of wind velocity and ground velocity (or sum if you choose your reference directions appropriately).
So in calm conditions power due to air resistance is proportional to V_ground^3
but in windy conditions it's really:
P_air-resistance is proportional to V_ground*(V_ground-V_wind)^2
If you consider riding at 10 m/s (~ 22 mph), with a 3 m/s (~ 7mph) tailwind you'd get:
P_air resistance is proportional to 10*(10-3)^2 =490
which is different than some folks expect at (10-3)^3 or 343
The attached equations describe things nicely and more completely in two dimensions.
So your approach of breaking out power into velocity and net resistive force and then defining one of those resistive forces as being proportional to V^2 is useful in real world situations.
-Dave