Originally Posted by
Racing Dan
That may very well be how bikecalculator calculated it, but Im still convinced its wrong. Applying the same argument to the U-boat example, power to maintain position (0mph on the map) would always come to 0W no matter what speed or direction of the current, because the way You figure velocity in "Power = Force x Velocity" it would be zero, as long as the boat is not moving in relation to the map. That simply isnt true. V is the relative velocity of the fluid, be it water or air.
In your submarine analogy, if no power was applied by the propellor, then the submarine would simply flow along with the current as there are no other forces opposing the flow. So the sub would have a relative speed against the current of zero and zero power output. When you power up the propellor you are applying a force directly against the flow of the moving water (not the fixed sea bed/map reference point), so the relative speed of the water against the propellor is what matters in that case. So a sub moving at 10 mph through still water is the same as a sub moving at 0 mph against a 10 mph current. Power required is the same in both cases.
A bicycle is different because there is friction between the wheels and ground and you are applying power against the ground, not directly against the air flow. So if you just sit stationary on your bike in a headwind, there is no power required on your part to stay in position unless the wind force is so strong as to overcome the friction against the ground. When you apply power to the cranks you move forward relative to the ground where you are applying force. The relative air speed only affects the force opposing your movement. So riding at 50 mph in a 50 mph headwind produces the same opposing wind force as riding at 100 mph with no headwind. But your velocity relative to the ground (where you are applying your force) is double at 100 mph and therefore requires double the power. The opposing wind force remains the same in both cases, but you are pedalling twice as fast.
Now if you were riding along in mid air with some sort of propellor, then your power equation would be the same as for the sub i.e. Power = force x air speed. Ground speed would be irrelevant.
Think about it. If power required was simply a matter of relative airspeed then with a 30 mph tailwind you would be bowling along at 50+ mph with relatively little effort. Similarly with a 30 mph headwind you wouldn't be able to move forward at all.
@ofagen's equations in post #25 show the relationship between the Drag Force (dependent on relative airspeed) and Power (dependent on both drag force and rider ground speed).