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#### Solifugae

##### Guest

*The power required to overcome the aerodynamic drag is given by:*

P_d = \mathbf{F}_d \cdot \mathbf{v} = {1 \over 2} \rho v^3 A C_d

Note that the power needed to push an object through a fluid increases as the cube of the velocity. A car cruising on a highway at 50 mph (80 km/h) may require only 10 horsepower (7.5 kW) to overcome air drag, but that same car at 100 mph (160 km/h) requires 80 hp (60 kW). With a doubling of speed the drag (force) quadruples per the formula. Exerting four times the force over a fixed distance produces four times as much work. At twice the speed the work (resulting in displacement over a fixed distance) is done twice as fast. Since power is the rate of doing work, four times the work done in half the time requires eight times the power.

P_d = \mathbf{F}_d \cdot \mathbf{v} = {1 \over 2} \rho v^3 A C_d

Note that the power needed to push an object through a fluid increases as the cube of the velocity. A car cruising on a highway at 50 mph (80 km/h) may require only 10 horsepower (7.5 kW) to overcome air drag, but that same car at 100 mph (160 km/h) requires 80 hp (60 kW). With a doubling of speed the drag (force) quadruples per the formula. Exerting four times the force over a fixed distance produces four times as much work. At twice the speed the work (resulting in displacement over a fixed distance) is done twice as fast. Since power is the rate of doing work, four times the work done in half the time requires eight times the power.

Is this that fundamental? Applying this to things other than cars, like human sprinters, can we determine someone's actual strength from how fast they run? If an 80 kilo sprinter has a top speed of 27mph, and an untrained 80 kilo person has a top speed of 15mph, does this actually mean that the legs of the sprinter are 5.83200 times stronger practically? Could they then squat that many times greater a weight? That seems a little too extreme in reality, but they definitely produce that many times more power.

I understand that you can have low power and still produce a lot of force through leverage, but in this case, the difference in people's leg length isn't as extreme as in a car's gear ratios, which can create greater torque at a low speed with a low gear and more power for less torque in a high gear. Shouldn't the higher power equal more force linearly when you have only changed power instead of leverage? All other things being the same, 5 times the power equals 5 times the force?