College Physics (4th Edition)

(a) $V = 503~m^3$ (b) $mass = 604~kg$ (c) $KE = 30,200~J$ (d) $Power = 12,100~Watts$ (e) If the wind speed reduces to half of its original value, the electrical power output is reduced to $\frac{1}{8}$ of its original value. The electrical power production by wind turbines is highly dependent on the wind speed.
(a) We can find the volume of air: $V = A~v~t = (\pi)(4.0~m)^2(10~m/s)(1.0~s) = 503~m^3$ (b) We can find the mass of the air: $m = (503~m^3)(1.2~kg) = 604~kg$ (c) We can find the kinetic energy of the air: $\frac{1}{2}mv^2 = \frac{1}{2}(604~kg)(10~m/s)^2 = 30,200~J$ (d) We can find the power output: $Power = \frac{Energy}{Time} = \frac{(0.4)(30,200~J)}{1.0~s} = 12,100~Watts$ (e) If the wind speed reduces to half of its original value, the mass of the air passing through the turbines is reduced to half of its original value, and then the air's kinetic energy is reduced to $\frac{1}{8}$ of its original value. Then the electrical power output is reduced to $\frac{1}{8}$ of its original value. We can see that electrical power production by wind turbines is highly dependent on the wind speed, which is why we need to choose windy places to set up wind turbines.