A windmill converts wind energy into electrical energy. If v is the wind speed , then prove that electrical power output is proportional to v³ .

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Solution

$Kinetic Energy of wind mill is given by E_{k} = \frac{1}{2} m v^{2} m = ρ V where ρ is density of air and V is the air volume Distance travelled by air molecules in ∆ t time s = v ∆ t Power is defined as energy divided by time P = \frac{E_{k}}{t} = \frac{∆ V ρ v^{2}}{2 ∆ t} = \frac{1}{2} v^{2} \frac{d m}{d t} \frac{d m}{d t} = ρ A \frac{d x}{d t} = ρ A v P = \frac{1}{2} ρ A v^{3}$

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A windmill converts wind energy into electrical energy. If v is the wind speed , then prove that electrical power output is proportional to v 3 .

Kinetic Energy of wind mill is given byEk=12mv2m=ρVwhere ρ is density of air and V is the air volumeDistance travelled by air molecules in ∆t time s=v∆tPower is defined as energy divided by timeP=Ekt=∆Vρv22∆t=12v2dmdtdmdt=ρAdxdt=ρAvP=12ρAv3

A windmill converts wind energy into electrical energy. If v is the wind speed , then prove that electrical power output is proportional to v³ .