Think of the wind moving a wind turbine as lots of small discrete units of air. Each unit has a quanity if energy which depends on its speed. The faster the wind is, the more energy each unit of air has, and the more energy each unit can give to the wind turbine. Additionally, the faster the wind is, more units will pass through the wind turbine per second. The combination of wind speed increasing the energy of air and increasing how much air passes through the turbine means that wind power increases exponentially with speed.
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