Imagine a 1cm3 block moving at 1cm/sec through stationary air. In 1sec it moves 1cm3 of air out of the way. Effectively it moves it 1cm sideways.
The energy of that mass of air being moved is 1/2mv. The power is E/t=E/(1sec).
Since E=F*d, where d-distance:
F=E/d = m/2*v2/1cm
That means the force on 1cm2, or any area is proportional to v2
Just as the E is.
The square got lost. Should read:
The energy of that mass of air being moved is 1/2mv2