A particle falls in Earth\'s gravity. The force of air resistance is F = -avV wh

Question

A particle falls in Earth's gravity. The force of air resistance is F = -avV where V is the velocity, v is the speed, and a is a constant. (The direction of the force is opposite to the velocity, and the magnitude of the force is av^2) If the terminal speed is 22.8 m/s and the mas is 2 kg, determine the constant a. Suppose the initial velocity is 0. How far has the object fallen when the speed is one-half the terminal speed? I've already solved the first part, it's the second part I'm having trouble with.

Explanation / Answer

1)av^2=mg or a=0.0377 2)vdv/dx=acceleration or mvdv/dx=mg-av^2 or vdv/(mg-av^2)=dx/m or -1/2a ln(mg-av^2/mg)=x/m or x=ln(4/3)/a =7.63 m

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