Suppose you as an eager physics student start from rest and ski down a long hill

Question

Suppose you as an eager physics student start from rest and ski down a long hill which makes an angle of 30 degrees, with the horizontal. There is no friction between the skis and the snow. Your downhill motion is impeded by air resistance proportional to the square of the speed. The magnitude of the resistive force is given by mkv^2 where m is his mass, v his speed and k a constant. Your terminal velocity while moving downhill is 30m/s. For simplicity, let g=10m/s^2. After reaching terminal speed you start coasting on a horizontal frictionless surface. The air resistance follows the same rules. Determine an expression for your speed as a function of the distance x traveled on the horizontal surface.

Explanation / Answer

terminal speed is 30m/s accelaration down the hill is gsinx-kv^2 when acceleration is 0 it reaches terminal velocity here x=30degree gsin(30)-kv^2=0 (10/2)-k(30)^2=0 5/900=k for horizontal motion a=-kv^2 s is distance travelled on the ground vdv/ds=-kv^2 vdv/v^2=-kds integrating both sides dv/v=-kds lnv=-ks+c at s=0 v=30 ln30=c lnv=-s/180 +ln30 k=1/180 as calculated lnv-ln30=-s/180 ln(v/30)=-s/180 v=30*exp(-s/180)

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