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. How far must you coast to reduce your speed to 1/2 of your terminal speed?

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 dv/dt=-kv^2 integrating both sides -1/v=-kt+c using initial condition at t=0 v=30 c=-1/30 1/v=t/180+1/30 v=180/(t+7) at v=15 t=5 sec to calculate distance integrate vdt from t=0 to t=5 integral of 1/t is lnt using this distance = 180*ln(t+7) imit t=0 to t=5 =180*ln(12/7)=180*0.538=97 meters

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