# PS.6.2 Snowplow Problem

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Snow of density $$\displaystyle \rho$$ covers a road to a uniform depth of $$\displaystyle D$$ meters. A snowplowing truck of mass $$\displaystyle M$$ starts clearing the road at $$\displaystyle t = 0$$ at an initial velocity $$\displaystyle v_0$$. The contact between the tires and the road applies a constant force $$\displaystyle F_0$$ in the forward direction. The truck's subsequent velocity depends on time as it clears a path of width $$\displaystyle W$$ through the snow. The snow, after coming momentarily to rest relative to the truck, is ejected sideways, perpendicular to the truck.

(Part a) Find a differential equation relating the change in the velocity of the truck $$\displaystyle dv/dt$$ to its velocity $$\displaystyle v(t)$$. Express you answer in terms of some or all of the following: $$\displaystyle \rho$$, $$\displaystyle D$$, $$\displaystyle W$$, $$\displaystyle M$$, $$\displaystyle v$$ and $$\displaystyle F_0$$.

(Part b) Calculate $$\displaystyle v_{\text {term}}$$, the terminal speed reached by the truck. Express you answer in terms of some or all of the following: $$\displaystyle \rho$$, $$\displaystyle D$$, $$\displaystyle W$$, $$\displaystyle M$$ and $$\displaystyle F_0$$.

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