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 \(\displaystyle F_0\).
