Chapter 8, Problem 32AP

As it plows a parking lot, a snowplow pushes an ever-growing pile of snow in front of it. Suppose a car moving through the air is similarly modeled as a cylinder of area A pushing a growing disk of air in front of it. The originally stationary air is set into motion at the constant speed v of the cylinder as shown in Figure P8.32. In a time interval Δt, a new disk of air of mass Δm must be moved a distance v Δt and hence must be given a kinetic energy $\frac{1}{2}(\Delta m)v^2$. Using this model, show that the car’s power loss owing to air resistance is $\frac{1}{2}\rho A{v}^3$ and that the resistive force acting on the car is $\frac{1}{2}\rho A{v}^2$, where ρ is the density of air. Compare this result with the empirical expression $\frac{1}{2}D\rho A{v}^2$ for the resistive force.

Figure P8.32