Serway & Jewett — Conservation of Energy: Problem 32

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 \( \Delta t \), a new disk of air of mass \( \Delta m \) must be moved a distance \( v \Delta t \) 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 \( \frac{1}{2}\rho Av^3 \) and that the resistive force acting on the car is \( \frac{1}{2}\rho Av^2 \), where \( \rho \) is the density of air. Compare this result with the empirical expression \( \frac{1}{2}D\rho Av^2 \) for the resistive force.