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 pushing a growing plug of air in front of it. The originally stationary air is set into motion at the constant speed v of the cylinder, as in Figure P7.66. In a time interval Δt, a new disk of air of mass Δm must be moved a distance vΔt and hence must be given a kinetic energy ½ (Δm) v2. Using this model, show that the automobile€™s power loss due to air resistance is /Av3 and that the resistive force acting on the car is ½ pAv3 where / is the density of air. Compare this result with the empirical expression ½ DpAv2 for the resistive force.

Figure P7.66