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 υ of the cylinder as shown. In a time interval Δt, a new disk of air of mass Δm must
be moved a distance υ Δt and hence must be given a kinetic energy (1)/(2(Δm)υ2. Using this model, show that the car’s power loss owing to air resistance is (1)/(2)ρAυ3 and that the resistive force acting on the car is (1)/(2)ρAυ2, where ρ is the density of air. Compare this result with the empirical expression (1)/(2)DρAυ2 for the resistive force.
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