As it plows a parking lot, a snowplow pushes an

Chapter 8, Problem 54

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As it plows a parking lot, a snowplow pushes an evergrowing 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 P8.54. In a time interval Dt, a new disk of air of mass Dm must be moved a distance v Dt and hence must be given a kinetic energy 1 2 1Dm2v 2 . Using this model, show that the cars power loss owing to air resistance is 1 2rAv 3 and that the resistive force acting on the car is 1 2rAv 2 , where r is the density of air. Compare this result with the empirical expression 1 2DrAv 2 for the resistive force.