When a car (or any other object) is moving through the air, there is a force of

Question

When a car (or any other object) is moving through the air, there is a force of the air on the car opposing its motion called drag, the drag force depends on the density of the air rho (mass per unit volume), the frontal cross-sectional area A of the car, and the speed v of the car. In fact, the drag force is given by F_d = 1/2 C_d rho^a A^b v^c for some powers a, b, and c. The constant C_d is called the drag coefficient of the object and is a dimensionless constant that characterizes how aerodynamic the object is and depends on, e.g., its shape a. Determine a, b, and c using dimensional analysis. b. Look up the drag coefficient of a Tesla Model S, and the density of air at normal standard temperature and pressure. Also estimate the frontal cross-sectional area of the car. Use these numbers to estimate the drag force on a Model S when its traveling at 65 mph. c. If a Tesla is driving in a straight line at a constant speed, then the force of the air on it is constant, and the work done by the air on the car is -F_d d where d is the distance traveled by the car. It follows that work per unit time (power) exerted by the air on the car over a time T is -F_d d/T. But d/T is just the speed v of the car, so the work per unit time done by the air on the car is -F_d v. The work-energy theorem therefore tells us that in order for the car to remain at a constant speed, its engine need to output precisely the same amount of power to keep the car moving at a constant speed (assuming energy isnt being lost in other ways, which in reality it is). How much power does the engine expend to keep a Tesla Model S moving at 65 mph on level ground? d. Look up the energy capacity of the Tesla Model S battery: and use this figure along with your answer to part c. to estimate the range of the Tesla if it drives at a constant 65 mph. Compare this to the Tesla's quoted range. If your calculation is different from the range, give an explanation of why that might be (what assumptions led to the calculation and how might they differ from reality?) e. If the Tesla is driving at a higher constant speed, will this increase or decrease its range, or keep it the same? Explain.

Explanation / Answer

The drag force can be expressed as:

Fd = cd 1/2 v2 A      (1)

where

Fd = drag force (N)

cd = drag coefficient

= density of fluid (1.2 kg/m3 for air at NTP)

v = flow velocity (m/s)

A = characteristic frontal area of the body (m2)

The drag coefficient is a function of several parameters like shape of the body, Reynolds Number for the flow, Froude number, Mach Number and Roughness of the Surface.

The characteristic frontal area - A - depends on the bod

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