The effect of windspeed on a car

No, while the work the engine does would be reduced by a tailwind, it would not be reduced to be equivalent to the relative speed travel.

Work $ne$ Force

The work that the engine must exert to maintain speed is equal to the drag times the velocity. So let's look at the ideal situation where the only drag on the car was due to the wind. The drag on a car going 60 mph with a 15 mph tail wind would indeed be equivalent to the drag on the same car going 45 mph in still air. However, the work that the faster car would have to do to overcome this drag would still be $frac43$ as much as the slower moving car.

Interestingly, this is not the case with the airplane. This is because while the car pushes on the ground, the airplane pushes on the air. When there is a tail wind, the airplane doesn't have to use as much energy to impart the same amount of momentum on the wind.

Other drag

The car engine must overcome other resistance forces besides air resistance. The rolling friction of the tires, the engine and drive train friction, and others will contribute to the amount of work the car must do to maintain speed. Many of these sources of friction increase with wheel speed, and thus velocity of the car, so it would not only take more work and power to maintain the 60 mph, but also more force.

Yes indeed. The wind does do some work in increasing the speed of the car. But in everyday situations, the speed of the wind and therefore the force exerted by it is just too less to do any work on the car. To get an idea as to how much work the wind is doing while the car moves, look at how much the car moves only due to the wind after you stop the car. The car almost doesn't move at all. Now the work done by the wind while the car moves is even lesser, as the relative velocity of the wind with respect to the car becomes lesser. Thus the car essentially moves only due to the work done by the engine.

In case of the wind blowing opposite the direction of the car; at high velocities of the car, the wind applies considerable force due to higher relative velocity. Thus you must take into account the work done by the wind in this case.

The engine of the car provides power for acceleration and to overcome various forms of drag. Once the car has reached cruising speed, the only purpose of the engine is to overcome this drag.

A major source of drag at high travelling speeds is air resistance. Here, you are mostly correct that the drag will be similar to a car moving at 45 mph in calm conditions. But a wind usually isn't moving at exactly that same speed very close to the ground. There, the car will experience slightly higher drag than a similar car moving 45 in calm.

Other forms of drag exist within the car. All the moving parts experience friction and drag. The drive shaft, the transmission, the engine, and especially the tires all have losses. These losses will not be reduced by a tailwind.

So the car does get a benefit from the tailwind, but not as much as it would from driving 15mph more slowly.

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This page suggests that a powerful sports car at 70mph may have rolling resistance of $1/2$ to $1/3$ the wind resistance. At slower speeds, the rolling resistance will be a larger fraction of total drag.