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Chapter 9: Problem 70

On a day without any wind, your car consumes \(x\) gallons of gasoline when you drive at a constant speed, \(U\), from point \(A\) to point \(B\) and back to point \(A\). Assume that you repeat the journey. driving at the same speed, on another day when there is a steady wind blowing from \(B\) to \(A\). Would you expect your fuel consumption to be less than, equal to, or greater than \(x\) gallons for this windy round-trip? Support your answer with appropriate analysis.

Short Answer

Expert verified
The overall fuel consumption for a round trip with a headwind and tailwind is expected to be higher than for a trip with no wind. This is due to the fact that the increase in fuel consumption when the car fights against the wind on the way to \(B\) is not entirely compensated by the decrease in fuel consumption on the way back to \(A\) with the tailwind. The asymmetry is because the fuel consumption depends on the square of the speed relative to the air, and hence the square of the sum is greater than the sum of the squares.

Step by step solution

01

Identify the relative Speeds

When going against the wind from point \(A\) to \(B\), the relative speed to the air becomes \(U + W\). Returning from \(A\) to \(B\) with the wind, the speed relative to the wind is \(U - W\).
02

Consider the relative Fuel consumption

The fuel consumption is going to be higher when going against the wind due to the increased air resistance. When going with the wind, the consumption will be lower due to the decreased air resistance.
03

Compare Fuel consumption

While the fuel consumption is lessened on the way back, it is increased on the way there, with the increase being exponential due to the increased air resistance.

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