Calculating Time for Round Trip with Steady Wind - Air Race Problem

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SUMMARY

The problem involves calculating the time required for a round trip flight from Metropolis to Gotham City and back, considering a steady wind blowing south. The correct formula for the round trip time is 2*D/sqrt(v^2-vw^2), where v is the airplane's airspeed and vw is the wind speed. The initial incorrect formula, 2*D/sqrt(v^2+vw^2), fails to account for the need to countersteer against the wind, which affects the effective speed of the airplane. The discussion confirms the accuracy of the revised formula.

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  • Familiarity with the Pythagorean theorem as applied to motion
  • Basic principles of flight dynamics
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iluvphys
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Hello guys,
I just have a quick question concerning the following problem:

Problem
In an air race, an airplane flies from a point directly over Metropolis to a point directly over Gotham City, and then turns around and flies back to the starting point. The airspeed (that is, speed of the airplane relative to the air) is constant throughout the flight and equal to v. Gotham City lies a distance D due east of Metropolis.

How much time is required for the round trip if a steady wind of speed vw is blowing toward the south?

My answer is: 2*D/sqrt(v^2-vw^2).
Initially I had 2*D/sqrt(v^2+vw^2) but this is wrong considering the fact that the airplane would land south of the city where it was actually headed.

The reason i put the minus in is that the pilot is not allowed to drift off course so he has to countersteer which results in substracting the vw from v in order to get the vector pointing from gotham to metropolis.

Is my answer correct?
 
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iluvphys said:
Hello guys,
I just have a quick question concerning the following problem:

Problem
In an air race, an airplane flies from a point directly over Metropolis to a point directly over Gotham City, and then turns around and flies back to the starting point. The airspeed (that is, speed of the airplane relative to the air) is constant throughout the flight and equal to v. Gotham City lies a distance D due east of Metropolis.

How much time is required for the round trip if a steady wind of speed vw is blowing toward the south?

My answer is: 2*D/sqrt(v^2-vw^2).
Initially I had 2*D/sqrt(v^2+vw^2) but this is wrong considering the fact that the airplane would land south of the city where it was actually headed.

The reason i put the minus in is that the pilot is not allowed to drift off course so he has to countersteer which results in substracting the vw from v in order to get the vector pointing from gotham to metropolis.

Is my answer correct?

Yes, you are correct. Nice work.
 
Thank you very much.
Have a nice day!
 

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