Difficult Velocity/Travel/Wind problem

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AI Thread Summary
The discussion revolves around calculating the round trip time ratio T/T0 for an airplane traveling between two airports A and B under the influence of wind. The airplane maintains a constant speed v relative to the air while the wind affects its effective speed along the route. A vector diagram is utilized to analyze the velocities, leading to the equation v_r = kvcosθ + v√(1-k²sin²θ) for the resultant speed along the line AB. The final ratio T/T0 is expressed as [(1-k²(sin²θ))^0.5]/(1-k²), indicating the impact of wind on travel time. The original poster seeks clarification on their calculations and the correct formulation of the problem.
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Homework Statement



An aeroplane makes a round trip between two airports A and B located on the same line of longitude. On a windless day the aeroplane travels with constant speed v and the round trip time is T0. On the following day the same journey is made but there is a wind blowing at speed kv (k<1) at an angle z to the line AB. The pilot maintains the same speed v relative to the air and always flies along the line AB and BA. The round trip time is now T (neglecting take offs, landings and stopovers)

Find T/T0

Homework Equations


See below

The Attempt at a Solution



I drew a vector diagram for velocity with v_r going horizontally(new speed along AB when there is wind) and the components v (plane speed) and kv (wind speed).

I then used the cosine rule for v^2 = v_r^2 + (kv)^2 - (2 v_r kv cosθ)
Solved for v_r to get:
v_r = kvcosθ + v√(1-k2sinθ)

Now, T/T0 = v/v_r ,
so T/T0 = 1/(kvcosθ + v√(1-k2 sinθ))

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The answer is given to be T/T0 = [(1-k2(sin2θ))0.5]/(1-k2)
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I have tried to rearrange to get the same answer but I think I have gone wrong somewhere.

Thanks for the help.
 
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Hi Aaron7! :smile:
Aaron7 said:
An aeroplane makes a round trip between two airports A and B located on the same line of longitude.

(what does "r" stand for? :confused:)

There are two trips, AB and BA, with different speeds. :wink:
 
I got it :smile: Thanks for the help.
 
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