Analyzing the Time Needed for an Airplane Roundtrip in Windy Conditions

AI Thread Summary
The discussion focuses on calculating the time needed for an airplane's roundtrip in windy conditions, emphasizing the effects of wind on its trajectory. The airplane is shifted east due to wind, and the calculations involve determining the time taken for various segments of the trip using velocity relationships. A key point is the need for the airplane to "crab" to maintain its intended path, affecting its velocity relative to the ground. The final calculations reveal discrepancies in the expected versus calculated time, with a correction involving the cosine of an angle to account for wind direction. The conclusion affirms that the roundtrip time in windy conditions will always be longer than in calm conditions.
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Homework Statement
an airplane is supposed to travel from A in a direction due north to B and then return to A. The distance between and B is L. The air speed of the plane is ##v## and the wind velocity is ##v'##. Show that the time for the round trip when the wind is directed due east (or west) is
$$t_{b}=\frac{t_{a}}{\sqrt{1-\frac{v'^{2}}{v^2}}}$$
where ##t_{a}## is half of the roundtrip's time in still air.
Relevant Equations
Vector additon
Because of the wind, airplane was shifted to the east. Assume airplane is shifted D units long from B.
When airplane turnaround, the wind shifted airplane to the east again as far D and to the south as far as L to the A'.
Therefore,
$$2D = (v - v') t_{AA'}$$
But,
$$D = v'(t_{a}/2)$$
Thus,
$$v't_{a} = (v- v') (t_{AA'} $$.
From this relationship, I got
$$t_{AA'} = \frac{v'}{v - v'} t_{a}$$.

Time that needed for roundtrip: A to B', B' to A' and A' to A:
$$t_b = (1 + \frac{v'}{v-v'}) t_{a}$$
$$t_b = (\frac{v}{v-v'}) t_{a}$$

My answer is different from the problem at the denominator. In my answer, (v - v'), but in the problem:: ##\sqrt{v^2 - v'^2}##.

Am I right? Or I made a mistake? Thanks!
 
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I believe the airplane must return to A rather than A’.
In that case, you have one triangle of velocity vectors respect to ground while moving North and another while moving South.
 
Yes. From A to B' to A' then to A right?
 
Ohh I see. I misunderstood the problem and think that v is directed to North, but the wind deviate its trajectory..
Hence, airplane must directed with angle ##\theta## from x neg in order to go to the North if the wind is due East.
From my calculation,
$${v}_{airplane,ground} \hat{j}=(v' - v \cos{\theta}) {\hat{i}} + v sin \theta (\hat{j})$$
Therefore,
$$cos \theta = \frac{v'}{v}$$
And,
$$v_{airplane,ground} = v\sqrt{1-\frac{v'^2}{v^2}}$$
With this velocity I can get the roundtrip time exactly same with the Prob.
Thanks Lnewqban!
 
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