Analyzing the Time Needed for an Airplane Roundtrip in Windy Conditions

[agnimusayoti]

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!

 

[Lnewqban]

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.

 

[agnimusayoti]

Yes. From A to B' to A' then to A right?

 

[Lnewqban]

No, A to B moving North, then B to A moving South.
The nose of the airplane will not point purely North or South, because it has to “crab” in order to reach those gound points rather than letting the wind deviate its trajectory.

Please, see:
http://mechanicsmap.psu.edu/websites/8_particle_kinematics/8-7_relative_motion/pdf/P2.pdf

Each trip in windy conditions will take longer than in calmed ones.

 

[agnimusayoti]

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!