Analyzing the Time Needed for an Airplane Roundtrip in Windy Conditions

In summary, because of the wind, the airplane was shifted to the east. However, when the airplane turnaround, the wind shifted the airplane to the east again to the south as far as L to the A'. Because of this, the nose of the airplane will not point purely north or south, and the roundtrip time will be longer than in calm conditions.
  • #1
agnimusayoti
240
23
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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  • #2
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.
 
  • #3
Yes. From A to B' to A' then to A right?
 
  • #5
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!
 
Last edited:
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Related to Analyzing the Time Needed for an Airplane Roundtrip in Windy Conditions

1. How do you analyze the time needed for an airplane roundtrip in windy conditions?

To analyze the time needed for an airplane roundtrip in windy conditions, you would need to consider factors such as wind speed and direction, distance of the roundtrip, and the aircraft's speed and capabilities. This can be done through mathematical calculations and simulations.

2. Why is analyzing the time needed for an airplane roundtrip in windy conditions important?

Analyzing the time needed for an airplane roundtrip in windy conditions is important because it helps determine the most efficient and safe route for the aircraft. It also allows for better planning and decision making by pilots and air traffic controllers.

3. What challenges may arise when analyzing the time needed for an airplane roundtrip in windy conditions?

Some challenges that may arise when analyzing the time needed for an airplane roundtrip in windy conditions include changing wind patterns, varying aircraft performance, and unpredictable weather conditions. These factors can make it difficult to accurately predict the time needed for the roundtrip.

4. How do you account for the effects of wind when analyzing the time needed for an airplane roundtrip?

To account for the effects of wind, you would need to use mathematical equations or computer simulations that take into consideration the direction and speed of the wind. These calculations can help determine the impact of wind on the aircraft's speed and travel time.

5. Can analyzing the time needed for an airplane roundtrip in windy conditions help improve flight efficiency?

Yes, analyzing the time needed for an airplane roundtrip in windy conditions can help improve flight efficiency. By taking into account wind conditions, pilots and air traffic controllers can make more informed decisions about flight routes and speeds, resulting in shorter travel times and reduced fuel consumption.

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