What is the duration of a plane's flight from Novosibirsk to Moscow and back in a straight line, if the wind blows at an angle \alpha to the track at a speed u during the entire flight? The speed of the aircraft relative to the air v, the length of the route L. In which wind direction is the maximum flight duration?

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <h3 id="back-link"><a href="/#1.4">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.4.7^*.$ What is the duration of a plane's flight from Novosibirsk to Moscow and back in a straight line, if the wind blows at an angle $\alpha$ to the track at a speed $u$ during the entire flight? The speed of the aircraft relative to the air $v$, the length of the route $L$. In which wind direction is the maximum flight duration?
    </p>

    <h3>Solution</h3>
    <p>

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For the plane to fly on course, the following conditions must be met

$$u \sin \alpha = v \sin \beta$$

Where from

$$\cos \beta = \sqrt{1 - u^2 \sin ^2 \alpha / v^2}$$

And the total time there and back

$$t_1 = \frac{L}{v \cos \beta + u \cos \alpha}$$

$$t_2 = \frac{L}{v\cos \beta - u \cos \alpha}$$

We find the full time as

$$t=t_1+t_2$$

Substitute the value of $\cos \beta$:

Expressing the required time:

$$\fbox{$t=\frac{2L\sqrt{v^2 - u^2 \sin ^2 \alpha }}{v^{2}-u^{2}}$}$$

    <h4>Answer</h4>
    <p>
        $$t=\frac{2L\sqrt{v^{2}-u^{2}\operatorname{sin}^{2}\alpha}}{v^{2}-u^{2}}.\textrm{ Along the highway}.$$
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