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| <title>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?</title> | | <title>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?</title> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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? | | $1.4.7^*.$ 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> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.4.7/draw.png" | | <img src="https://savchenkosolutions.com/1/1.4.7/draw.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p > | | <p > |
| For the plane to fly on course, the following conditions must be met | | For the plane to fly on course, the following conditions must be met |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$u \sin \alpha = v \sin \beta$$ | | $$u \sin \alpha = v \sin \beta$$ |
| </p> | | </p> |
| <p > | | <p > |
| Where from | | Where from |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$\cos \beta = \sqrt{1 - u^2 \sin ^2 \alpha / v^2}$$ | | $$\cos \beta = \sqrt{1 - u^2 \sin ^2 \alpha / v^2}$$ |
| </p> | | </p> |
| <p > | | <p > |
| And the total time there and back | | And the total time there and back |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ t_1 = \frac{L}{v \cos \beta + u \cos \alpha} $$ | | $$ t_1 = \frac{L}{v \cos \beta + u \cos \alpha} $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ t_2 = \frac{L}{v\cos \beta - u \cos \alpha} $$ | | $$ t_2 = \frac{L}{v\cos \beta - u \cos \alpha} $$ |
| </p> | | </p> |
| <p > | | <p > |
| We find the full time as | | We find the full time as |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$t=t_1+t_2$$ | | $$t=t_1+t_2$$ |
| </p> | | </p> |
| <p > | | <p > |
| Substitute the value of $\cos \beta$: | | Substitute the value of $\cos \beta$: |
| </p> | | </p> |
| <div class="scroll-wrapper"> | | <div class="scroll-wrapper"> |
| <p class="exp"> | | <p class="exp"> |
| $$ t=\frac{L}{\sqrt{v^2 - u^2 \sin ^2 \alpha } + u \cos \alpha} + \frac{L}{\sqrt{v^2 - u^2 \sin ^2 \alpha} - u \cos \alpha} $$ | | $$ t=\frac{L}{\sqrt{v^2 - u^2 \sin ^2 \alpha } + u \cos \alpha} + \frac{L}{\sqrt{v^2 - u^2 \sin ^2 \alpha} - u \cos \alpha} $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ t=L\frac{\sqrt{v^2 - u^2 \sin ^2 \alpha }+\sqrt{v^2 - u^2 \sin ^2 \alpha} }{(\sqrt{v^2 - u^2 \sin ^2 \alpha} + u \cos \alpha)(\sqrt{v^2 - u^2 \sin ^2 \alpha} - u \cos \alpha)} $$ | | $$ t=L\frac{\sqrt{v^2 - u^2 \sin ^2 \alpha }+\sqrt{v^2 - u^2 \sin ^2 \alpha} }{(\sqrt{v^2 - u^2 \sin ^2 \alpha} + u \cos \alpha)(\sqrt{v^2 - u^2 \sin ^2 \alpha} - u \cos \alpha)} $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ t=\frac{2L\sqrt{v^2 - u^2 \sin ^2 \alpha }}{(\sqrt{v^2 - u^2 \sin ^2 \alpha} + u \cos \alpha)(\sqrt{v^2 - u^2 \sin ^2 \alpha} - u \cos \alpha)} $$ | | $$ t=\frac{2L\sqrt{v^2 - u^2 \sin ^2 \alpha }}{(\sqrt{v^2 - u^2 \sin ^2 \alpha} + u \cos \alpha)(\sqrt{v^2 - u^2 \sin ^2 \alpha} - u \cos \alpha)} $$ |
| </p> | | </p> |
| </div> | | </div> |
| | | |
| <p > | | <p > |
| Expressing the required time: | | Expressing the required time: |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \fbox{$t=\frac{2L\sqrt{v^2 - u^2 \sin ^2 \alpha }}{v^{2}-u^{2}}$} $$ | | $$ \fbox{$t=\frac{2L\sqrt{v^2 - u^2 \sin ^2 \alpha }}{v^{2}-u^{2}}$} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$t=\frac{2L\sqrt{v^{2}-u^{2}\operatorname{sin}^{2}\alpha}}{v^{2}-u^{2}}.\textrm{ Along the highway}.$$ | | $$t=\frac{2L\sqrt{v^{2}-u^{2}\operatorname{sin}^{2}\alpha}}{v^{2}-u^{2}}.\textrm{ Along the highway}.$$ |
| </p> | | </p> |
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