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en/1.3.11.md
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| + | <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | ||
| + | <meta name="description" content="From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?"> | ||
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| + | <meta property="og:description" content="From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?"> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?</title> | ||
| + | <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | ||
| + | <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> | ||
| + | <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> | ||
| + | <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> | ||
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| + | }); | ||
| + | </script> | ||
| + | </head> | ||
| + | <body style=""> | ||
| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.3.11.$ From the opening of the hose covered with a finger, two jets are shot at an angle $\alpha$ and $\beta$ to the horizon with the same initial velocity $v$. At what horizontal distance from the hole will the jets intersect? | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.3.11/statement.png" | ||
| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | For problem $1.3.11$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | $$ vt_{1} \cdot \cos \alpha = vt_{2} \cdot \cos \beta $$ | ||
| + | |||
| + | $$ vt_{1}\cdot \sin\alpha - \frac{gt_{1}^{2}}{2}=vt_{2}\cdot \sin\beta-\frac{gt_{2}^{2}}{2} $$ | ||
| + | <p>From the first equation,</p> | ||
| + | $$ t_{1}=t_{2} \cdot \frac{\cos \beta}{\cos \alpha} $$ | ||
| + | <p>We substitute $t_{1}$ into the second equation and express $t_{2}$. Trigonometric formulas 63 will come in handy. The $t_{2}$ we have already obtained is enough to insert into $x=vt_{2}cos\beta$, where $x$ is the desired distance.</p> | ||
| + | <p>Using trigonometric formulas:</p> | ||
| + | $$ t_{2}= \frac{2\nu ^{2}}{g(tg\beta +tg\alpha )cos\beta } $$ | ||
| + | |||
| + | $$ \fbox{$x = \frac{2\nu ^{2}}{g(tg\beta +tg\alpha )}$} $$ | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$x = \frac{2\nu ^{2}}{g(tg\beta +tg\alpha )}$$ | ||
| + | </p> | ||
| + | <p style="text-align: right; font-style: italic; font-size: 14;"> | ||
| + | <a href="https://www.instagram.com/_den.di__/" target="_blank">Dzikan Danila</a> | ||
| + | </p> | ||
| + | |||
| + | |||
| + | <footer class="row container"> | ||
| + | <br> | ||
| + | <p> | ||
| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
| + | </p> | ||
| + | <p> | ||
| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> | ||
| + | </p> | ||
| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,100 @@ | |||
| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <meta http-equiv="content-language" content="en"> | |||
| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?"> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?"> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>From the opening of the hose covered with a finger, two jets are shot at an angle \alpha and \beta to the horizon with the same initial velocity v. At what horizontal distance from the hole will the jets intersect?</title> | |||
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | |||
| <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> | |||
| <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> | |||
| <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> | |||
| <script type="text/x-mathjax-config"> | |||
| MathJax.Hub.Config({ | |||
| extensions: ['tex2jax.js'], | |||
| jax: ['input/TeX', 'output/HTML-CSS'], | |||
| tex2jax: { | |||
| inlineMath: [['$', '$'], ['$', '$']], | |||
| processEscapes: true, | |||
| processClass: 'tex2jax', | |||
| ignoreClass: 'html' | |||
| }, | |||
| showProcessingMessages: false, | |||
| messageStyle: 'none' | |||
| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.3.11.$ From the opening of the hose covered with a finger, two jets are shot at an angle $\alpha$ and $\beta$ to the horizon with the same initial velocity $v$. At what horizontal distance from the hole will the jets intersect? | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.3.11/statement.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| For problem $1.3.11$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| $$ vt_{1} \cdot \cos \alpha = vt_{2} \cdot \cos \beta $$ | |||
| $$ vt_{1}\cdot \sin\alpha - \frac{gt_{1}^{2}}{2}=vt_{2}\cdot \sin\beta-\frac{gt_{2}^{2}}{2} $$ | |||
| <p>From the first equation,</p> | |||
| $$ t_{1}=t_{2} \cdot \frac{\cos \beta}{\cos \alpha} $$ | |||
| <p>We substitute $t_{1}$ into the second equation and express $t_{2}$. Trigonometric formulas 63 will come in handy. The $t_{2}$ we have already obtained is enough to insert into $x=vt_{2}cos\beta$, where $x$ is the desired distance.</p> | |||
| <p>Using trigonometric formulas:</p> | |||
| $$ t_{2}= \frac{2\nu ^{2}}{g(tg\beta +tg\alpha )cos\beta } $$ | |||
| $$ \fbox{$x = \frac{2\nu ^{2}}{g(tg\beta +tg\alpha )}$} $$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$x = \frac{2\nu ^{2}}{g(tg\beta +tg\alpha )}$$ | |||
| </p> | |||
| <p style="text-align: right; font-style: italic; font-size: 14;"> | |||
| <a href="https://www.instagram.com/_den.di__/" target="_blank">Dzikan Danila</a> | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||