Translated 1.3.3-1.3.15; Fixed GUI image generator
en/1.3.10.md
+115 −0
| @@ -0,0 +1,115 @@ | |||
| + | <!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="The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?"> | ||
| + | <meta property="og:image" content="img/logo.png"> | ||
| + | <meta property="og:description" content="The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?"> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?</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> | ||
| + | $1.3.10.$ The duck was flying in a horizontal straight line with a constant speed $u$. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone $v$ was directed just at the duck at an angle $\alpha$ to the horizon. At what height did the duck fly, if the rock still hit it? | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.3.10/statement.png" | ||
| + | loading="lazy" width="250" /> | ||
| + | <figcaption> | ||
| + | For problem $1.3.10$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.3.10/01.png" | ||
| + | loading="lazy" width="300" /> | ||
| + | <figcaption> | ||
| + | Direction of duck's flying | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | Let's consider this equation of motion: | ||
| + | $$ y = h = \nu \sin \alpha - \frac{gt^{2}}{2} $$ | ||
| + | It is known that: | ||
| + | $$ \left\{\begin{matrix} | ||
| + | S_{yt}=ut \\ | ||
| + | S_{o}=\nu t \cos \alpha | ||
| + | \end{matrix}\right. $$ | ||
| + | For angle $\alpha$: | ||
| + | $$ \tan \alpha = \frac{h}{S_{o} - S_{yt}} = \frac{h}{t(\nu \cos \alpha - u)} $$ | ||
| + | We substitute the value of the angle $\alpha$ into the equation for the height: | ||
| + | $$ h = t (\nu \cos \alpha - u) \tan \alpha = \nu t \sin \alpha - \frac{gt^{2}}{2} $$ | ||
| + | From this equation we can express time $t$: | ||
| + | $$ t = \frac{2u \tan \alpha}{g} $$ | ||
| + | We substitute the time value $t$ back into the equation for height: | ||
| + | $$ \fbox{$h = \frac{2u \tan^{2} \alpha}{g} (\nu \cos \alpha - u)$} $$ | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$h=\frac{2utg^{2}\alpha}{g}(\nu cos\alpha -u)$$ | ||
| + | </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> | ||
| + | </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 +1,115 @@ | |||
| <!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="The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?"> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?"> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>The duck was flying in a horizontal straight line with a constant speed u. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone v was directed just at the duck at an angle \alpha to the horizon. At what height did the duck fly, if the rock still hit it?</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> | |||
| $1.3.10.$ The duck was flying in a horizontal straight line with a constant speed $u$. An inexperienced "hunter" threw a stone at it, and the throw was made without pre-emption, i.e. at the time of the throw, the speed of the stone $v$ was directed just at the duck at an angle $\alpha$ to the horizon. At what height did the duck fly, if the rock still hit it? | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.3.10/statement.png" | |||
| loading="lazy" width="250" /> | |||
| <figcaption> | |||
| For problem $1.3.10$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.3.10/01.png" | |||
| loading="lazy" width="300" /> | |||
| <figcaption> | |||
| Direction of duck's flying | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Let's consider this equation of motion: | |||
| $$ y = h = \nu \sin \alpha - \frac{gt^{2}}{2} $$ | |||
| It is known that: | |||
| $$ \left\{\begin{matrix} | |||
| S_{yt}=ut \\ | |||
| S_{o}=\nu t \cos \alpha | |||
| \end{matrix}\right. $$ | |||
| For angle $\alpha$: | |||
| $$ \tan \alpha = \frac{h}{S_{o} - S_{yt}} = \frac{h}{t(\nu \cos \alpha - u)} $$ | |||
| We substitute the value of the angle $\alpha$ into the equation for the height: | |||
| $$ h = t (\nu \cos \alpha - u) \tan \alpha = \nu t \sin \alpha - \frac{gt^{2}}{2} $$ | |||
| From this equation we can express time $t$: | |||
| $$ t = \frac{2u \tan \alpha}{g} $$ | |||
| We substitute the time value $t$ back into the equation for height: | |||
| $$ \fbox{$h = \frac{2u \tan^{2} \alpha}{g} (\nu \cos \alpha - u)$} $$ | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$h=\frac{2utg^{2}\alpha}{g}(\nu cos\alpha -u)$$ | |||
| </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> | |||
| </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> | |||