| <!DOCTYPE html> | | <!DOCTYPE html> |
| <html lang="en"> | | <html lang="en"> |
| | | |
| <head> | | <head> |
| <meta charset="utf-8"> | | <meta charset="utf-8"> |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | | <meta name="viewport" content="width=device-width, initial-scale=1.0"> |
| <meta http-equiv="content-language" content="en"> | | <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="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> |
| <meta name="description" content="The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?"> | | <meta name="description" content="The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?"> |
| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
| <meta property="og:title" content="The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?"> | | <meta property="og:title" content="The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?"> | | <meta property="og:description" content="The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?</title> | | <title>The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed v, forming at some point in time an angle \alpha with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time?</title> |
| <link rel="stylesheet" href="../../css/css-latex/style.css"> | | <link rel="stylesheet" href="../../css/css-latex/style.css"> |
| <link rel="icon" href="../../img/logo.png" type="image/png"> | | <link rel="icon" href="../../img/logo.png" type="image/png"> |
| <script src="../../js/jquery-1.10.1.min.js"></script> | | <script src="../../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 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"> | | <script type="text/x-mathjax-config"> |
| MathJax.Hub.Config({ | | MathJax.Hub.Config({ |
| extensions: ['tex2jax.js'], | | extensions: ['tex2jax.js'], |
| jax: ['input/TeX', 'output/HTML-CSS'], | | jax: ['input/TeX', 'output/HTML-CSS'], |
| tex2jax: { | | tex2jax: { |
| inlineMath: [['$', '$'], ['$', '$']], | | inlineMath: [['$', '$'], ['$', '$']], |
| processEscapes: true, | | processEscapes: true, |
| processClass: 'tex2jax', | | processClass: 'tex2jax', |
| ignoreClass: 'html' | | ignoreClass: 'html' |
| }, | | }, |
| showProcessingMessages: false, | | showProcessingMessages: false, |
| messageStyle: 'none' | | messageStyle: 'none' |
| }); | | }); |
| </script> | | </script> |
| </head> | | </head> |
| <body style=""> | | <body style=""> |
| <header style="text-align:center;"> | | <header style="text-align:center;"> |
| <a href="../../" style="text-decoration: none;"> | | <a href="../../" style="text-decoration: none;"> |
| <div id="logo"> | | <div id="logo"> |
| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
| </div> | | </div> |
| </a> | | </a> |
| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.13.$ The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed $v$, forming at some point in time an angle $\alpha$ with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time? | | $1.5.13.$ The rope tied to the boat is pulled by the free end so that it does not sag. The boat moves at a constant speed $v$, forming at some point in time an angle $\alpha$ with the length of rope located between the pole and the boat. How fast should the free end of the rope be pulled at this point in time? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.13/statement.png" | | <img src="https://savchenkosolutions.com/1/1.5.13/statement.png" |
| loading="lazy" width="210" /> | | loading="lazy" width="210" /> |
| <figcaption> | | <figcaption> |
| For problem $1.5.13$ | | For problem $1.5.13$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.13/draw.png" | | <img src="https://savchenkosolutions.com/1/1.5.13/draw.png" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p>Если веревка не провисает, то</p> | | <p>Если веревка не провисает, то</p> |
| $$ \fbox{$u=\upsilon\cos\alpha$} $$ | | $$ \fbox{$u=\upsilon\cos\alpha$} $$ |
| <p>And since all points of the rope have the same speed, the rope must be pulled with exactly the speed $u$.</p> | | <p>And since all points of the rope have the same speed, the rope must be pulled with exactly the speed $u$.</p> |
