| <!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="A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?"> | | <meta name="description" content="A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?"> |
| <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="A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?"> | | <meta property="og:title" content="A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?"> | | <meta property="og:description" content="A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?</title> | | <title>A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?</title> |
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | | <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"> | | <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 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 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;"> |
| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| $1.1.20^*.$ A ball is launched along a pool table with sides $a$ and $b$ from the middle of side $b$. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement? | | $1.1.20^*.$ A ball is launched along a pool table with sides $a$ and $b$ from the middle of side $b$. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement? |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| For problem $1.1.20^*$ | | For problem $1.1.20^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| <p> | | <p> |
| By analogy with <a href="../1.1.18">1.1.18</a>, we can use the Image Method and represent elastic walls as optical mirrors. | | By analogy with <a href="../1.1.18">1.1.18</a>, we can use the Image Method and represent elastic walls as optical mirrors. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="sol.png" alt="1.1.20" | | <img src="sol.png" alt="1.1.20" |
| loading="lazy" width="400" /> | | loading="lazy" width="400" /> |
| <figcaption> | | <figcaption> |
| Arrangement of image-dots | | Arrangement of image-dots |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p> | | <p> |
| To get to the starting position, all you have to do is hit any of the picture holes. | | To get to the starting position, all you have to do is hit any of the picture holes. |
| </p> | | </p> |
| <p> | | <p> |
| The coordinates of the holes are described by the expression: | | The coordinates of the holes are described by the expression: |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $x=2ma$ и $y=nb$, где $m$ и $n$ — any integers | | $x=2ma$ и $y=nb$, где $m$ и $n$ — any integers |
| </p> | | </p> |
| <p> | | <p> |
| Whence the desired angle:" | | Whence the desired angle:" |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $\alpha = arctg (2ma/(nb))$ | | $\alpha = arctg (2ma/(nb))$ |
| </p> | | </p> |
| | | |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $\tan \alpha = 2ma/(nb)$, where $m$ and $n$ are any integers | | $\tan \alpha = 2ma/(nb)$, where $m$ and $n$ are any integers |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <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> | | <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> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
