| <!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 flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic."> | | <meta name="description" content="A ball flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic."> |
| <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 flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic."> | | <meta property="og:title" content="A ball flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="A ball flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic."> | | <meta property="og:description" content="A ball flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>A ball flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic.</title> | | <title>A ball flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic.</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;"> |
| <div id = "logo"> | | <div id = "logo"> |
| <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.3.16^*.$ A ball flies into a tube of length $l$, inclined at an angle $\alpha$ to the horizon, with a horizontal velocity $v$. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic. | | $1.3.16^*.$ A ball flies into a tube of length $l$, inclined at an angle $\alpha$ to the horizon, with a horizontal velocity $v$. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.3.16/statement.png" | | <img src="https://savchenkosolutions.com/1/1.3.16/statement.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem $1.3.16^*$ | | For problem $1.3.16^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>a) Find the condition under which there will be no collisions$(x=0)$</p> | | <p>a) Find the condition under which there will be no collisions$(x=0)$</p> |
| <p>From the law of conservation of energy:</p> | | <p>From the law of conservation of energy:</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \frac{mv_0^2}{2} = mgl \cdot \sin \alpha $$ | | $$ \frac{mv_0^2}{2} = mgl \cdot \sin \alpha $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ v_{0}\leq\frac{\sqrt{2gl\sin \alpha}}{\cos\alpha} $$ | | $$ v_{0}\leq\frac{\sqrt{2gl\sin \alpha}}{\cos\alpha} $$ |
| </p> | | </p> |
| <p>In this case, time will pass</p> | | <p>In this case, time will pass</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ t = \frac{2v_0}{g} \text{ctg} \alpha $$ | | $$ t = \frac{2v_0}{g} \text{ctg} \alpha $$ |
| </p> | | </p> |
| <p>b) Now let's look at those cases when touching occurs:</p> | | <p>b) Now let's look at those cases when touching occurs:</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ v_{0}>\frac{\sqrt{2gl\sin \alpha}}{\cos\alpha} $$ | | $$ v_{0}>\frac{\sqrt{2gl\sin \alpha}}{\cos\alpha} $$ |
| </p> | | </p> |
| <p>Path between two touches:</p> | | <p>Path between two touches:</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ L=v_{0}\cos\alpha t-\frac{g\sin\alpha t^{2}}{2} $$ | | $$ L=v_{0}\cos\alpha t-\frac{g\sin\alpha t^{2}}{2} $$ |
| </p> | | </p> |
| <p>When we receive the required time, we choose the one that is smaller, because we need to find the time when it will come out</p> | | <p>When we receive the required time, we choose the one that is smaller, because we need to find the time when it will come out</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ t=\frac{v_{0}\cos\alpha-\sqrt{v_{0}^{2}\cos^{2}\alpha-2g\sin\alpha L}}{g\sin\alpha} $$ | | $$ t=\frac{v_{0}\cos\alpha-\sqrt{v_{0}^{2}\cos^{2}\alpha-2g\sin\alpha L}}{g\sin\alpha} $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $\begin{aligned}&t=\frac{2v}{g}\operatorname{ctg}\alpha\text{ with }v\cos\alpha<\sqrt{2gl\sin\alpha};\\&t=\frac vg\operatorname{ctg}\alpha\bigg(1-\sqrt{1-\frac{2gl\operatorname{tg}\alpha}{v^2\cos\alpha}}\bigg)\text{ with }v\cos\alpha>\sqrt{2gl\sin\alpha}.\end{aligned}$ | | $\begin{aligned}&t=\frac{2v}{g}\operatorname{ctg}\alpha\text{ with }v\cos\alpha<\sqrt{2gl\sin\alpha};\\&t=\frac vg\operatorname{ctg}\alpha\bigg(1-\sqrt{1-\frac{2gl\operatorname{tg}\alpha}{v^2\cos\alpha}}\bigg)\text{ with }v\cos\alpha>\sqrt{2gl\sin\alpha}.\end{aligned}$ |
| </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> alex@savchenkosolutions.com <br></small> | | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> |
| </p> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
