Translated 1.3.16-1.3.30
en/1.3.17.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="In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting."> | ||
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| + | <meta property="og:description" content="In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting."> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting.</title> | ||
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| + | </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> | ||
| + | $1.3.17.$ In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is $\Delta t$. The bottom of the box forms an angle $\alpha$ with the horizon. Find the speed of the ball immediately after hitting. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.3.17/statement.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | <figcaption> | ||
| + | For problem $1.3.17$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <p>The most important thing in this problem is that the ball bounces along the same trajectory. Indeed, in an elastic collision, the angle of incidence is equal to the angle of rebound. If the ball falls on the wall of the box so that its speed is not perpendicular to the wall, then when it bounces, its trajectory would differ from the trajectory of its approach - I showed this with an additional small drawing. Therefore, $v_2$ is perpendicular to the wall. And this means that the velocity vectors $v_1$ and $v_2$ are perpendicular to each other. | ||
| + | </p> | ||
| + | <p> | ||
| + | Let's draw a triangle of velocities. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.3.17/drawing.png" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Velocity triangle | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | </p>It follows that</p> | ||
| + | $$v_1 = g \tau \cos \alpha$$ | ||
| + | $$v_2 = g \tau \sin \alpha$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$v_{1}=g\Delta t\sin\alpha\text{, }v_{2}=g\Delta t\cos\alpha$$ | ||
| + | </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> aliaksandr@savchenkosolutions.com <br></small> | ||
| + | </p> | ||
| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,101 @@ | |||
| <!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="In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is \Delta t. The bottom of the box forms an angle \alpha with the horizon. Find the speed of the ball immediately after hitting.</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: [['$', '$'], ['$', '$']], | |||
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| 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> | |||
| $1.3.17.$ In a rectangular box, elastically hitting the bottom and the right wall, a ball jumps back and forth along one trajectory. The time interval between hitting the bottom and the wall is $\Delta t$. The bottom of the box forms an angle $\alpha$ with the horizon. Find the speed of the ball immediately after hitting. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.3.17/statement.png" | |||
| loading="lazy" width="200" /> | |||
| <figcaption> | |||
| For problem $1.3.17$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <p>The most important thing in this problem is that the ball bounces along the same trajectory. Indeed, in an elastic collision, the angle of incidence is equal to the angle of rebound. If the ball falls on the wall of the box so that its speed is not perpendicular to the wall, then when it bounces, its trajectory would differ from the trajectory of its approach - I showed this with an additional small drawing. Therefore, $v_2$ is perpendicular to the wall. And this means that the velocity vectors $v_1$ and $v_2$ are perpendicular to each other. | |||
| </p> | |||
| <p> | |||
| Let's draw a triangle of velocities. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.3.17/drawing.png" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Velocity triangle | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| </p>It follows that</p> | |||
| $$v_1 = g \tau \cos \alpha$$ | |||
| $$v_2 = g \tau \sin \alpha$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$v_{1}=g\Delta t\sin\alpha\text{, }v_{2}=g\Delta t\cos\alpha$$ | |||
| </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> aliaksandr@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||