Translated 1.4.1-1.4.18

astrosander edited
View solution History
revision #5471 GitHub 54d1163 newer →
en/1.4.12.md +114 −0
@@ -0,0 +1,114 @@
+<!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="A body flies horizontally at speed v into the space between two vertical walls that move at speed u. Determine the speed of the body after the n-th impact on the front wall. Distance between walls L. The blows are absolutely elastic.">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="A body flies horizontally at speed v into the space between two vertical walls that move at speed u. Determine the speed of the body after the n-th impact on the front wall. Distance between walls L. The blows are absolutely elastic.">
+ <meta property="og:image" content="img/logo.png">
+ <meta property="og:description" content="A body flies horizontally at speed v into the space between two vertical walls that move at speed u. Determine the speed of the body after the n-th impact on the front wall. Distance between walls L. The blows are absolutely elastic.">
+ <meta name="yandex-verification" content="6cfda41f74038368">
+ <title>A body flies horizontally at speed v into the space between two vertical walls that move at speed u. Determine the speed of the body after the n-th impact on the front wall. Distance between walls L. The blows are absolutely elastic.</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="../../#1.4">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $1.4.12.$ A body flies horizontally at speed $v$ into the space between two vertical walls that move at speed $u$. Determine the speed of the body after the $n$-th impact on the front wall. Distance between walls $L$. The blows are absolutely elastic.
+</p>
+<center>
+ <figure>
+ <img src="https://savchenkosolutions.com/1/1.4.12/statement.png"
+ loading="lazy" width="180" />
+ <figcaption>
+ For problem $1.4.12$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ <center>
+<figure>
+<img src="https://savchenkosolutions.com/1/1.4.12/animation.gif"
+loading="lazy" width="350" />
+<figcaption>
+Motion animation
+</figcaption>
+</figure>
+</center>
+
+<p>Let's consider how the velocity $v_x$ changes along the horizontal axis $Ox$</p>
+<p>When the body flies to the right wall, $v_x = v$. Then, after an elastic collision, the velocity changes to $v_x = v - 2u$. For more details on why this happens, see <a href="../1.4.9" target="_blank">1.4.9</a>.</p>
+<p>Similarly, after hitting the left wall, $v_x$ will be equal to </p>
+<p class="exp">$$v_x = (v - 2u) + 2u = v$$</p>
+<p>Thus, always after hitting the right wall, the speed of $v_x$ will be equal to $v - 2u$</p>
+<p class="exp">$$v_x = v - 2u$$</p>
+<p>And the time between $n$ and $n-1$ hits</p>
+<p class="exp">
+$$ t_n - t_{n-1} = \frac{L}{v - 2u} + \frac{L}{v} $$
+</p>
+ <p class="exp">
+$$ t_n - t_{n-1} = L \frac{2(v-u)}{v(v-2u)} $$
+</p>
+<p>Since there are no collisions in the vertical plane, the velocity $v_y$ depends on the time $t$ as</p>
+<p class="exp">$$v_y = gt$$</p>
+<p>And in the general case:</p>
+<p class="exp">$$v_y = g(2n-1)(t_n - t_{n-1})$$</p>
+<p class="exp">
+$$ \fbox{$v_y = (2n − 1)g L \frac{2(v-u)}{v(v-2u)}$} $$
+</p>
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ The projection of the velocity on the horizontal direction $v_x = v −2u$; the projection of the velocity on the vertical direction $$v_y = (2n − 1)g L \frac{2(v-u)}{v(v-2u)}$$
+ </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>