Added 1.2.11-1.2.14

astrosander edited
View solution History
revision #2905 GitHub bc0d954 newer →
en/1.2.14.md +117 −0
@@ -0,0 +1,117 @@
+<!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="According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}.">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}.">
+ <meta property="og:image" content="img/logo.png">
+ <meta property="og:description" content="According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}.">
+ <meta name="yandex-verification" content="6cfda41f74038368">
+ <title>According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}.</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="../#1.2">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $1.2.14.$ According to the graph of acceleration versus time, set the speed at times $4$ and $15$ s, if at time $1$ s the speed is $3$ $\frac{m}{s}$.
+
+</p>
+<center>
+ <figure>
+ <img src="statement.png"
+ loading="lazy" width="230" />
+ <figcaption>
+ For problem $1.2.14$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ Acceleration depends on time, as:
+$$\left\{\begin{matrix}
+a(t) = 0\;m/s^2,\; 0\text{ s} \leq t \leq 2\text{ s}\\
+a(t) = 20t,\; 2\text{ s} \leq t \leq 5\text{ s}\\
+a(t) = 60\;m/s^2,\; 5\text{ s} \leq t \leq 9\text{ s}\\
+a(t) = 60-20(t-9),\; 9\text{ s} \leq t \leq 12\text{ s}\\
+a(t) = 0, t> 12\text{ s}
+\end{matrix}\right.$$
+
+Given that the area under the graph of acceleration vs. time is velocity vs. time. Then the velocity depends on time as:
+
+$$\left\{\begin{matrix}
+v(t) = 3\;m/s,\; 0\text{ s} \leq t \leq 2\text{ s}\\
+v(t) = 3+20t^2/2,\; 2\text{ s} \leq t \leq 5\text{ s}\\
+v(t) = 93+60(t-5),\; 5\text{ s} \leq t \leq 9\text{ s}\\
+v(t) = 333 - 20 (t - 9)^2/2 + 60 (t - 9),\; 9\text{ s} \leq t \leq 12\text{ s}\\
+v(t) = 423\;m/s, t> 12\text{ s}
+\end{matrix}\right.$$
+
+The graph of this dependence is presented below:
+</p>
+ <center>
+ <figure>
+ <img src="graph.png" alt="1.2.14"
+ loading="lazy" width="350" />
+ <figcaption>
+ Dependence of velocity on time
+ </figcaption>
+ </figure>
+</center>
+<p>
+It follows that at $4$ and $15\text{ s}$ the velocity is $43$ and $423\;m/s$, respectively.
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$v_1=43\;m/s; \;v_2=423\;m/s.$$
+ </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> astrosander01@gmail.com <br></small>
+ </p>
+ </footer>
+ </body>
+
+</html>