Translated 1.5.11-1.5.20

astrosander edited Sep 17, 2024, 12:43 AM

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		+	<meta name="description" content="Plot an approximate graph of the speed of point B as a function of time, if the speed v_A of point A is constant. Find the formula for this relationship if x(0) = 0.">
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		+	<meta name="date" content="2023-10" scheme="YYYY-MM">
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		+	<title>Plot an approximate graph of the speed of point B as a function of time, if the speed v_A of point A is constant. Find the formula for this relationship if x(0) = 0.</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="../../#1.5">$\leftarrow$Back</a></h3>
		+
		+	<h3> Statement </h3>
		+	<p>
		+	$1.5.15^*.$ Plot an approximate graph of the speed of point $B$ as a function of time, if the speed $v_A$ of point $A$ is constant. Find the formula for this relationship if $x(0) = 0$.
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/1/1.5.15/statement.png"
		+	loading="lazy" width="250" />
		+	<figcaption>
		+	For problem $1.5.15^*$
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+	<center>
		+	<figure>
		+	<img src="https://savchenkosolutions.com/1/1.5.15/draw.png"
		+	loading="lazy" width="300" />
		+	<figcaption>
		+	Velocity distribution on threads
		+	</figcaption>
		+	</figure>
		+	</center>
		+
		+	<p>NO: Before viewing the solution to this problem, I advise you to familiarize yourself with the solution <a href="../1.5.14">1.5.14</a></p>
		+
		+	<p>At time $t$, the height to which the point dropped</p>
		+	$$ x = v_A t\;(1) $$
		+	<p>Let's consider the change in the length of the thread over a small period of time $dt$
		+	</p>
		+	$$ dl = \sqrt{L^2 + (x+dx)^2}-\sqrt{L^2 + x^2} $$
		+
		+	$$ dl = \sqrt{L^2 + x^2}\cdot \left(\sqrt{1 + \frac{2xdx}{L^2 + x^2}}-1\right) $$
		+	<p>We will use the formula for small quantities $(1+x)^\alpha \approx 1+\alpha x$, where $x\rightarrow 0$:</p>
		+	$$ dl = \frac{xdx}{\sqrt{L^2 + x^2}} $$
		+	<p>Given that $v_B = \frac{dl}{dt}$ and $v_A = \frac{dx}{dt}$</p>
		+	$$ v_B = \frac{x}{\sqrt{L^2 + x^2}} \frac{dx}{dt} $$
		+
		+	$$ v_B = v_A\frac{x}{\sqrt{L^2 + x^2}} $$
		+	<p>Substitute $(1):$ </p>
		+	$$ \fbox{$v_B = \frac{v_A^2t}{\sqrt{L^2 + v_A^2t^2}}$} $$
		+	<p>NO: A more detailed and beautiful problem with a similar idea can be found in <a href="https://belphol.github.io/books/LongProblemsPart1.pdf" target="_blank">"Very Long Physics Problems"</a> by A.I. Slobodyanyuk (Problem 2)</p>
		+	</p>
		+
		+	<h4>Answer: </h4>
		+	<p>$$v_B = \frac{v_A^2t}{\sqrt{L^2 + v_A^2t^2}}$$</p>
		+
		+
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		+	<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> alex@savchenkosolutions.com <br></small>
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	@@ -0,0 +1,110 @@
			<!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="Plot an approximate graph of the speed of point B as a function of time, if the speed v_A of point A is constant. Find the formula for this relationship if x(0) = 0.">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="Plot an approximate graph of the speed of point B as a function of time, if the speed v_A of point A is constant. Find the formula for this relationship if x(0) = 0.">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="Plot an approximate graph of the speed of point B as a function of time, if the speed v_A of point A is constant. Find the formula for this relationship if x(0) = 0.">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>Plot an approximate graph of the speed of point B as a function of time, if the speed v_A of point A is constant. Find the formula for this relationship if x(0) = 0.</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">
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			<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="../../#1.5">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$1.5.15^*.$ Plot an approximate graph of the speed of point $B$ as a function of time, if the speed $v_A$ of point $A$ is constant. Find the formula for this relationship if $x(0) = 0$.
			</p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/1/1.5.15/statement.png"
			loading="lazy" width="250" />
			<figcaption>
			For problem $1.5.15^*$
			</figcaption>
			</figure>
			</center>
			<p>
			</p>

			<h3>Solution</h3>
			<p>
			<center>
			<figure>
			<img src="https://savchenkosolutions.com/1/1.5.15/draw.png"
			loading="lazy" width="300" />
			<figcaption>
			Velocity distribution on threads
			</figcaption>
			</figure>
			</center>

			<p>NO: Before viewing the solution to this problem, I advise you to familiarize yourself with the solution <a href="../1.5.14">1.5.14</a></p>

			<p>At time $t$, the height to which the point dropped</p>
			$$ x = v_A t\;(1) $$
			<p>Let's consider the change in the length of the thread over a small period of time $dt$
			</p>
			$$ dl = \sqrt{L^2 + (x+dx)^2}-\sqrt{L^2 + x^2} $$

			$$ dl = \sqrt{L^2 + x^2}\cdot \left(\sqrt{1 + \frac{2xdx}{L^2 + x^2}}-1\right) $$
			<p>We will use the formula for small quantities $(1+x)^\alpha \approx 1+\alpha x$, where $x\rightarrow 0$:</p>
			$$ dl = \frac{xdx}{\sqrt{L^2 + x^2}} $$
			<p>Given that $v_B = \frac{dl}{dt}$ and $v_A = \frac{dx}{dt}$</p>
			$$ v_B = \frac{x}{\sqrt{L^2 + x^2}} \frac{dx}{dt} $$

			$$ v_B = v_A\frac{x}{\sqrt{L^2 + x^2}} $$
			<p>Substitute $(1):$ </p>
			$$ \fbox{$v_B = \frac{v_A^2t}{\sqrt{L^2 + v_A^2t^2}}$} $$
			<p>NO: A more detailed and beautiful problem with a similar idea can be found in <a href="https://belphol.github.io/books/LongProblemsPart1.pdf" target="_blank">"Very Long Physics Problems"</a> by A.I. Slobodyanyuk (Problem 2)</p>
			</p>

			<h4>Answer: </h4>
			<p>$$v_B = \frac{v_A^2t}{\sqrt{L^2 + v_A^2t^2}}$$</p>


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			<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> alex@savchenkosolutions.com <br></small>
			</p>
			</footer>
			</body>

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