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		@@ -33,7 +33,7 @@
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		−	<h2>Solutions of Savchenko P~~hysics Textbook~~</h2>
		+	<h2>Solutions of Savchenko Problems in Physics</h2>
			<p class="author">
			Aliaksandr Melnichenka <br/>
			October 2023
			</p>
			</header>

			<h3 id="back-link"><a href="../#1.1">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$1.1.22.$ Inside a fixed smooth-walled cylinder of radius $R$ a small ball is flying, elastically reflecting from the walls so that the minimal distance from it to the cylinder axis is $h$. What fraction of time is the distance from the cylinder axis less than $r$ but greater than $h$?

			</p>
			<center>
			<figure>
			<img src="statement.png"
			loading="lazy" width="200" />
			<figcaption>
			For problem $1.1.22$
			</figcaption>
			</figure>
			</center>
			<p>
			</p>

			<h3>Solution</h3>
			<p>

			<center>
			<figure>
			<img src="animation.gif" alt="1.1.22"
			loading="lazy" width="250" />
			<figcaption>
			Impact on cylinder walls
			</figcaption>
			</figure>
			</center>

			<p>
			Let's consider the motion of the ball described in the problem condition
			</p>
			<center>
			<figure>
			<img src="drawing.png" alt="1.1.22"
			loading="lazy" width="250" />
			<figcaption>
			The path of the ball between impacts
			</figcaption>
			</figure>
			</center>
			<p>
			Since the impact velocity does not change between impacts, and impacts are absolutely elastic, we can consider the velocity unchanged. Consequently, for equal time intervals the body passes equal distances.
			</p>
			<p>
			Then the required ratio $\frac{\Delta t}{t}$ is found as the ratio $\frac{BC}{AD}$
			</p>
			<p style="text-align: center;">
			$$\frac{\Delta t}{t} = \frac{BC}{AD}$$
			</p>
			<p>
			By Pythagoras' theorem:
			</p>
			<p style="text-align: center;">
			$BC = 2 \sqrt{r^2-h^2}$
			</p>
			<p style="text-align: center;">
			$AD = 2 \sqrt{R^2-h^2}$
			</p>
			<p>
			From where
			</p>
			<p style="text-align: center;">
			$$\frac{BC}{AD} = \frac{\sqrt{r^2-h^2}}{\sqrt{R^2-h^2}}$$
			</p>
			<p>
			Or
			</p>
			<p style="text-align: center;">
			$$\frac{\Delta t}{t} = \frac{\sqrt{r^2-h^2}}{\sqrt{R^2-h^2}}$$
			</p>
			</p>

			<h4>Answer</h4>
			<p>
			$$\Delta t/t=\sqrt{(r^2-h^2)/(R^2-h^2)}$$
			</p>


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			<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
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		unchanged lines 97

	<!DOCTYPE html>		<!DOCTYPE html>
	<html lang="en">		<html lang="en">

	<head>		<head>
	<meta charset="utf-8">		<meta charset="utf-8">
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	<meta name="yandex-verification" content="6cfda41f74038368">		<meta name="yandex-verification" content="6cfda41f74038368">
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	@@ -33,7 +33,7 @@
	</head>		</head>
	<body style="">		<body style="">
	<header style="text-align:center;">		<header style="text-align:center;">
	<h2>Solutions of Savchenko P~~hysics Textbook~~</h2>		<h2>Solutions of Savchenko Problems in Physics</h2>
	<p class="author">		<p class="author">
	Aliaksandr Melnichenka <br/>		Aliaksandr Melnichenka <br/>
	October 2023		October 2023
	</p>		</p>
	</header>		</header>

	<h3 id="back-link"><a href="../#1.1">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../#1.1">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>
	<p>		<p>
	$1.1.22.$ Inside a fixed smooth-walled cylinder of radius $R$ a small ball is flying, elastically reflecting from the walls so that the minimal distance from it to the cylinder axis is $h$. What fraction of time is the distance from the cylinder axis less than $r$ but greater than $h$?		$1.1.22.$ Inside a fixed smooth-walled cylinder of radius $R$ a small ball is flying, elastically reflecting from the walls so that the minimal distance from it to the cylinder axis is $h$. What fraction of time is the distance from the cylinder axis less than $r$ but greater than $h$?

	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="statement.png"		<img src="statement.png"
	loading="lazy" width="200" />		loading="lazy" width="200" />
	<figcaption>		<figcaption>
	For problem $1.1.22$		For problem $1.1.22$
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	</p>		</p>

	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>

	<center>		<center>
	<figure>		<figure>
	<img src="animation.gif" alt="1.1.22"		<img src="animation.gif" alt="1.1.22"
	loading="lazy" width="250" />		loading="lazy" width="250" />
	<figcaption>		<figcaption>
	Impact on cylinder walls		Impact on cylinder walls
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>

	<p>		<p>
	Let's consider the motion of the ball described in the problem condition		Let's consider the motion of the ball described in the problem condition
	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="drawing.png" alt="1.1.22"		<img src="drawing.png" alt="1.1.22"
	loading="lazy" width="250" />		loading="lazy" width="250" />
	<figcaption>		<figcaption>
	The path of the ball between impacts		The path of the ball between impacts
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	Since the impact velocity does not change between impacts, and impacts are absolutely elastic, we can consider the velocity unchanged. Consequently, for equal time intervals the body passes equal distances.		Since the impact velocity does not change between impacts, and impacts are absolutely elastic, we can consider the velocity unchanged. Consequently, for equal time intervals the body passes equal distances.
	</p>		</p>
	<p>		<p>
	Then the required ratio $\frac{\Delta t}{t}$ is found as the ratio $\frac{BC}{AD}$		Then the required ratio $\frac{\Delta t}{t}$ is found as the ratio $\frac{BC}{AD}$
	</p>		</p>
	<p style="text-align: center;">		<p style="text-align: center;">
	$$\frac{\Delta t}{t} = \frac{BC}{AD}$$		$$\frac{\Delta t}{t} = \frac{BC}{AD}$$
	</p>		</p>
	<p>		<p>
	By Pythagoras' theorem:		By Pythagoras' theorem:
	</p>		</p>
	<p style="text-align: center;">		<p style="text-align: center;">
	$BC = 2 \sqrt{r^2-h^2}$		$BC = 2 \sqrt{r^2-h^2}$
	</p>		</p>
	<p style="text-align: center;">		<p style="text-align: center;">
	$AD = 2 \sqrt{R^2-h^2}$		$AD = 2 \sqrt{R^2-h^2}$
	</p>		</p>
	<p>		<p>
	From where		From where
	</p>		</p>
	<p style="text-align: center;">		<p style="text-align: center;">
	$$\frac{BC}{AD} = \frac{\sqrt{r^2-h^2}}{\sqrt{R^2-h^2}}$$		$$\frac{BC}{AD} = \frac{\sqrt{r^2-h^2}}{\sqrt{R^2-h^2}}$$
	</p>		</p>
	<p>		<p>
	Or		Or
	</p>		</p>
	<p style="text-align: center;">		<p style="text-align: center;">
	$$\frac{\Delta t}{t} = \frac{\sqrt{r^2-h^2}}{\sqrt{R^2-h^2}}$$		$$\frac{\Delta t}{t} = \frac{\sqrt{r^2-h^2}}{\sqrt{R^2-h^2}}$$
	</p>		</p>
	</p>		</p>

	<h4>Answer</h4>		<h4>Answer</h4>
	<p>		<p>
	$$\Delta t/t=\sqrt{(r^2-h^2)/(R^2-h^2)}$$		$$\Delta t/t=\sqrt{(r^2-h^2)/(R^2-h^2)}$$
	</p>		</p>


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	<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>		<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>
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