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+ <header style="text-align:center;">
+ <h2>Solutions of Savchenko Physics Textbook</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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