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| <title>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?</title> | | <title>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?</title> |
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| <h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3> |
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| <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 $h$? |
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| <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 $1.1.22$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
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| <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> |
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| <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> |
| $$\Delta t/t=\sqrt{(r^2-h^2)/(R^2-h^2)}$$ | | $$\Delta t/t=\sqrt{(r^2-h^2)/(R^2-h^2)}$$ |
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