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| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
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| <title>The shooter tries to hit a disk of radius R, which moves from one wall to another at constant modulo velocity so fast that it cannot be tracked. Draw a graph of the probability of hitting the disk as a function of the distance between the aiming point and the left wall.</title> | | <title>The shooter tries to hit a disk of radius R, which moves from one wall to another at constant modulo velocity so fast that it cannot be tracked. Draw a graph of the probability of hitting the disk as a function of the distance between the aiming point and the left wall.</title> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.1.23^*.$ The shooter tries to hit a disk of radius $R$, which moves from one wall to another at constant modulo velocity so fast that it cannot be tracked. Draw a graph of the probability of hitting the disk as a function of the distance between the aiming point and the left wall. | | $1.1.23^*.$ The shooter tries to hit a disk of radius $R$, which moves from one wall to another at constant modulo velocity so fast that it cannot be tracked. Draw a graph of the probability of hitting the disk as a function of the distance between the aiming point and the left wall. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| For problem $1.1.23^*$ | | For problem $1.1.23^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| The shots are fired at height $R$ from the floor perpendicular to the direction of motion of the disc. At what point is the least and most probable shot? What is their value? Consider the cases $L > 4R$, $4R > L > 2R$, where $L$ is the distance between the walls. | | The shots are fired at height $R$ from the floor perpendicular to the direction of motion of the disc. At what point is the least and most probable shot? What is their value? Consider the cases $L > 4R$, $4R > L > 2R$, where $L$ is the distance between the walls. |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="animation.gif" alt="1.1.23" | | <img src="animation.gif" alt="1.1.23" |
| loading="lazy" width="300" /> | | loading="lazy" width="300" /> |
| <figcaption> | | <figcaption> |
| Wheels moving | | Wheels moving |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p> | | <p> |
| When the target is moving, a bystander is less likely to see the target from the edge, as it stays there for a shorter period of time. | | When the target is moving, a bystander is less likely to see the target from the edge, as it stays there for a shorter period of time. |
| </p> | | </p> |
| <p> | | <p> |
| Thus, the maximum probability will be observed on the interval: | | Thus, the maximum probability will be observed on the interval: |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$\left\{\begin{matrix} | | $$\left\{\begin{matrix} |
| 0 \leq x < 2R\\ | | 0 \leq x < 2R\\ |
| L-2R < x \leq L | | L-2R < x \leq L |
| \end{matrix}\right.$$ | | \end{matrix}\right.$$ |
| </p> | | </p> |
| <p> | | <p> |
| Since the velocity of the target is constant, and finding the target in the range $2R \leq x \leq L-2R$ is equally likely, the probability of finding it in this range is equal to the ratio of the diameter to the length of this range: $\frac{2R}{L-2R}$. | | Since the velocity of the target is constant, and finding the target in the range $2R \leq x \leq L-2R$ is equally likely, the probability of finding it in this range is equal to the ratio of the diameter to the length of this range: $\frac{2R}{L-2R}$. |
| </p> | | </p> |
| <p> | | <p> |
| Graphs corresponding to these modes: | | Graphs corresponding to these modes: |
| </p> | | </p> |
| | | |
| <center> | | <center> |
| <figure> | | <figure> |
| <img id="sol" src="sol.png" alt="1.1.23" | | <img id="sol" src="sol.png" alt="1.1.23" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Probability of a hit as a function of aiming distance | | Probability of a hit as a function of aiming distance |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| <a href="#sol">See Fig.</a> Zero at the walls. The largest at any place at a distance from the walls, | | <a href="#sol">See Fig.</a> Zero at the walls. The largest at any place at a distance from the walls, |
| greater than $2R$, and equal to $2R/(L-2R)$ at $L > 4R$; at any place at a distance from the walls greater than $L - 2R$, and equal to one at $4R > L > 2R$. | | greater than $2R$, and equal to $2R/(L-2R)$ at $L > 4R$; at any place at a distance from the walls greater than $L - 2R$, and equal to one at $4R > L > 2R$. |
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