<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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.">
<meta property="og:title" content="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.">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="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>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>
$1.1.23^*.$ 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>
<center>
<figure>
<img src="statement.png"
loading="lazy" width="230" />
<figcaption>
For problem $1.1.23^*$
</figcaption>
</figure>
</center>
<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.
</p>
<h3>Solution</h3>
<p>
<center>
<figure>
<img src="animation.gif" alt="1.1.23"
loading="lazy" width="300" />
<figcaption>
Wheels moving
</figcaption>
</figure>
</center>
<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.
</p>
<p>
Thus, the maximum probability will be observed on the interval:
</p>
<p style="text-align: center;">
$$\left\{\begin{matrix}
0 \leq x < 2R\\
L-2R < x \leq L
\end{matrix}\right.$$
</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: $\frac{2R}{L-2R}$.
</p>
<p>
Graphs corresponding to these modes:
</p>
<center>
<figure>
<img id="sol" src="sol.png" alt="1.1.23"
loading="lazy" width="350" />
<figcaption>
Probability of a hit as a function of aiming distance
</figcaption>
</figure>
</center>
</p>
<h4>Answer</h4>
<p>
<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 $4R > L > 2R$.
<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>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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.">
<meta name="description" content="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.">
<meta property="og:title" content="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.">
<meta property="og:title" content="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.">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="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.">
<meta property="og:description" content="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>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>
$1.1.23^*.$ 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.
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
<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 $4R > L > 2R$.
<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>
<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>
