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.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

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

    <h3> Statement </h3>
    <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.

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.

    <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.$$

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: $\frac{2R}{L-2R}$.

Graphs corresponding to these modes:

          <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 $4R > L > 2R$.

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