A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?

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>
        $1.1.20^*.$ A ball is launched along a pool table with sides $a$ and $b$ from the middle of side $b$. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?

    <h3>Solution</h3>
    <p>
        
        <p>
          By analogy with <a href="/1.1.18">1.1.18</a>, we can use the Image Method and represent elastic walls as optical mirrors.  
        </p>
          <center>
          <figure>
            <img src="sol.png" alt="1.1.20" 
              loading="lazy" width="400" />
            <figcaption>
              Arrangement of image-dots
            </figcaption>
          </figure>
          </center>

        <p>
          To get to the starting position, all you have to do is hit any of the picture holes. 
        </p>
        <p>
          The coordinates of the holes are described by the expression: 
        </p>
        <p style="text-align: center;">
          $x=2ma$ и $y=nb$, где $m$ и $n$ — any integers
        </p>
        <p>
          Whence the desired angle:" 
        </p>
        <p style="text-align: center;">
          $\alpha = arctg (2ma/(nb))$ 
        </p>
        
    </p>

    <h4>Answer</h4>
    <p>
        $\tan \alpha = 2ma/(nb)$, where $m$ and $n$ are any integers
    </p>


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