A ball flies into a tube of length l, inclined at an angle \alpha to the horizon, with a horizontal velocity v. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic.

Savchenko Solutions

Solutions of Savchenko Problems in Physics
knowledge must be free

    <h3 id="back-link"><a href="/#1.3">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.3.16^*.$ A ball flies into a tube of length $l$, inclined at an angle $\alpha$ to the horizon, with a horizontal velocity $v$. Determine the time of the ball's stay in the pipe, if the ball hits its walls elastic.

    <h3>Solution</h3>
    <p>
        <p>a) Find the condition under which there will be no collisions$(x=0)$</p>

From the law of conservation of energy:

$$\frac{mv_0^2}{2} = mgl \cdot \sin\alpha$$

$$v_{0}\leq\frac{\sqrt{2gl\sin\alpha}}{\cos\alpha}$$

In this case, time will pass

$$t = \frac{2v_0}{g} \text{ctg} \alpha$$

b) Now let's look at those cases when touching occurs:

$$v_{0}>\frac{\sqrt{2gl\sin\alpha}}{\cos\alpha}$$

Path between two touches:

$$L=v_{0}\cos\alpha t-\frac{g\sin\alpha t^{2}}{2}$$

When we receive the required time, we choose the one that is smaller, because we need to find the time when it will come out

$$t=\frac{v_{0}\cos\alpha -\sqrt{v_{0}^{2}\cos^{2}\alpha -2g\sin\alpha L}}{g\sin\alpha}$$

    <h4>Answer</h4>
    <p>
        $\begin{aligned}&t=\frac{2v}{g}\operatorname{ctg}\alpha\text{ with }v\cos\alpha <\sqrt{2gl\sin\alpha};\\&t=\frac vg\operatorname{ctg}\alpha\bigg(1-\sqrt{1-\frac{2gl\operatorname{tg}\alpha}{v^2\cos\alpha}}\bigg)\text{ with }v\cos\alpha >\sqrt{2gl\sin\alpha}.\end{aligned}$
    </p>



    <p style="text-align: right; font-style: italic; font-size: 14;">   
      Aliaksandr Melnichenka<br>
    </p>
        <footer class="row container">
  <br>
    <p>
        <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
    </p>
    <p>
        <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>
    </p>
</footer>