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+ <header style="text-align:center;">
+ <h2>Solutions of Savchenko Problems in Physics</h2>
+ <p class="author">
+ Aliaksandr Melnichenka <br/>
+ October 2023
+ </p>
+ </header>
+
+ <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.
+</p>
+<center>
+ <figure>
+ <img src="https://savchenkosolutions.com/1/1.3.16/statement.png"
+ loading="lazy" width="200" />
+ <figcaption>
+ For problem $1.3.16^*$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ <p>a) Find the condition under which there will be no collisions$(x=0)$</p>
+<p>From the law of conservation of energy:</p>
+<p class="exp">
+$$ \frac{mv_0^2}{2} = mgl \cdot \sin \alpha $$
+</p>
+ <p class="exp">
+$$ v_{0}\leq\frac{\sqrt{2gl\sin \alpha}}{\cos\alpha} $$
+</p>
+<p>In this case, time will pass</p>
+<p class="exp">
+$$ t = \frac{2v_0}{g} \text{ctg} \alpha $$
+</p>
+<p>b) Now let's look at those cases when touching occurs:</p>
+<p class="exp">
+$$ v_{0}>\frac{\sqrt{2gl\sin \alpha}}{\cos\alpha} $$
+</p>
+<p>Path between two touches:</p>
+<p class="exp">
+$$ L=v_{0}\cos\alpha t-\frac{g\sin\alpha t^{2}}{2} $$
+</p>
+<p>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</p>
+<p class="exp">
+$$ t=\frac{v_{0}\cos\alpha-\sqrt{v_{0}^{2}\cos^{2}\alpha-2g\sin\alpha L}}{g\sin\alpha} $$
+</p>
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $\begin{aligned}&t=\frac{2v}{g}\operatorname{ctg}\alpha\text{ with }v\cos\alpha<\sqrt{2gl\sin\alpha};\\&t=\frac vg\operatorname{ctg}\alpha\bigg(1-\sqrt{1-\frac{2gl\operatorname{tg}\alpha}{v^2\cos\alpha}}\bigg)\text{ with }v\cos\alpha>\sqrt{2gl\sin\alpha}.\end{aligned}$
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