Translated 1.3.3-1.3.15; Fixed GUI image generator

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+ <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>
+ $1.3.15.$ A ball is launched along the inner surface of a smooth vertical cylinder of radius $R$ at an angle $\alpha$ to the vertical. What initial velocity must be given to it so that it returns to the starting point?
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+ <p>We will divide the movement into vertical and in the plane of the cross section of the cylinder</p>
+<p class="exp">
+$x = vt \sin \alpha$
+</p>
+<p class="exp">
+$$ y = vt \cos \alpha - \frac{gt^2}{2} $$
+</p>
+<p>We take into account that the displacement along $Ox$ is equal to the circumference, and along $Oy$ — 0:</p>
+
+<p class="exp">
+$$ vt \cos \alpha - \frac{gt^2}{2} = 0 $$
+</p>
+$$vt \sin \alpha = 2\pi n R,\,n\in \mathbb{N}$$
+We find time $t$ as
+$$ t = \frac{2\pi n R}{v \sin \alpha} $$
+$$ 2v \cos \alpha = g\frac{2\pi n R}{v \sin \alpha} $$
+<p>From here we express $v$:
+<p class="exp">
+$$\fbox{$v=\sqrt{2\pi Rgn/\sin2\alpha}$}$$
+</p>
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $v=\sqrt{2\pi Rgn/\sin2\alpha}$, where $n$ is any natural number; when $\alpha = 0$, the speed can be any in absolute value.
+
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