$1.3.16^*.$ 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.
a) Find the condition under which there will be no collisions$(x=0)$
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
$$ t=\frac{v_{0}\cos\alpha-\sqrt{v_{0}^{2}\cos^{2}\alpha-2g\sin\alpha L}}{g\sin\alpha} $$
$\begin{aligned}&t=\frac{2v}{g}\operatorname{ctg}\alpha\text{ with }v\cos\alpha<\sqrt{2gl\sin\alpha};\\&t=\frac vg\operatorname{ctg}\alpha\bigg(1-\sqrt{1-\frac{2gl\operatorname{tg}\alpha}{v^2\cos\alpha}}\bigg)\text{ with }v\cos\alpha>\sqrt{2gl\sin\alpha}.\end{aligned}$