# Problem № 5 (15 points)

A person was building a house. Only the roof was left to be done. The builder wanted to achieve the goal that raindrops falling on the roof would slide off it as quickly as possible. Determine the angle of inclination of the roof necessary to achieve this goal. Neglect the friction of the raindrops on the roof.

![](https://cdn.mathpix.com/cropped/2024_05_06_3676ffec89688f4f2c70g-1.jpg?height=477&width=825&top_left_y=1258&top_left_x=610)

Answer: $\alpha=45^{\circ}$

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# Solution and Evaluation Criteria:

Newton's second law for a rolling drop: $\quad m g \sin \alpha=m a$.

Distance traveled by the drop: $s=\frac{1}{2} \frac{x}{\cos \alpha}$, where $x-$ is the width of the house.

(4 points)

Time of rolling: $t=\sqrt{\frac{2 S}{a}}=\sqrt{\frac{2 \cdot \frac{1}{2} \frac{x}{\cos \alpha}}{g \sin \alpha}}=\sqrt{\frac{2 x}{g \sin 2 \alpha}}$.

The rolling time will be minimal if $\sin 2 \alpha=1$, i.e., $\alpha=45^{\circ}$.