# Problem №1 (10 points)

Raindrops fall in windless weather at a speed of $v=2 m / s$. It is known that the rear window of a car is inclined at an angle $\alpha=60^{\circ}$ to the horizontal. At what speed $u$ should the car travel along a horizontal flat road so that its rear window remains dry

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# Solution:

To keep the car's glass dry during movement, the drops relative to the glass must move parallel

![](https://cdn.mathpix.com/cropped/2024_05_06_a34127a82f1949864e11g-4.jpg?height=656&width=625&top_left_y=1371&top_left_x=110)

We get that:

$$
u=\frac{v}{\operatorname{tg} 60^{\circ}}=\frac{2}{\sqrt{3}} \approx 1.15 \mathrm{M} / \mathrm{c}
$$

We find that the car must travel at a speed of no less than $1.15 \mathrm{M} / \mathrm{c}$

To transport a steel pipe across a lake, it was welded on both sides to make it watertight. Determine the minimum internal diameter at which a pipe weighing $100 \kappa g$ and 5 m long will not sink. The density of steel $\rho_{C T}=7800 \kappa g / \mathrm{m}^{3}$, the density of water $\rho_{b}=1000 \kappa g / \mathrm{m}^{3}$.

## Solution:

For a floating pipe:

$F_{A}=m g$

$\rho_{B} g\left(V_{C T}+V_{\text {Air }}\right)=m g$

$\rho_{B} g\left(\frac{m}{\rho_{C T}}+V_{\text {Air }}\right)=m g$

We get that the volume of air inside the pipe:

$V_{\text {Air }}=\frac{m}{\rho_{B}}-\frac{m}{\rho_{C T}}=\frac{100}{1000}-\frac{100}{7800}=\frac{68}{780} \mathrm{M}^{3}$

This volume:

$V_{\text {Air }}=l \cdot S=l \cdot \frac{\pi D^{2}}{4}$

Thus, the required diameter:

$D=\sqrt{\frac{4 V_{\text {Air }}}{l \cdot \pi}} \approx 0.149 \cdot \mu \approx 15 \mathrm{~cm}$