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## hydromechanics

### 1. Series 35. Year - 4. fall to the seabed

A cylindrical capsule (Puddle Jumper) with a diameter $d = 4 \mathrm{m}$, a length $l = 10 \mathrm{m}$ and with a watertight partition in the middle of its length is submerged below the ocean surface and falls to the seabed at a speed of $v = 20 \mathrm{ft\cdot min^{-1}}$. At the depth $h = 1~200 ft$, the glass on the front base breaks and the corresponding half of the capsule is filled with water. At what speed will it fall now? How long will it take for the capsule to sink to the bottom at the depth $H=3~000 ft$? Assume that the walls of the capsule are very thin against its dimensions.

Dodo watches Stargate Atlantis.

### 4. Series 34. Year - P. Fykos bird on vacation

How would aviation work on other planets (with atmosphere)? Consider mainly jet aircraft. Which planetary parameters would influence the aviation positively and which negatively, compared to Earth's?

Karel visited the museum of aviation in Košice.

### 2. Series 34. Year - 3. a car at the bottom of a lake

There are several movie scenes where a car falls into water together with its passengers. Calculate the torque with which a person must push the door in order to open it at the bottom of a lake if the bottom of the door's frame is $8,0 \mathrm{m}$ deep underwater. Assume that the door is rectangular with dimensions $132 \mathrm{cm}$ x $87 \mathrm{cm}$ and opens along the vertical axis.

Katarína likes dramatic scenes on cliffs.

### 6. Series 33. Year - 2. under pressure

The water level in bath reaches height $15{,} \mathrm{cm}$. The plug has a shape of a conical frustum which perfectly fits the hole in the base of the bath. Its radii of bases are $16,0 \mathrm{mm}$ and $15,0 \mathrm{mm}$ and its mass is $11,0 \mathrm{g}$. What force does the bath bottom act on the plug? Assume that the drain pipe below contains air of atmospheric pressure.

Jindra felt pressure to think simple problems up.

### 6. Series 33. Year - E. viscosity

Measure viscosity (in $\textrm{Pa}\cdot\textrm{s}$) of two different oils using Stokes' method.

Jáchym stole Jirka's idea to steal this problem from labs.

### 5. Series 33. Year - S. min and max

*We are sorry. This type of task is not translated to English.*

They had to wait a lot for Karel.

### 4. Series 33. Year - 3. uuu-pipe

What period of small oscillations will water in a glass container (shown on the picture) have? The dimensions of the container and the equilibrium position of water are shown. Assume that there is room temperature and standard pressure and that water is perfectly incompressible.

Karel was thinking about U-pipes again.

### 3. Series 33. Year - 1. fountain with nozzle

Let us have a fountain with $N$ nozzles of the same cross section. These nozzles are fed by one pump with constant volumetric flow rate, which leads to water streaming to the height $h$. Find this height in case of all nozzles with the exception of one being blocked.

Lukáš experimented in the town square.

### 3. Series 33. Year - 3. umbrella

As you probably have noticed, water flow creates a mushroom-like shape when a teaspoon is placed against it (e.g. while washing the dishes). Assume (for simplification) planar round-shaped teaspoon of small radius. When placed perpendicularly to the flow, falling from rest from the height $h$, a wonderful rotational paraboloid would form. Find the optimal height to put the teaspoon in to maximalise the distance from the original flow axis to the place, where the falling water touches the surface (e.g. of the sink). Assume water to be an ideal liquid (uncompressible, zero viscosity, no inner friction).

**Bonus:** Find optimal height to maximalise the volume of the paraboloid.

Matěj washed the dishes.

### 3. Series 33. Year - 5. probability density of water

Imagine a container from which continually and horizontally flows out water stream with constant cross-section area. Velocity of the stream randomly fluctuate with uniform distribution from $v_1$ to $v_2$. Water from the container continually freely falls onto a horizontal floor below. Figure out arbitrary area of the floor to which falls exactly $90 \mathrm{\%}$ of water.

Another from a list of problems, which crossed Jachym's mind while being on a toilet.