# 3. Series 33. Year

* Post deadline: 30 December 2019*

* Upload deadline: 31 December 2019 11:59:59 PM CET (local time in Czech Republic)*

### 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.

### 2. …boom

A jet fighter has flown directly overhead at constant velocity parallel to the ground. We have heard a sonic boom at $t=1{,}50 \mathrm{s}$ after that, when the fighter has been at zenith distance $\theta =30.0\dg $. Find out the height of the figther above the ground.

**Bonus:** Also find the direction from which we have heard the boom relative to the place where we have seen it.

Dodo \uv {is looking forward} to aviation days.

### 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.

### 4. ladybird on a rubber

Ladybird moves with velocity $4 \mathrm{cm\cdot s^{-1}}$. When we place the ladybird onto a rubber, she comes through it in $10 \mathrm{s}$. What happens when the ladybird starts moving and we start prolonging the rubber the way that its length will be increasing with velocity $5 \mathrm{cm\cdot s^{-1}}$? Is the ladybird able to come through the whole rubber to its end? If yes, how long will it take? Consider that the rubber prolongs uniformly and never breaks.

Matej was watching Vsauce.

### 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.

### P. meteor swarm

Is it possible that droplet of rain evaporates earlier than it hits the ground? Think up suitable model of evaporating of rain droplets during their fall and show under what conditions (some of the relevant parameters are initial radius, behaviour of outdoor temperature in relation to height above sea level) the droplet can evaporate completely. You can assume that the droplet arises suddenly in particular height $h_0$ with initial radius $r_0$ and in first approximation it falls through dry atmosphere. And when is it possible that the droplet freezes?

Mirek waited for rain.

### E. dense measurement

Construct a hydrometer (for example from straw and plasteline) and measure dependence of water density on the concetration of salt dissolved in it.

Bouyant Matěj.

### S.

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