# 6th Round of 26th year

Series

### 1. ... disgusting water(2 points)

Many years ago you drank 2 dcl of water. Imagine that since then all the water on the Earth has mixed. If you drink 2 dcl of water today, how many molecules from the original water you drank does it contain?

### 2. ... stupid wire(2 points)

What is the minimal length of a wire so that if you hang it from a ceiling, it will break due to its own mass? The wire's density is $ρ$ = 7900 kg·m^{ − 3}, it has a diameter $D$ = 1 mm, and it breaks at $σ$_{max} = 400 MPa. Assume that everything takes place in a homogeneous gravitational field $g$ = 9.81 m·s^{ − 2}. Bonus   If the wire's length is maximal possible so that it does not break, how much will it stretch (in percents)? Young's modulus of the wire's material is $E$ = 200 GPa.

### 3. ... a drowned lens(4 points)

If an object is placed a distance $p$ from a thin glass lens (index of refraction $n$_{s}), we can see its image on a screen that is placed a distance $d$ from the lens. Without altering any distances, we immerse this system into a liquid (index of refraction $n$). Under what conditions can we still observe the object's image on the screen, and how far from lens would this image be?

### 4. ... filling a tank(4 points)

Imagine a large tank containg tea with a little opening at its bottom so that one can pour it into a glass. When open, the speed of the flow of tea from the tank is $v$_{0}. How will this speed change if, while pouring a glass of tea, someone is filling the tank by pouring water into it from its top? Assume that the diameter of the tank is $D$, the diameter of the flow of tea into the tank is $d$, and that of the flow of tea out of the tank is much smaller than $D$. The tea level is height $H$ above the lower opening, and the tank is being filled by pouring a water into it from height $h$ above the tea level. You are free to neglect all friction.

### 5. ... baseball(4 points)

Let us consider the following model of a baseball player hitting a ball. Baseball bat is a thin homogeneous rod of length $L$ and mass $m$. The bat can only rotate around an axis perpendicular to the axis of the bat that is located at its end. The bat is rotating with an angular velocity $ω$. How far from the end of the bat should the player hit the ball in order to minimize the force with which the bat acts on the player's hands?

### P. ... turn it of – I can't!(5 points)

How many people per second can be killed by a nuclear reactor without any protective walls?

### E. ... a balloon accident(8 points)

A loaded falling balloon will eventually reach certain constant terminal velocity. Measure how does this velocity depend on the balloon size, and on the mass of its load.

### S. ... series(6 points)

* Calculate the time a tokamak COMPASS can store an energy for. The energy of its plasma is 5 kJ, and its ohmic heating is 300 kW.

• Calculate the alpha heating in tokamak COMPASS if it used a DT mixture. Typical plasma temperature is 1 keV, hustota 10^{20} m^{ − 3}, and the volume of the plasma 1 m&sup3;. Assuming the ohmic heating from the preceeding question, calculate $Q$.
• Using the picture from the main text and knowledge of the DD reaction ^{2}_{1}D + ^{2}_{1}D → ^{3}_{2}He + n + 3,27 MeV (50 %),

{2}_{1}D + {2}_{1}D →

&frac34; energie v of the energy in the first reaction are carried off by a neutron, calculate the total plasma heating that will occure during one DD reaction (assume that it is followed by a DT fusion with the product of the second reaction). Also estimate the requirements on the confinment time assuming density odhadněte nároky na dobu udržení při hustotě 10^{20} m^{ − 3} a teplotě 10 keV.

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