# Series 6, Year 26

### (2 points)1. disgusting water

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?

Karel is afraid of cholera.

### (2 points)2. stupid wire

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 $ρ=7900\;\mathrm{kg}\cdot \mathrm{m}^{-3}$, it has a diameter $D=1\;\mathrm{mm}$, and it breaks at $σ_{max}=400MPa$. Assume that everything takes place in a homogeneous gravitational field $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}$.

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 $E=200GPa$.

Karel stuck a wire into his eye

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

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?

Pikos went swimming

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

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.

### (4 points)5. baseball

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?

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

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

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

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.

Pikos

### (6 points)S. series

• 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. 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.

Robin 