# 1th Round of 29th year

### 1. ... densifying Hofmann(2 points)

During the electrolysis in a Hofmann voltameter the electolyte is a soluton of sulfuric acid in water. The mass of the acid in the solution is practicaly constant but as the name says the water slowly dissolves into hydrogen and oxygen. So the concentration of the acid in the solution rises. How long will take for the mass fraction of the acid in the solution to rise to twice the original amount if there was a current of $I$ = 1 A passing through the solution, the original mass fraction of the acid in the solution was $w$_{0} = 5 % and the volume of the solution in the container was$V$_{0} = 2 l?

### 2. ... jumping out of a train(2 points)

In a train, that can move without friction on rails, stand 2 people, both with a mass $m$. In which of the two following situations shall the train reach a higher speed? When both jump out at the same time or when they will jumping outone after another? A person can jump out the train with a relative speed $u$ (the speed of a person jumping out the train versus the speed of the train).

### 3. ... golden sphere(3 points)

A golden sphere has a mass of $m$_{1} = 96,25 g while in the air. When sunken into water it is balanced out by a weight with the mass $m$_{2} = 90,25 g. Determine whether the item is hollow. If it is then determine the volume of the hollow part.The density of gold is $ρ$_{$Au$} = 19,25 g·cm^{ − 3}, density of water $ρ$_{$H_{2}O$} = 1,000 g·cm^{ − 3}. The graviational acceleration is $g$ = 9,81 m·s^{ − 2}.

### 4. ... the lethal lens(4 points)

Imagine that around the Sun on a circular orbit is a convex lens with a diameter that is equal to the diameter of the Sun, the focal point of which orbits with a sufficient precision on the orbit of Earth. Determine how the lens will burn the Earth during one of its orbit (ie. how much solar energy will be given to Earth by the lens), if it orbits at the distance of Mercury and compare it with the state where it will be as far as Venus.

*Bonus* Consider the eclipse that the lens will cause during its orbit.

### 5. ... chernobyl(4 points)

If someone would eat 5 µg of the isotope of cesium ^{137}Cs, how long would it take for them to have only0,04 % of the original amount of this isotope? Assume that cesium ^{137}Cs has a half-life of 30,42 let and a biological half-life (the time it takes for half of the original amount of the material to leave the body) is approximately 5 days. Determine also, how many of the particles will have decayed in the body up till then.

### P. ... decompression illness(5 points)

Surely you have heard before (at least in a movie) about how it is dangerous to go diving in great depths and then to immediately travel by plane. If a person were to do this he might suffer from the so called. decompression illnes. Describe as accurately as possible what physical processes the human body will undergo during this sickness(how precisely do the phyiscal laws act in this case) and say why precisely they are dangerous for a person. Is it dangerous for people to attempt the opposite sequence of events (ie. first travelling by plane then diving) Je pro lidi nebezpecná i opacná posloupnost akcí, tedy cestování letadlem a následné potápení? (when solving this you can utilise all available resources but in the end you must use your own words to describe it)

### E. ... small g(8 points)

Measure the local gravitational acceleration with at least two different methods. Then compare these two methods in detail.

### S. ... Ideal Gas(6 points)

* As a little warm-up, to help you understand the numbers we'll be using, try to find height to what should be an average person (70 kg), lifted in order to use up all the energy of a standard Mars bar ( 250 cal for 50 g bar). Determine also what is the energy equivalent to $k$_{B}$T$ at room temperature and express it in electronvolts (i.e. the unit of energy equivalent to the kinetic energy electron gains when accelerated at potential difference of 1 V. Explicitly 1 eV = 1,602 · 10^{-19} J).

- The Ideal Gas Law can be modified in many ways. If you rewrite it using amount of substance, instead of number of particles, you get

$$pV = n N_\mathrm{A} k_\mathrm{B} T\,,$$

, where $N$_{A}$k$_{B} together is labeled as $R$ and is called universal gas constant. Express its value. Then modify the equation once again using mass of the gas and third time into a form containing gas density.

- Evaluate the volume of a single mole of gas at room temperature. It is useful to remember this number.
- And finally, a small consideration. Notice, when we were discussing the work of ideal gas, we automatically reached for the inner gas pressure value. Try to reason this choice of pressure. We might be objecting we should use the surrounding pressure or even the pressure difference between the inner and outer pressure. $Evaluation of this section will be moderate, do not be afraid to write whatever you think of yourself..</a>