# Series 2, Year 35

* Upload deadline: 23rd November 2021 11:59:59 PM, CET (local time in Czech Republic)*

### (3 points)1. chasing the light

Jindra walks down a long, lit corridor. His eyes are at a height of $1,7 \mathrm{m}$ above the floor, the light on the ceiling is at a height of $3,4 \mathrm{m}$. Jindra is now at a distance of $10 \mathrm{m}$ (horizontally) from the nearest light and is approaching it at a speed of $3 \mathrm{km\cdot h^{-1}}$. He sees a reflection of the light on the polished floor. How fast is the reflection approaching Jindra at this point?

Jindra remembered walking down the corridor at the elementary school.

### (3 points)2. fixed station

We have two carabiners anchored in a rock, both at the same height and at a distance of $d$ from each other. We snap a loop with the length $l$ into the carabiners. Then we snap another carabiner on the loop, from which we would like to abseil, while applying a downward force of $F = 2{,} \mathrm{kN}$. Calculate the tension in the loop and the force that will act on the carabiners, in cases where the abseil carabiner is slung on one and on both parts of the loop. In which case is the force acting on the loop lesser and which case is safer?

Dodo dreamed about rock climbing.

### (6 points)3. model of friction

What would be the coefficient of static friction between the body and the surface if we considered a model in which there were wedges with a vertex angle $\alpha $ and a height $d$ on the surface of both bodies? Try to compare your results with real coefficients of friction.

Karel took inspiration from KorSem.

### (7 points)4. tea tap

Matěj wants to pour some tea from a bevarage dispenser into a glass of mass $M$. He uses one hand to hold the glass and second hand to control the faucet, which changes the volume of the current of the tea. The speed of the outflow $v$ is constant (we can assume that the speed at the contact with the glass is identical). Since Matěj does not want to overstrain himself, he would like to hold the glass with a constant force from the start of the pouring to its end. What is the value of the volume of the current as a function of time that satisfies this requirement? How long will it take to fill the whole glass?

Matěj likes tea from a dispenser

### (8 points)5. Shkadov thruster

A long time ago in a galaxy far, far away, one civilisation decided to move its whole solar system. One of the possibilities was to build a „Dyson half-sphere“, i. e. a megastructure which would capture approximately half of the radiation output of the start and reflect it in a single direction. An ideal shape would therefore be a paraboloid of revolution. What would be the relation between the radiation output of the star, surface mass density of such a mirror and its distance from the star such that this distance is constant?

Karel watches Kurzgesagt.

### (10 points)P. le bomb

What is the maximum power of a nuclear bomb?

Karel was thinking about the american presidents.

### (13 points)E. light or dense ethanol

Measure the dependance of the density of an alcohol solution in water on its volume concentration in water. Include also the measurement of pure alcohol and pure water for comparison.

Be careful when mixing alcohol and water – remember that the volume of the mixture is not exactly the sum of their original volumes.

Karel was thinking that the participants might have a little sniff.

### (10 points)S. compressing

What energy must a laser impulse lasting $10 \mathrm{ns}$ have in order for the shock wave generated by it to be able to heat the plasma to a temperature at which a thermonuclear fusion reaction can occur? What will be the density of the compressed fuel? **Note:** Assume that the initial plasma is a monatomic ideal gas.