# 1. Series 33. Year

* Post deadline: 7th October 2019*

* Upload deadline: 8th October 2019 11:59:59 PM CEST (local time in Czech Republic)*

### 1. D1

A truck driving on a highway has a $2 \mathrm{\%}$ higher speed than a bus in front of it. The driver of the truck decides to overtake the bus, but when the truck is exactly next to the bus, a right curve begins on the highway, making the path of the truck longer. As a consequence, the two vehicles drive next to each other all the way along the curve, whilst a notable traffic jam starts to build up behind them. Determine the radius of the curve (at the middle of the inner driving lane) if the separation between the centers of the lanes is $3{,}75 \mathrm{m}$.

Matej doesn't like trucks on highways.

### 2. battery issue on holidays

How long does it take for a fully charged car battery ($12 \mathrm{V}$, $60 \mathrm{Ah}$) to run out, when someone forgets to turn off the daytime running lights, locks the car and walks away? Specifically we are interested in a situation with two head lights H4 (each running with $55 \mathrm{W}$) and two rear lights P21/5W (each running with $5 \mathrm{W}$). For simplicity, assume no transport losses between the battery and the lights, that there is no other significant consumption of power and that the voltage on the battery stays constant.

Karel. Don't even ask.

### 3. infra sauna

Dano continues with equiping of his mansion with another sauna—this time an infra sauna. He wants to place a tube lamp right underneath the ceiling of the sauna which is $H=2,5 \mathrm{m}$ above the ground. Suppose the source of radiation emits energy with the power per unit length of $p = 1,2 \mathrm{kW\cdot m^{-1}}$, a radiation of what intensity and total energy would reach the skin of a person situated approximately $h=50 \mathrm{cm}$ above ground? The lamp is a straight tube, shines in a homogeneous manner and reaches from wall to wall just under the middle of the ceiling.

**Hint:** For simplicity, approximate the sauna to be a room where the sides touching the lamp and the ceiling are mirrors and the other two sides and the floor absorb the light without remitting it back into the room.

Karel visited wellness in Slovakia.

### 4. discoball

Once upon a time, Mišo wanted to throw the biggest party of all time. You need a proper disco ball for that, so he had the Moon tiled with mirrors reflecting solar light, making it into the biggest disco ball ever. It is clear how his party ended up, but we are interested in the minimum difference of magnitudes of the disco ball and the Sun, when viewed directly from the Earth.

Matěj had a ball.

### 5. generally relativistic

Before he set off on his flight towards Mars, the Starman in his Tesla Roadster arranged with Musk that once he reaches the distance $r=5 \cdot 10^{6} \mathrm{km}$ from the centre of mass of the Earth, Musk will shine a powerful green laser at him. The wavelength of the laser increases under the influence of the gravitational field of Earth. Compare this change of the wavelength to the electromagnetic Doppler effect. Study each of these effects separately. Assume that the Starman is moving away from Earth with velocity $v=4 \mathrm{km\cdot s^{-1}}$.

Vašek wants to go on a trip with Starman.

### P. planet destroyer

How small could a weapon capable of destroying a planet be? We are interested in the smallest and the lightest such weapons. The process should be reasonably fast, at least shorter than a human lifetime, and the faster it is, the better.

Karel watches sci-fi too much, this time the intro of Men in Black II.

### E. bottled

How does the frequency of the sound made by blowing over a glass bottle depend on the volume of the liquid in the bottle? Discuss also the influence of the shape of the bottle on this frequency.

Legolas can't play any instrument, so he is playing hell on us

### S. slow start-up

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

Karel wants to have the longest problem assignment.