Problem Statement of Series 1, Year 33

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 $3.75\,\mathrm{m}$.

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.

Dano continues with equiping of his mansion with another sauna–-this time an infrasauna. 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 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 walls touching the lamp and the ceiling are mirrors and the other two walls and the floor absorb the light without remitting it back into the room.

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.

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 $v=4\,\mathrm{km\cdot s^{-1}}$.

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.

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.

1. Express the following physical quantities\footnote{They might have no physical meanings.} using only SI base units.
   1. $\mathrm{F}\cdot\Omega$, where $\mathrm{F}$ is the farad and $\Omega$ is the ohm
   2. $\mathrm{N}\cdot\mathrm{Pa}$, where $\mathrm{N}$ is the newton and $\mathrm{Pa}$ is the pascal
   3. $\dfrac{\mathrm{C}\cdot\mathrm{V}}{\mathrm{J}}$, where $\mathrm{C}$ is the coulomb, $\mathrm{V}$ is the volt a $\mathrm{J}$ is the joule
   4. $\dfrac{\mathrm{T}\cdot\mathrm{Wb}}{\mathrm{H}\cdot\mathrm{Sv}}$, where $\mathrm{H}$ is the henry, $\mathrm{Sv}$ is the sievert, $\mathrm{T}$ is the tesla and $\mathrm{Wb}$ is the weber
2. Find all the mistakes in the following statements and explain why they are mistakes. (2 points)
   1. $s = vt^2/2 = 5.2 \cdot 1.2^2 /2 = 3.744\,\mathrm{m} \,. $
   2. $y_{\mathrm{m}} \sin \left( 2 \pi \omega \right) = 15 cm \cdot \sin \left( 2 \cdot 3.141 \cdot 50 Hz \right) \doteq 0 cm $
   3. We successfully used a radiation measurement set in an experiment. Based on a measurement of radioactive decay of uranium in uraninite, we found out that the activity of the sample was exactly 532,24 becquerels.
   4. $s = 1.23\,\mathrm{m}$, $t = 2.7\,\mathrm{s} \Rightarrow v = s/t \doteq 0.46\,\mathrm{m\cdot s^{-1}}$, $m = 240\,\mathrm{g}$, $E = mv^2/2 \doteq 25\,\mathrm{J}$, $P = E/t \doteq 9.3\,\mathrm{W}$
3. What is the force exerted by wind on the crown of a tree? We know that it is related to the wind velocity $v$, the area of the crown's cross-subsubsection $S$ and the density of air $\rho$. Find an expression for this force using dimensional analysis.
4. Compose a characteristic number corresponding to the flow of a liquid along a characteristic length $l$ under a pressure gradient $\dfrac{\d p}{\d x}$ (you may imagine this as the ratio of pressure decrease to travelled distance, $\dfrac{\Delta p}{\Delta x}$). The liquid has density $\rho$ and kinematic viscosity $\nu$. Determine which variants of this characteristic number exist. Pick one of them and try to interpret it.

5. Bonus Make up a Planck unit which is as original as possible (a quantity composed from the reduced Planck constant $\hbar$, the gravitational constant $G$, speed of light $c$, the Boltzmann constant $k_{\mathrm{B}}$ and the Coulomb constant $k_{\mathrm{e}}$ - not necessarily all of them). Describe how you derived it and comment on its value. We will mention the most interesting units in the leaflet along with the solution to this problem.