Series 3, Year 36

Danka attached a garden hose with an inner diameter of $1{,}5 \mathrm{cm}$ to a tap in her dorm room and placed the other end on the edge of a window on the eighth floor, $23 \mathrm{m}$ above the ground. What is the necessary volumetric flow rate of the water tap so that Danka can spray a stream of water on the people disturbing the night's silence? They are standing below the window at a horizontal distance $9 \mathrm{m}$ from the building. Is Danka able to achieve this if water is being sprayed horizontally from the hose and there is no wind?

Bonus: Where is the farthest these people can stand so Danka can still spray them if the volumetric flow rate of the tap is $0{,}4 \mathrm{l\cdot s^{-1}}$? Danka can now set the end of the hose so that water sprays at an arbitrary angle to the horizontal plane.

The FYKOS-bird arrived at his cottage in the middle of winter with only $T_1 = 12 \mathrm{\C }$ indoors. So he lit a fire in the fireplace by using wood of heating value $Q_0 = 14{,}23 \mathrm{MJ\cdot kg^{-1}}$. How much wood does he need to burn to heat the air inside to $T_2 = 20 \mathrm{\C }$? The cottage is in the shape of a rectangular cuboid with dimensions $a = 6 \mathrm{m}$, $b = 8 \mathrm{m}$ and $c = 3 \mathrm{m}$. A roof is in the shape of an irregular recumbent triangular prism with a height of $v = 1{,}5 \mathrm{m}$, the upper edge of which is the axis of the cottage layout. The air occupies $87 \mathrm{\%}$ of the volume of the cottage and its specific heat capacity is $c_v = 1 \mathrm{007 J\cdot kg^{-1}\cdot K^{-1}}$. Does the result match the expectation? Discuss the simplicity of the model used.

Matěj and David are sliding on bobsleds down the hill. The hill with a slope of $\alpha =29 \mathrm{\dg }$ turns into the horizontal ground at the bottom of it. Both of them started from rest from the same height. Matěj's bobsled always travels the same distance $l$ on an inclined plane as well as in a horizontal part. Since the bobsled digs deeper into the snow at higher loads, assume the coefficient of friction to be proportional to the normal force as $f(F)=kF$, where $k$ is a positive constant. Determine how many times Matěj will travel farther from the bottom of the hill than David if David's mass (including the bobsled) is $12 \mathrm{\%}$ greater than Matěj's. Also, assume that bobsledders don't lose any energy at the bottom of the hill.

Since our Sun will explode one day, it will be necessary to organize the construction of an evacuation spacecraft, in which at least $0{,}000~001 \%$ of humanity can escape. To escape, people choose the star Tau Ceti which is $12 \mathrm{ly}$ away at that time. They build engines that accelerate the spacecraft to a cruising speed of $v = 0{,}75 c$ in very little time. Unfortunately, at half a distance to their destination, they observe both the Sun's and Tau Ceti's explosions. How long before the observation the explosions happened in an inertial coordinate frame of the spacecraft? And when in the coordinate system fixed with the stars? Presume that the Sun and Tau Ceti are static relative to each other.

Assume you have a guitar that is perfectly tuned at room temperature. By how many semitones (in tempered tuning) will the individual strings be out of tune if we move to a campfire, where it is cooler by $10 \mathrm{\C }$? Will the guitar still sound in tune? The distance between the string attachment points is $d = 65 \mathrm{cm}$. The strings have a density $\rho = 8~900 \mathrm{kg.m^{-3}}$, a Young's modulus of elasticity $E = 210 \mathrm{GPa}$ and a thermal expansion coefficient $\alpha = 17 \cdot 10^{-6} \mathrm{K^{-1}}$.

What phenomena can affect the measurement of gravitational acceleration using a pendulum? Estimate how many valid digits your result would have to contain to measure them. Consider also the phenomena that you usually neglect.

Charge the object by rubbing it and then measure the dependence of its self-discharge on time. Determine the electrical conductivity of air. Consider that the magnitude of the charge varies according to \[\begin{equation*} Q = Q_0 \eu ^{-\frac {\sigma }{\varepsilon }t} , \end {equation*}\] where $Q_0$ is the initial charge, $\varepsilon $ is the permeability of the air, and $\sigma $ is the conductivity we are looking for. Hint: Hang two small metallic objects (e.g. nuts) at the same height on thin, long filaments. Then take a straw, rub it to charge it, and transfer some of the charges to the objects. They should begin to repel away from the straw. Afterwards, you can determine the product of the charges and the conductivity from their relative distances.

1. Similarly to the series, use the Hückel method to create the Hamiltonian matrix for the cyclobutadiene molecule and verify that its eigenvalues are $\alpha +2\beta $, $\alpha $, $\alpha $, $\alpha -2\beta $. Sketch the diagram of the final energies in the resulting orbitals. And show how the electrons will occupy them. $(4~b)$
   Bonus: What is the main difference in the characterics of these orbitals and their occupancy compared to a benzene molecule we showed in the series? What are the consequences for the cyclobutadiene molecule? $(2~b)$
   
2. Try going back to the beta-carotene molecule and calculate again at what wavelength it should absorb using the Hückel method. What should the value of the parameter $\beta $ be equal to in order to be consistent with the experimental results
   Alternative: If you encounter a problem with the diagonalisation of the hamiltonian, solve the problem statement with the hexa-1,3,5-triene molecule. The experimentally determined absorption value in this case is at a wavelength of $250 \mathrm{nm}$. $(4~b)$
   
3. What happens to a molecule (a molecule with only simple bonds is sufficient) if we use UV light to excite an electron from the $\sigma $ to the $\sigma ^\ast $ orbital? $(2~b)$