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## mathematics

### (9 points)5. Series 37. Year - 5. tuning a circuit

Consider a series circuit with a resistor of resistance $R$, a coil, and a capacitor with the capacitance $C$. AC voltage sources with identical amplitudes $U$ are connected in series with these components. These sources vary in frequency by being multiples of $\omega _0$, where $n$ represents an integer. What frequency, denoted by $\omega _0$, would allow us to find a coil possessing an inductance $L$, such that voltages with frequencies different from $N \omega _0$ are suppressed by at least $90 \mathrm{\%}$ on the resistor? $N$ is a positive natural number known in advance (i.e., the value of $\omega _0$ may depend on it), and we do not want to suppress the voltage with frequency $N\omega _0$ by more than $90 \mathrm{\%}$.

Jarda wanted to have as many different sources in the circuit as possible.

### (7 points)6. Series 36. Year - 4. light faster than light

There is a laser in the distance $L$ from a large screen. Initially, the laser shines on the screen so that the distance from a laser spot on the screen to the laser is $R > L$. Then at the time $t=0 \mathrm{s}$, we begin to rotate the laser at a uniform angular speed $\omega $. Consequently, the distance of the spot on the screen from the laser decreases to $L$ and then increases back to $R$. What is the speed of this laser spot relative to the screen? Is it possible that the spot moves at a speed greater than the speed of light in a vacuum? Is there a limit, can it be infinite? How (qualitatively) does this speed depend on the spot's position on the screen? The whole apparatus is in a vacuum.

Marek J. wanted to verify statements about the apparent surpassing of the speed of light.

### (6 points)5. Series 36. Year - 3. waiting for an elevator

Karel uses an elevator in a building with a ground floor and $12$ floors above it, while the height between floors is $h=3{.}0 \mathrm{m}$. Consider that the elevator accelerates half the time and decelerates half the time at a constant acceleration of $a=1{.}0 \mathrm{m\cdot s^{-2}}$ and that there is a $50\mathrm{\%}$ probability that the elevator is stationary on the ground floor. The rest of the probability is evenly distributed among the other floors. What is the expected waiting time for the elevator on each floor of the building? Neglect the time needed for opening doors.

**Bonus:** Let us have $2$ elevators in a twelve-story building. One elevator is always recalled to the ground floor. To which floor should we send the second one to minimize the average waiting time? Similarly, assume that half of the rides will start on the ground floor and the other half, with equally distributed probability, will start on any other floor.

Karel often waits for an elevator.

### (8 points)5. Series 36. Year - 5. xenon was wandering

A once positively ionized xenon atom flew out from the center of a large cylindrical coil with velocity $v=7 \mathrm{m\cdot s^{-1}}$ and began to move through a homogeneous magnetic field, which is in a plane perpendicular to the magnetic lines of force. At a certain point the coil is disconnect from the source, thus its induction begins to decrease exponentially according to the following equation $\f {B}{t}=B_0\eu ^{-\Omega t}$, in which $B_0=1,1 \cdot 10^{-4} \mathrm{T}$ and $\Omega =600 \mathrm{s^{-1}}$. What is the deviation from the initial direction after the atom is stabilized?

Vojta spent several hours thinking about a reasonable problem assignment with a clever solution, but ultimately, it ended horrendously. And he has yet to see the solution.

### (8 points)1. Series 36. Year - 4. mountain transport

There is a town on the slope of a hill whose shape is a cone with apex angle $\alpha = 90\dg $. On the other side of the cone, right opposite to the town in the same altitude, lies a train station. Mayor of the town decided to build a road to the station. They can either drill a tunnel or build a road on the surface of the hill. What is the maximum ratio of per-kilometer prices for the tunnel and for the surface road, so that building a tunnel is cheaper? The road can be built anywhere on the hill.

Matěj builds Semmeringbahn.

### (10 points)6. Series 35. Year - 5. fly rocket, fly

We have constructed a small rocket weighing $m_0 = 3 \mathrm{kg}$, from which $70\%$ is fuel. The exhaust velocity is $u = 200 \mathrm{m\cdot s^{-1}}$ and the initial flow of the exhaust fumes is $R = 0,1 \mathrm{kg\cdot s^{-1}}$ and both these values remain constant during the flight. The rocket is equipped with stabilization elements, so it does not deviate from its desired trajectory. It has been launched from the rest position vertically. Assume that the friction force of the air is proportional to the velocity $F\_o = -bv$, where $b = 0,05 \mathrm{kg\cdot s^{-1}}$, $v$ is the velocity of the rocket and the sign minus means that the force exerts against the direction of the motion. What height above the ground level does the rocket fly in time $T = 25 \mathrm{s}$ from the engine startup?

Jindra got a homework to deliver a satellite onto the Low Earth orbit.

### (3 points)3. Series 35. Year - 1. Where my center of gravity is?

We can find an unofficial interpretation that the red, blue and white colors on the Czech flag symbolize blood, sky (i.e. air) and purity. Find the position of the center of mass of the flag interpreted in this way, assuming that purity is massless. The aspect ratio is $3:2$ and the point where all three parts meet is located exactly in the middle. Look up the blood and air densities.

**Bonus:** Try to calculate the position of the center of mass of the Slovak flag as accurately as possible. You can use different approximations.

Matěj likes to have fun with flags.

### (8 points)3. Series 35. Year - 4. gentle tide

Close to the shore, the speed of sea waves is influenced by the presence of the sea bed. Assume that the speed of waves $v$ is a function of the gravity of Earth $g$ and the water depth $h$. We have $v = C g^\alpha h^\beta $. Using dimensional analysis, determine the speed of the waves as a function of the depth. Constant $C$ is dimensionless, and cannot be determined using this method.

Besides the speed of the waves, swimming Jindra is also interested in the direction of incidence of the waves. Let's define a system of coordinates, where the water surface lies in the $xy$ plane. The shoreline follows the equation $y = 0$, the ocean lies in the $y > 0$ half-plane. The water depth $h$ is given as a function of distance from the shore $h = \gamma y$, where $\gamma = \const $. On the open ocean, where the speed of the waves is constant $c$ (not influenced by the depth), plane waves are propagating at incidence angle $\theta _0$ to the $x$ axis. Find a differential equation \[\begin{equation*} \der {y}{x} = \f {f}{y} \end {equation*}\] describing the shape of the wavefront close to the shore, but do not attempt to solve it, it is far from trivial. Calculate the incidence angle of the wavefront at the shoreline.

**Bonus:** Solve the differential equation and find the shape of the wavefront close to the shore.

Jindra loves simple dimensional analysis and complicated differential equations.

### (6 points)2. Series 35. Year - 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.

### (10 points)4. Series 34. Year - S. Oscillations of carbon dioxide

We will model the oscillations in the molecule of carbon dioxide. Carbon dioxide is a linear molecule, where carbon is placed in between the two oxygen atoms, with all three atoms lying on the same line. We will only consider oscillations along this line. Assume that the small displacements can be modelled by two springs, both with the spring constant $k$, each connecting the carbon atom to one of the oxygen atoms. Let mass of the carbon atom be $M$, and mass of the oxygen atom $m$.

Construct the set of equations describing the forces acting on the atoms for small displacements along the axis of the molecule. The molecule is symmetric under the exchange of certain atoms. Express this symmetry as a matrix acting on a vector of displacements, which you also need to define. Furthermore, determine the eigenvectors and eigenvalues of this symmetry matrix. The symmetry of the molecule is not complete – explain which degrees of freedom are not taken into account in this symmetry.

Continue by constructing a matrix equation describing the oscillations of the system. By introduction of the eigenvectors of the symmetry matrix, which are extended so that they include the degrees of freedom not constrained by the symmetry, determine the normal modes of the system. Determine frequency of these normal modes and sketch the directions of motion. What other modes could be present (still only consider motion along the axis of the molecule)? If there are any other modes you can think of, determine their frequency and direction.

Štěpán was thinking about molecules