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### (12 points)6. Series 34. Year - E. spilled glass

Take a glass, can or any other cylindrically symmetrical container. Measure the relationship between the angle of inclination of the container when it tips over and the amount of water inside of it. We recommend to use a container with greater ratio of its height to the diameter of its base.

### (9 points)2. Series 34. Year - P. costly ice hockey

Estimate how much the complete glaciation of an ice hockey rink costs.

Danka doesn't like ice hockey, but she likes figure skating.

### (10 points)6. Series 33. Year - S.

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

### (10 points)4. Series 33. Year - P. climate changes feat. airplanes

Travel by airplane affects the atmosphere not only by well-known carbon emissions. Discuss how the aircraft industry affects warming of the atmosphere of Earth.

### (10 points)2. Series 33. Year - S.

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

### (10 points)1. Series 33. Year - 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.

### (10 points)1. Series 33. Year - S. slow start-up

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

Karel wants to have the longest problem assignment.

### (3 points)5. Series 32. Year - 2. warm reachability into the ball

Imagine you have subcooled homogeneous metal ball that you have just taken out of a freezer set to very low temperature. You want to find out how fast the ball temperature will increase if you put it in a warm room. It would be a university-level problem. Because of that, we made it easier for you. We ask about how deep into the ball will the „warm area“ reach. You can estimate it using dimensional analysis. We know relevant parameters of the ball - its density $\left [ \rho \right ] = \jd {kg.m^{-3}}$, specific heat capacity $\left [c\right ] = \jd {J.kg^{-1}.K^{-1}}$, thermal conductivity of the ball $\left [ \lambda \right ] = \jd {W.m^{-1}.K^{-1}}$ and we are interested in dependence on time $\left [t\right ] = \jd {s}$.

Karel inspired himself by a problem from Eötvös Competition.

### (10 points)6. Series 31. Year - S. Matrices and populations

- Simulate the dynamics of a predator-prey system using Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} &= r\_x x - D\_x xy ,\\ \frac{\d y}{\d t} &= r\_y xy - D\_y y . \end {align*}\] where $x$ and $y$ are the population sizes of prey and predator respectively, the parameters $r\_x$ and $r\_y$ represent the populations’ growth and the parameters $D\_x$ and $D\_y$ represent the shrinking of the populations. Set the parameters to be $r\_x = 0{.}8$, $D\_x0= 1{.}0$, $r\_y = 0{.}75$, $D\_y = 1{.}5$. Run the simulations for several different value pairs for initial population sizes $x = 0{.}5$ and $y = 2{.}0$; $x = 1{.}5$ and $y = 0{.}5$; $x = 1{.}95$ and $y = 0{.}75$. Plot the predator population size as a function of the prey population size. Discuss the results.

**Bonus:**Find the solutions for the same situations analytically (by integrating the differential equations). - Using the competitive Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} = r\_x x \(1 - \(\frac {x + I\_{xy} y}{k\_x}\)\) ,

\frac{\d y}{\d t} = r\_y y \(1 - \(\frac {y + I\_{yx} x}{k\_y}\)\) . \end {align*}\] simulate the dynamics of two competing populations (e.g. hawks and eagles) for the following values of parameters: $r\_h = 0{.}8$, $I\_{he} = 0{.}2$, $k\_h = 2{.}0$, $r\_e = 0{.}6$, $I\_{eh} = 0{.}3$, $k\_e = 1{.}0$. Set the initial population sizes to be $h = 0{.}01$, $e = 1{.}0$. Then, simulate the same situation, but change the interaction coefficients to $I\_{he} = 1{.}5$ a $I\_{eh} = 0{.}6$. Plot the results in one graph - the sizes of populations vs time. Discuss the results. - Verify the importance of pivoting.

Solve the system of linear equations \[\begin{equation*} \begin{pmatrix} 10^{-20} & 1\\ 1 & 1 \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \end{pmatrix} \end {equation*}\] at first exactly (on paper), then using LU factorization with partial pivoting (you may utilize some Python module, e.g.`scipy.linalg.lu()`

), and finally, solve the system using LU factorization without pivoting. Compare the resultant $\vect {x}$ obtained from the three methods and the results of matrix multiplication $L^{-1}\cdot U$ ($P\cdot L^{-1}\cdot U$ in the case of pivoting). - Consider an infinite parallel-plate capacitor. The gap between plates has a thickness $L=10 \mathrm{cm}$ and the voltage between the plates is $U=5 \mathrm{V}$. Between the plates of the capacitor grounded electrode in the shape of an infinitely long prism with square base of side length $a=2 \mathrm{cm}$, whose center lies $l=6{,}5 \mathrm{cm}$ away from the grounded plane of the original capacitor. The prism is oriented such that one of its short sides is perpendicular to the capacitor plates. Find the distribution of electric potential in the condensator. Since the problem has a translational symmetry in the direction of the infinite side of the prism, it is sufficient to solve it only in the plane parallel to the plates, i.e. it is a 2D problem. Render the potential distribution in this plane. You may utilize the code attached to this task.

**Bonus:**Calculate and render the distribution of the electric field strength $\vect {E}$.

Mirek and Lukáš fill matrices with atto-foxes.

### (9 points)5. Series 31. Year - P. floating mercury

Try to invent as much „physics tricks“ as possible thanks to which mercury would float on the liquid water for at least a limited time. The more permanent solution you find, the better.