# Serial of year 21

You can find the serial also in the yearbook.

*We are sorry, this serial has not been translated.*

## Tasks

### 1. Series 21. Year - S. gravitation

Download program Planeta.pas, ThreeStars-TriHvezdy.pas. As an excerscice try to understand the programs, modify them and change them.

- First task is to add into at least two of programs something new. For example change origin conditions, add another planet or star. Also try to change gravitational law – calculate with force $F=A⁄R+B⁄R$, where $A$ and $B$ are constants etc.
- Assume two same mass stars, which orbit around each other on circle. In axis of this circle the third star is approaching, at the beginning with the same speed as the two stars are orbiting. It has also the same mass. Create simulation what will happen.

As solution of both problems please send us pictures with extensive commentary. State at least short explanation what is at the images. Also mention how have you proceed during calculations and which computation system have you used.

Zadal autor seriálu Lukáš Stříteský.

### 2. Series 21. Year - S. cutting of wild plains

### Uranium storage

Very important question is storing of radioactive waste. Usually it is stored in cylindrical containers immersed in water, which keeps the surface at constant temperature 20 °C. Your task is to find the temperature distribution inside containers of square base of edge length 20 cm. Container is relatively long, therefore just temperature distribution in horizontal cross section is of interest. Uranium will be in block of square base of edge 5 cm. From the experience with cylindrical capsules we know, that it will have constant temperature of about 200 °C.

### Heating wire

Lets have a long wire of circular cross section and radius $r$ from a material of heat conductivity $λ$ and specific conductivity $σ$. Then a electric field is applied. Lets the electric field inside the wire is constant and parallel with the axis of the wire and the strength is $E$. Then the current through wire will be $j=σE$ and will create Joule's heat with volume wattage $p=σE$.

Because the material of the wire has non-zero temperature conductivity, some equilibrium gradient of temperature will form. The gradient fulfills Poisson's equation $λΔT=-p$. Assume, that the end of wire is kept at temperature $T_{0}$. This gives a border condition needed to solve the equation. Due to symmetry we can take into account only two dimensions: on cross section of wire (temperature will be independent of shift along the axis of symmetry). Now it is easy to solve the problem with methods described in text.

However, we will make our situation little bit more complex and will assume, that specific electrical conductivity $σ$ is function of temperature. So we will have a equation of type Δ$T=f(T)$.

Try to solve this equation numerically and solve it for some dependency of conductivity on temperature (find it on internet, in literature of just pick some nice function) and found temperature profile in wire profile. Try to change intensity of electric field $E$ and plot volt-amper characteristics, you can try more than one temperature dependency. $σ(T)$ (e.g. semiconductor which conductivity increase with temperature, or metal, where conductivity is decreasing) etc.

Do not limit your borders, we would be glad for any good idea.

### Capacity of a cube

Calculate capacity of ideally conductive cube of edge length 2$a$ (2Ax2Ax2A). If you think, it is simple, try to calculate for cuboid (AxBxC) or other geometrical shapes.

**Hint:**
Capacity is a ration of the charge on the cube to the potential on the surface of cube (assuming that the potential in infinity is zero). Problem can be solved by selecting arbitrary potential of cube and solving Laplace equation Δ$φ=0$ outside of the cube and calculating total charge in cube using Gauss law. E.g. calculating intensity of electrical field and derivating potential and calculation of flow through nicely selected surface around the cube.

Final solution is finding a physical model, its numerical solution and realization on computer. More points you will get for deeper physical analysis and detailed commentar. For algorithm you can also get extra points.

Zadal spoluautor seriálu Lukáš Stříteský.

### 3. Series 21. Year - S. wandering of a sailor, pi-circuit and epidemic in Prague

### Integral

Integrate using Monte Carlo method function e^{$-x}$ on interval [ $-100,100]$. Try numerically find value of this integration interval from −∞ till +∞.

**Hint:** Function is symmetrical in origin, therefore it is sufficient to integrate on interval [ 0, +∞ ) . Make substitution $x=1⁄t-1$, where you change limits of integration from 0 to 1.

### Wandering of sailor

Drunken sailor stepped out onto pier of length 50 steps and wide 20 steps. He goes to land. At each step forward looses balance and makes one step left or right. Find, what is probability of reaching land and what is probability of falling off the pier into the sea.

Sailor was lucky and survived. However the second night he goes (again drunken) from ship to land. This time there is strong wind of speed of 3 m\cdot s^{−1}, which causes change of probability of stepping to the left to 0.8 and 0.2 to the right. Again, find the probability, that he reaches the other side or will fall into the sea.

Third night the situation repeats again. The wind is blowing randomly, following normal distribution with mean value 0 m\cdot s^{−1} and dispersion 2 m\cdot s^{−1}. Find the probability of sailor reaching land. You can assume, that sailor walks slowly and inertia of wind is negligible, therefore wind is uncorrelated between individual steps.

### Pi-circuit

Having 50 resistor of resistance 50 Ω we want to create a circuit with the resistance in Ohms closest to number π. Solve it using simulated annealing.

For this task you can adapt our program, which can be found on our web pages.

If you do not feel like solving this problem, try to solve problem of „traveling salesman“ with introducing curved Earth surface into a model and find solution for concrete set of towns, e.g. capitals of European countries.

### Epidemic in Prague

Investigate evolution of epidemic in Prague. Assume 1 million inhabitants. Intensity of infection $β$ is 0$,4⁄1000000$ per day, cure $γ$ is ( four days )^{$-1}$. At the beginning there is 100 infected people. Compare the evolution with case of vaccinated population of 20% of population. Also compare with vaccination during the epidemic, where 0.5% population is vaccinated per day. The end of epidemic is, when less than 20 people are ill.

There is a lot of data you can get from computer simulation. Apart from the mean value also plot a graph, where you will show five random simulations. You can also observe fluctuations. Compare your results with deterministic model which does not assume randomness of process of infection. The number of points, which we will give out will reflect how many interesting data you will process.

Zadali autoři seriálu Marek Pechal a Lukáš Stříteský.

### 4. Series 21. Year - S. quantum harmonic oscillator

Calculate time dependence of wave function of particle, which is located in potential $V(x)=\frac{1}{2}kx$ and which is at time $τ=0$ described by wave function

$ψ_{R}(X,0)=\exp(-((X-X_{0}))⁄4)$,

ψ$_{I}(X,0)=0$.

It is wave packet with the center not in the origin. We can tell you, that this is so called *coherent stat of harmonic oscillator* and wave packet should oscillate around origin with angular frequency
√( $k⁄m)$ same as classical particle.

If you can calculate the previous, then you can try what will be the behaviour of wave packet of different width (e.g. denominator in exponential is different from 4) of how the behaviour will look like with different potential.

Zadal autor seriálu Marek Pechal.

### 5. Series 21. Year - S. sequence, hot orifice and white dwarf

- Derive Taylor expansion of exponential and for $x=1$ graphically show sequence of partial sums of series \sum_{$k=1}^{∞}1⁄k!$ with series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$.

Using the same method compare series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$ and series of partial sums of series \sum_{$k=1}^{∞}x^{k}⁄k!$, therefore series {\sum_{$k=1}^{n}x^{k}⁄k!}_{n=1,2,\ldots}$, now for $x=-1$.

- The second task is to find concentration of electrons and positrons on temperature with total charge $Q=0$ in empty and closed cavity (you can choose value of $Q.)$ Further calculate dependence of ration of internal energy $U_{e}$ of electrons and positrons to the total internal energy of the system $U$ (e.g. the sum of energy of electromagnetic radiation and particles) on temperature and find value of temperature related to some prominent temperature and ratios (e.g. 3 ⁄ 4, 1 ⁄ 2, 1 ⁄ 4, …; can this ratio be of all values?).

Put your results into a graph – you can try also in 3-dimensions.

To get the calculation simplified, it could help to take some unit-less entity (e.g. $βE_{0}$ instead of $β$ etc.).

- Solve the system of differential equations for $M(r)$ and $ρ(r)$ in model of white dwarf for several well chosen values of $ρ(0)$ and for every value find the value which it get close $M(r)$ at

$r→∞$. This is probably equal to the mass of the whole star. Try to find the dependence of total weight on $ρ(0)$ and find its upper limit. Compare the result with the upper limit of mass for white dwarf (you will find it in literature or internet). Assume, that the star consists from helium.

Zadali autoři seriálu Marek Pechal a Lukáš Stříteský.