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biophysics

1. Series 29. Year - 5. chernobyl

If someone would eat 5&nbspµg of the isotope of cesium ^{137}Cs, how long would it take for them to have only0,04 % of the original amount of this isotope? Assume that cesium ^{137}Cs has a half-life of 30,42 let and a biological half-life (the time it takes for half of the original amount of the material to leave the body) is approximately 5 days. Determine also, how many of the particles will have decayed in the body up till then.

Kiki was hungry after her toxikology exam.

1. Series 29. Year - P. decompression illness

Surely you have heard before (at least in a movie) about how it is dangerous to go diving in great depths and then to immediately travel by plane. If a person were to do this he might suffer from the so called. decompression illnes. Describe as accurately as possible what physical processes the human body will undergo during this sickness(how precisely do the phyiscal laws act in this case) and say why precisely they are dangerous for a person. Is it dangerous for people to attempt the opposite sequence of events (ie. first travelling by plane then diving) Je pro lidi nebezpecná i opacná posloupnost akcí, tedy cestování letadlem a následné potápení? (when solving this you can utilise all available resources but in the end you must use your own words to describe it)

Mirek felt a sudden decrease in pressure after a successfully done exam in mathematical analysis.

6. Series 28. Year - 1. ...au

The Turtle A'Tuin, on the shell of which the four elephants that carry on their backs Discworld stand, isn't tiny. Let us assume that we would be bored with the sphericity of our Earth and we would want to exchange it for a circular disc with the same mass and density and with the width $h=1\;\mathrm{km}$ carried by the same turtle-elephant band. In case that the turtle would have hit with the tip of its tale into a planetoid, how long would it take for it to notice the impulse of pain, given that her tail and her brain are connected by a very long neuron? (This neuron is approximately as long as the diameter of the disc) How much earlier/later would A'tuin realise this pain (the length of the neuron is equivalent to its length 18 000 km)? For a numerical estimate assume that the speed of the spread of the signal in the large animals is the same as with normal land animals who experience a speed of $v≈120\;\mathrm{m}\cdot \mathrm{s}^{-1}$.

víte snad o lepším vymyšleném světě, než je Zeměplocha? Kiki ne!

4. Series 28. Year - P. unnamed snack bar

Considering biochemical processes in the human body and its mechanics estimate, how much energy is used by a cyclist to rise a thousand vertical meters if the average gradient is 5 %.

Mirek and Tomas were thinking about how many hills can be climbed with a certain snack bar (not naming one due to the lack of funding).

4. Series 23. Year - 1. Green or Blue?

The main source of atmospheric oxygen is photosynthesising plants. Imagine they become extinct. How long would the world’s oxygen stock last given that the consumption of the planet stays unchanged? The pieces of information that you may need can be found on the internet.

by Aleš while reading cheap sci-fi

1. Series 22. Year - E. what is smelling here?

Measure difference of density of fresh and spoiled egg and find out its time dependence! Assume statistical approach.

Hint: Eggs get spoiled very quickly if exposed to sun light and heat.

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

<h3>Integral</h3>

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.

<h3>Wandering of sailor</h3>

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.

<h3>Pi-circuit</h3>

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.

<h3>Epidemic in Prague</h3>

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.

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