3th Round of 21th year

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Series

1. ... English and Scots( points)

Find the change in rotation speed of the Earth, if English and Scots would start to drive on right instead on left. Make just an estimation.

2. ... lift to the skies( points)

Find the properties of material, which you need to make a rope for a lift from geostationary orbit to the Earth surface. Is such material available on Earth?

3. ... jumping on inclined plane(0 points)

figure

A small ball is thrown in horizontal direction onto a inclined plane. The ball starts to jump on the plane and after $N$ contacts it falls at right angle onto the plane. An example of such trajectory for $N$ = 4 is in figure. What is the inclination angle of the plane $α$? Assume, the ball bounce elastically, do not assume rotation.

4. ... particle in field( points)

Assume constant electrostatic field in time. Lets insert a charged particle into the field with zero speed. When we record the trajectory of particle, we see that it does not depend on its mass. Can you explain it?

P. ... high and low tide( points)

High tide and low tide are caused by tidal forces, mainly gravitational force of the Moon. High tide repeats every 12 hours and 25 minutes, however on the Earth we always see two high tides on opposite sides of the Earth. It means that one high tide circles the Earth in approximately twice the time which is 25 hours. Therefore on the equator of length 40 000 km the high tide is moving at speed approximately 40 000/25 km ⁄ h = 1 600 km ⁄ h. This is even more that the speed of sound in air.

However, from the experience we know, that water in ocean does not move at this speed, at the ships does not shipwreck regularly and bring bananas from Kostarika. Is there some mistake in calculation, or do we have to interpret it differently?

E. ... experimenting with Sun( points)

Make a measurement of the height of Sun above horizon at noon time and time from sunrise (middle of Sun disk) till its sunset. Then you can calculated theoretical duration of day and compare with reality and comment on differences.

S. ... wandering of a sailor, pi-circuit and epidemic in Prague( points)

<h3>Integral</h3>

Integrate using Monte Carlo method function e^{$~-~x$&sup2;} 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·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·s^{−1} and dispersion 2 m·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 ⁄ 1 000 000 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.

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