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### (6 points)4. Series 28. Year - S. Ljapunovian

• Assume a pen of length 10 cm with a center of mass precisely in the middle and $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}.Now$ imagine that you put the pen on the table with a null deviation $δx$ with an accuracy of $ndecimal$ places and with a null velocity. How long after making the pen stand can you be sure with just $n-decimal$ places of the nullness of the displacement?
• Consider a model of weather with the biggest Ljapun's exponent $λ=1.16\cdot 10^{-5}s^{-1}$. The weather forecast stops being useful if its error becomes bigger than 20 %. If you had determined the state of the weather with an accuracy of 1 %, how long do you estimate that your forecast would be good for? Give the answer in days and hours.
• Take Lorenz's model of convection from the last part, copy the function $f(xi,t)$ amd simulate and draw the values of the parameters $X(t)$ for two different trajectories using the commands X01=1;

Y01=2;

Z01=5;

X02=…;

Y02=…;

Z02=…;

nastaveni = odeset('InitialStep', 0.01,'MaxStep',0.1);

pocPodminka1=[X01,Y01,Z01];

reseni1=ode45(@f,[0,45],pocPodminka1,nastaveni);

pocPodminka2=[X02,Y02,Z02];

reseni2=ode45(@f,[0,45],pocPodminka2,nastaveni);

plot(reseni1.x,reseni1.y(:,1),reseni2.x,reseni2.y(:,1));

pause()

</pre> Instead of three dots $X02,Y02,Z02you$ have to give the initial conditions for the second trajectory. Run the code for at least five different orders of magnitude that are all still small and note the time, in which the second trajectory shall differ qualitatively from the first(ie will go in the opposite direaction). Don't decrease the deviation under cca 10^{$-8}$, because then the imprecision's of numerical integration start to show. Chart the dependency of the ungluing time on the order of magnitude of the deviation.

Bonus: Attempt to use the gained dependency of the ungluing time on the size of the deviation estimate Ljapun's exponent. You will need more than five runs and you can assume that at the moment of ungluing it will always overcome some constant $Δ_{c}$.

### (6 points)3. Series 28. Year - S. numerical

• Look at the equations of the Lorenz model and write a script to simulate them in Octave (maybe even refresh your knowledge of the second part of series). Together with the sketching command your script should have the following form: …

function xidot = f(t,xi)

xdot=…;

ydot=…;

zdot= …;

xidot = [xdot;ydot;zdot];

endfunction

config = odeset('InitialStep', 0.01,'MaxStep',0.1);

initialCondition=[0.2,0.3,0.4];

solution=ode45(@f,[0,300],initialCondition,config);

plot3(solution.y(:,1),solution.y(:,2),solution.y(:,3)); </pre> Just instead of three dots fill in the rest of the code (just as in the second part of the series) and use $σ=9,5$, $b=8⁄3.Then$ figure out with a precision of at least units for what positive $r$ the system goes from asymptomatic stopping to chaotic oscillation(it is independent of the initial conditions).

• Here is the full text of the Octave script for simulating and visualising the movement of a particle in a gravitational field of a massive object in the plane $xy$, where all the constants and parameters are equal to one: clear all

function xidot = f(t,xi)

alfa=0.1;

vx=xi(3);

vy=xi(4);

r=sqrt(xi(1)^2+xi(2)^2);

ax=-xi(1)/r^3;

ay=-xi(2)/r^3;

xidot = [vx;vy;ax;ay];

endfunction

config = odeset('InitialStep', 0.01,'MaxStep',0.1);

x0=0;

y0=1;

vx0=…;

vy0=0;

initialCondition=[x0,y0,vx0,vy0];

solution=ode45(@f,[0,100],initialCondition,config)

plot(solution.y(:,1),solution.y(:,2));

pause()</pre>

• Choose initial conditions $x0=0,y0=1,vy0=0$ and and a nonzero initial velocity in the direction $x$ such that the particle will be bound (ie. it won't escape the center.)
• Add to the gravitational force the following force $-α\textbf{r}⁄r^{4}$, where $αis$ a small positive number. Choose gradually increasing $α$ beginning with $α=10^{-3}$ and and show that they cause quasiperiodic movement.

### (2 points)2. Series 28. Year - 2. hedonistic cell

Estimate on the basis of macroscopically measureable quantities the number of cells in the human body and the number of particles in one mole , how many molecules of oxygen„are used“ daily by a human body cell. Find the relevant information needed for the calculation and don't forget to cite your sources properly.

Karel was thinking will taking the underground.

### (5 points)2. Series 28. Year - P. problem of the travelling salesman

When digital mobile phones started to be more common there was often an issue with accepting calls in a car. Nowadays this issue is mostly connected with trains. What factors influence the transmission of data in the GSM network and how can they influence the availability of the signal of the provider? How can one combat this problem?

Ales P. was travelling first class by train and he thought something up.

### (2 points)4. Series 27. Year - 1. Another unsharpened one

A freshly sharpened pencil 6B has a tip in the shape of a cone and the radius of the cone's base is $r=1\;\mathrm{mm}$ a and its height is $h=5\;\mathrm{mm}$. How long will the line that we are able to make with it be if the distance of two graphite layers is $d=3,4Å$ and the track of the pencil has on average $n=100$ such layers?

Mirek was calculating how long his pencils will last.

### (5 points)4. Series 27. Year - P. the true gravitational acceleration

Faleš wanted to determine the gravitational acceleration from an experiment in Prague(V Holešovickách 2 in the first floor/ground floor). In the experiment he was dropping a round ball from a height of a couple of meters above the Earth. Think about what kind of corrections he had to apply when analysing the data. Then think up your own experiment to determine g and discuss its accuracy.

Karel was thinking about the difference between gravitational acceleration and gravitational force

### (5 points)3. Series 27. Year - P. solar power

Can a plane fly using a solar power?

Dominika with Honza and Michal wanted to fly away to warmer countries.

### (2 points)1. Series 27. Year - 1. golden dam

How many bricks of 24-karat gold can you fit into the Orlík dam? What would be the pressure acting on a brick placed at the deepest point? The dimensions of a brick are 10 cm, 3 cm a 1 cm.

Karel wants to be rich.

### (4 points)5. Series 26. Year - P. Prague is flooded!

In 2002 Prague experienced serious floodings. Try to estimate the amount of water that can fit into the Prague subway system. All the important parameters of the subway system like the train sizes, number of stations, length of the tunnels etc. can be found online.

Karel was drowning.

### (2 points)4. Series 26. Year - 1. Who cares about Einstein?

Who is your favorite physicist except of Albert Einstein? What were her contributions to physics? Why do you consider her to be so brilliant? Why should she be famous? Write a short essay about her life and the discoveries she made.

Karel loves history.

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