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### (8 points)6. Series 29. Year - E. Malicious coefficient of restitution

If we drop a bouncing ball or any other elastic ball on an appropriate surface, it starts to bounce. During every hit on the surface some kinetic energy of the ball is dissipated (into heat, sound, etc.) and the ball doesn't return to its initial height. We define the coefficient of restitution as the ratio of the kinetic energy after and before the hit. Is there any dependence between the coefficient of restitution and the height which the ball fell from? Choose one suitable ball and one suitable surface (or several if you want) for which you determine the relation between the coefficient of restitution and the height of the fall. Describe the experiment properly and perform a sufficient number of measurements.

### (6 points)6. Series 29. Year - P. iApple

Think up and describe a device that can deduce its orientation relative to gravitational acceleration and convert this information to an electrical signal. Come up with as many designs as you can. (An accelerometer-like device that is in most smart phones.)

### (7 points)5. Series 29. Year - E. photographic

With the aid of a digital camera measure the frequency of the AC voltage in the electrical grid. A smart phone with an app supporting manual shutter speed should be a sufficient tool.

### (3 points)4. Series 29. Year - 3. Save the woods

We have a toilet paper roll with the diameter $R=8\;\mathrm{cm}$ with an inside hollow tube of diameter $r=2\;\mathrm{cm}$. Every layer of the paper has the thickness $d=200µm$ and the layers lies perfectly on top of each other. By how many does the number of pieces of the paper differ had we used a piece of the length $l_{1}=9\;\mathrm{cm}$ instead of $l_{2}=13\;\mathrm{cm}?$ A part of the solution has to be an estimate of the approximation error (if you use one).

**Bonus:** Calculate the precise length of the spiral the toilet paper makes.

### (8 points)4. Series 29. Year - E. Break it down

Measure the tensile strenght of office paper. Use a common office paper with the density 80 g\cdot m^{−2}.

### (5 points)4. Series 29. Year - P. Diet tower

How tall could be a tower built from aluminium cans of diet soft drink?

### (2 points)6. Series 28. Year - 2. breathe deeply

Mage Greyhald celebrated his 100th birthday a long time ago and has begun to fear that Death will pay him a long delayed visit. He decided thus that he will ecase himself into a magic chest, where Death can't reach him. Unfortuntely he forgot to tell the craftsmen to add breathing holes. Air in the chest takes up a volume;$V_{0}=400l$, the percentage of the volume that is oxygen is $φ_{0}=0,21$. With every breath he uses up only $k=20%$ of the volume's oxygen $V_{d}=0,5l$. The frequency of breaths of the mage after the closing of the chest rises according to the relation

$$\\f(t)=f_0 \cdot \frac{\varphi_0}{\varphi (t)}\,,$ wheref_{0}=15breaths\cdot min^{−1}is$ the initial brath frequency is $φ(t)$ and the percetage of the volume that is oxygen at time case $t$. Determine how long until Death will come for Greyhald if the minimum volume of oxygen in the air required for survival is $φ_{s}=0,06$.

DARK IN HERE, ISN'T IT? (Aneb Mirek a jeho kamarád Smrť.)

### (8 points)6. Series 28. Year - E. alchemial

On Discworld it is not unusual to be an alchemist. So we have decided that you should try it. Imagine that you are sitting an exam to enter the guild of alchemists. Together with the brochure of the series you got three wrapped pieces of metal. They are thin plates of metal so be careful with them so that yu won't destroy them and ideally don't touch them. It is your task to find out which (precious?) metals we sent you. We don't require you return the metals and so you can use whatever procedures to determine that, even destructive processes but we shall acknowledge only the sufficiently scientific ones. Your solution will be the description of the procedure required and to determine as precisely as possible the cmposition of the individual specimens and you should menntion the label that was on their packaging. Don't forget that it is even good to determine what metals they aren't.

*Note* If someone wouldlike to become a new participant in this seminar and they would like to solve this task then they should write an email to alchymie@fykos.cz and they will recieve the package from a week later up to 10 days.

Karel wanted to send out the bought gold, platinum and palladium.

### (6 points)6. Series 28. Year - S. mixing

Copy the function $iterace_stanMap$ from the series and using the following commands choose ten very close initial conditions for some $K$.

K=…;

X01=…;

Y01=…;

Iter1 = iterace_stanMap(X01,Y01,1000,K);

…

X10=…;

Y10=…;

Iter10 = iterace_stanMap(X10,Y10,1000,K);

</pre> Between $Iter1$ and $Iter10$ there are hidden a thousand iterations of given initial conditions using the Standard map. As to see how the ten points look after the $nth$ iteration, you have to write

n=…;

plot(Iterace1(n,1),Iterace1(n,2),„o“,…,Iterace10(n,1),Iterace10(n,2),„o“)

xlabel („x“);

ylabel („y“);

axis([0,2*pi,-pi,pi],„square“);

refresh;

</pre> we write $"o"$ into $plot$ so that the points will draw themselves as circles. The rest of the commands is then included so that the graph will include the whole square and that it would have the correct labels.

- Set some strong kicks, $Kat$ least approx. -0,6, and place the 10 initial conditions very close to each other somewhere in the middle of the chaotic region (ie for example „on the tip of a pen“). How do the ten iteration's distances with respect to each other change? Document on graphs. How do the ten initially very close initial conditions change after 1 000 iterations? What can we learn from this about the „willingness to mix“of the given area?
- Take again a large kick and set your ten initial conditions along the horizontal equilibrium of the rotor ie $x=0$, $y=0$. How will these ten initial conditions change in time with respect to each other? What can we say about their distance after a large amount of kickso?

- *Bonus:** Try to code and plot the behaviour of some other map. (For inspiration you can look at the sample solution of the last series.)

### (6 points)5. Series 28. Year - S. mapping

- Show that for arbitrary values of parameters $K$ and $T$ you can express the Standard map from the series express as

$$x_{n} = x_{n-1} y_{n-1},$$

$$\\ y_n = y_{n-1} K \sin(x),$$

where $x$, y$ are somehow scaled d$φ⁄dt,φ$. Show that the physical parameter $K$, x, y$$.

- Look at the model of the kicked rotor from the series and take this time the passed impuls$I(φ)=I_{0}$, after the period $T$ then $I(φ)=-I_{0}$, after another one $I_{0}$ and this way keep on kicking the rotor on and on.
- Make a map $φ_{n},dφ⁄dt_{n}$ on the basis of values $φ_{n-1},dφ⁄dt_{n-1}$ before the doublekick ± $I$ Why not?
- Solve $φ_{n},dφ⁄dt_{n}$ on the basis of some initial conditions $φ_{0},dφ⁄dt_{0}$ for an arbitrary $n$.
- *Bonus:** Try using the ingeredients from this series to design kicking which $will$ result in chaotic dynamics. Take care though because $φ$ is periodic with a period 2π and shouldn't d$φ⁄dt$ unscrew forever through kicking.