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### (12 points)3. Series 30. Year - E. reflective snap band

Measure as many characteristics of a high-vis snap band as you can. We are specifically interested in:

- The band contains a piece of metal on the inside, which can be bent lengthwise (when coiled) or along the shorter edge (when straight). What are the radii of curvature of these bents if there is no external force?
- If the band is straight and we start bending it in one place, at what angle will it snap into the bent state? At what angle does it become straight again? (Do we see any hysteresis?)
- What is the torque required to bend the band?
- Is one of the states (bent or straight) more energetically favourable? Estimate by how much. Unfortunatelly, we are unable to mail these bands abroad, we therefore ask that you obtain one yourself and include pictures of the band you used in your solution.

Erik could not bend his …

### (10 points)3. Series 30. Year - S. limiting

- Try to, in your own words, describe the method for creating interval estimations of expected value of a general distribution of measured data (it is sufficient to describe the following: central limit theorem (CLT), covariance, correlation (Pearson correlation coefficient), multidimensional CLT, law of propagation of uncertainty and its uses.) It’s not necessary to describe the concepts mathematically, a brief description in your own words is sufficient.
- In the attached datafile
*mereni3-1.csv*there are measurements of a certain physical quantity $v$. Assume we cannot be sure whether the measured data have a normal distribution. Find the uncertainty (standard deviation) of the measurements (neglect the type B uncertainty), set up the interval estimations using CLT and briefly interpret their meaning. How would the results (and interpretation) change if only the first quarter of the data was available? - Suppose our aim is to measure a physical quantities $x$ and $y$, which we will then plug into the equation \[\begin{equation*} v= \frac {1}{2} x y^2 . \end {equation*}\] and suppose that we are certain that all measurements are independent and we already have measured a significant amount of data, processed them and there are the results \[\begin{align*} x &= (5,2\pm 0.1) , \\ y &= (12{,}84\pm 0.06) . \end {align*}\] Estimate the value of $v$ and its uncertainty.

**Hint:**These equations may come in handy $$\frac{\partial}{\partial x} \( \frac {1}{2} x y^2 \) = \frac {1}{2} y^2\, ,$$ $$\frac{\partial}{\partial y} \( \frac {1}{2} x y^2 \) = x y \, .$$ - Using a computer simulation demonstrate the validity of central limit theorem i.e. generate $n$-tuples (sequences of $n$ real numbers) of independent realizations of a random variable, which does not have a normal distribution (use the exponential, uniform and Poisson distributions with arbitrary parameters) and show, using a histogram, that applying the transformation \[\begin{equation*} \sqrt {n}\frac {\overline {x_n - \mu }}{S_n} , \end {equation*}\] to the data will (approximately) yield a normal distribution $N(0, 1)$.

**Bonus:** Suppose our aim is to measure physical quantities $x$ and $y$, which we will then plug into \[\begin{equation*} v= x^2 \sin y .
\end {equation*}\] Assume the most general model of measurement (i.e. the measured data do not have a normal distribution and the measurements of $x$ and $y$ may not be independent. In the datafile *mereni3-2.csv* you may find the results of measurements of $x$ and $y$, determine the uncertainty of $v$ and construct an interval estimation of $v$.

For data processing and creating of plots use the *R* programming language. In the attached scripts is explained all necessary syntax.

### (12 points)2. Series 30. Year - E. one full fat milk, please

Milk with higher fat content should be „whiter“ – more light is scattered and less is transmitted. Conduct a measurement of the fat content of milk with the help of a color scale (contact us at fykos@fykos.czto get the pdf with the scale – you have to print it yourself). The difference in whiteness is most apparent when you add a dye to each glass of milk. You can use e.g. black ink or any other dye, but with different colors you have to create your own color scale which you have to add to your solution. Use different types of milk and mixture of milk with water. Discuss the reliability of this method of measurement.

Mára byl bledý jako stěna.

### (10 points)2. Series 30. Year - S. guessing problem

- Describe in your own words the purpose of interval estimation of mean of a normal distribution and explain its physical interpretation (it is sufficient to describe, in your own words, the following concepts: physical interpretation of the estimation of expected value, difference between point and interval estimation, measurement uncertainty). It’s not necessary to state the exact mathematical derivations. It’s sufficient to briefly explain the concepts and their properties.
- Attached to this task, in the file
*mereni1.csv*there are measured values of a certain physical quantity (assume type B uncertainty of B $s\_B = 0{,}1$). Create both the point and interval estimations of the measured physical quantity and try to interpret their meaning. - Suppose we measure a certain physical quantity and we know that due to the method being used, the measured values will have a variance equal to a constant $c$ (ignore the type B uncertainty). How many measurements do we need to make to achieve an uncertainty below $s$?
- In the attached file
*mereni2.csv*there are data of measurements one physical quantity two different ways (neglect type B uncertainty). Which method used more precise measurement equipment? Which method produced a more precise results Briefly give reasons for your answers.

**Bonus:** Try to rigorously derive that in a normal distribution the sample variance is an unbiased estimate of the real variance (i.e. the mean of sample variance is equal to the real variance). For the solution of this problem you may use any and all sources (if you cite them correctly).

For data processing and creating the plots, you may use the *R* programming language. Most of these tasks can be solved by slightly altering the attached scripts.

Michal guessed the optimal wording of the problem, let's hope he was right.

### (5 points)1. Series 30. Year - 3. Bouncy bounce

Let's have an ideal bouncy ball (with coefficient of restitution equal to one and negligible dimensions). We throw this bouncy ball down an infinitely long staircase, where a step has height $h$ and length $l$. The bounces happen without any influence from friction. Describe the relation between the maximum height reached (measured from the first step) after $n-th$ bounce and the initial parameters.

Lubošek potkal v městské dopravě Mikuláše.

### (7 points)1. Series 30. Year - 5. On a walk

Katka decided to go for a walk with her pet rat. They arrived on a flat meadow and when the rat was at a distance $x_{1}=50\;\mathrm{m}$ from Katka, she threw him a ball with the speed $v_{0}=25\;\mathrm{m}\cdot \mathrm{s}^{-1}$ and an angle of elevation $α_{0}$. In that moment, he started running towards her with the speed $v_{1}=5\;\mathrm{m}\cdot \mathrm{s}^{-1}$. Find a general formula for an angle $φ$ as a function of time, where the angle $φ(t)$ is the angle between the horizontal plane and the line between the rat and the ball. Draw this function into a graph and, based on the graph, determine, whether it's possible for the ball to obscure the Sun for the rat, when the Sun is situated $φ_{0}=50°$ above the horizon in the direction of the running rat. Use the acceleration due to gravity $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}$ and for simplicity imagine we are throwing the ball from a zero height.

Mirek pozoroval, co se děje v trávě.

### (12 points)1. Series 30. Year - E. Pechschnitte

Does bread always falls on the side that has the spread on it? Explore this Murphy's law experimentally with emphasis on statistics! Does it depend on the dimensions of the slice, or the composition and the thickness of the spread? Try to explain the experimental results with a theory. Use a sandwich bread.

Terka má stůl ve špatné výšce.

### (10 points)1. Series 30. Year - S. random one

- Try to explain in your own words what is a random variable and what are its properties (explanations of following concepts are required: random variable, distribution of a random variable, realization of a random variable, mean, variance, histogram).
- Generate graphs of probability distribution functions for the following distributions of random variable: normal, exponential, uniform (continuous) and Poisson. Describe what happens when you alter the parameters of aforementioned distributions.
- From the data set attached to this task, generate histograms and try to determine the associated distributions.
- Suppose we define a random variable $X$ as a result of a „fair“ (all outcomes are equally probable) six-sided dice roll. Determine the distribution function of the random variable $X$ and calculate $\mathrm {E} X$ and $\mathrm {var} X$.

**Bonus:** Name two different distributions of random variables with the same mean and variance.

For data processing and creating the plots, you may use the *R* programming language. Most of these tasks can be solved by slightly altering the attached scripts.

Michal created a random problem, hopefully it won't be too hard.

### (2 points)6. Series 29. Year - 1. It's about what's inside of us

In the year 2015, a Nobel prize for Physics was given for an experimental confirmation of the oscillation of neutrinos. You have probably already heard about neutrinos and maybe you know that they interact with matter very weakly so they can pass without any deceleration through Earth and similar large objects. Try to find out, using available literature and Internet sources, how many neutrinos are at any instant moment in an average person. Don't forget to reference the sources.

### (4 points)6. Series 29. Year - 3. Going downhill

We are going up and down the same hill with the slope $α$, driving at the same speed $v$ and having the same gear (and therefore the same RPM of the engine), in a car with mass $M$. What is the difference between the power of the engine up the hill (propulsive power) and down the hill (breaking power)?