Problem Statement of Series 1, Year 30

Three substances: water, steel and rum are put in a pot, which effectively doesn't convect any heat. The water has mass $m_v=0,5\mathrm{kg}$, temperature $t_v=90°F$ and specific heat capacity $c_v=1kcal\cdot {\mathrm{kg}}^{-1}\cdot K^{-1}$. The steel is in form of a cylinder, that has mass $m_o=200g$, temperature $t_o=60°\mathrm{C}$ and specific heat capacity $c_o=0,260kJ\cdot {\mathrm{kg}}^{-1}\cdot °F^{-1}$. Rum has mass $m_r=100000mg$, temperature $t_r=270K$ and specific heat capacity $c_r=3,5J\cdot g^{-1}\cdot {°\mathrm{C}}^{-1}$. What will be the temperature (in degrees centigrade) of the system when it reaches balance?

Petr likes to ride a bike on a flat road with a speed $v=10\mathrm{m}\cdot {\mathrm{s}}^{-1}$, and his smart bike tells him that his average power is $P=100W$. After an accident, his breaks are bent and they now persistently act on a wheel with a friction force $F_t=20N$ near the circumference. For how long ($t\prime )$ he needs to cycle now (with the same speed $v)$, to do the same amount of work as before, in time $t?$

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.

An observer is on a ship in the open sea, in the height $h$ above the sea surface. There is a horizontal railing in the distance $d$ from him, in such a position, that when he looks directly, perpendicularly at it, the bottom edge of the railings touches the horizon. However, when he looks at the part of the railing that is distance $l$ to the left of the original point, the horizon appears to be $s\pm s_s$ lower. What is the radius of the Earth.

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 ${\alpha }_0$. 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 $\phi$ as a function of time, where the angle $\phi (t)$ 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 ${\phi }_0=50°$ 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.

Did you ever think about, why the clouds simply don't fall down, when they consist of water, which is much denser than air? The raindrops fall to the ground in minutes, so why not clouds? Try to physically explain this. Support all of your claims with calculations.

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.

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1. 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).
2. 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.
3. From the data set attached to this task, generate histograms and try to determine the associated distributions.
4. 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.