Series 4, Year 31

Estimate how many grams of ice-cream is possible to be made out of $5 \mathrm{l}$ of liquid oxygen with temperature $-196 \mathrm{\C }$ and unlimited amount of milk and cream with room temperature $22 \mathrm{\C }$? Let's suppose that ice-cream consists of milk and cream only (same mass of both ingredients) and the temperature of the ice-cream should be $-5 \mathrm{\C }$. Use average heat capacity $c\_m = 3{,}45 \mathrm{kJ\cdot kg^{-1}\cdot K^{-1}}$ for milk and $c\_s = 4{,}45 \mathrm{kJ\cdot kg^{-1}\cdot K^{-1}}$ for cream (despite the fact that changes considerably in this temperature range). Find other needed quantities on the internet by yourself.

What is the least number of fidget spinners such that the day on Earth is extended by $1 \mathrm{ms}$ when we spin all of them? Try to guess all the missing quantities.

We have a cylindrical glass with a small hole at the bottom of the glass. The surface area of the hole is $S$. The glass is filled with water and the water flows into a second glass by itself. The second glass has no holes. What shape should the second glass have so that the water level grows linearly inside it? The glass is supposed to have cylindrical symmetry.

Bonus: The bottom of both glasses is at the same high and the glasses are connected by the hole.

We have a black box with three outputs (A, B, and C). We know that it consists of $n$ resistors with the same resistance but we don't know the circuit diagram. So we measure the resistance between each pair of outputs $R\_{AB} = 3 \mathrm{\Omega }$, $R\_{BC} = 5 \mathrm{\Omega }$ a $R\_{CA} = 6 \mathrm{\Omega }$. Your task is to find the minimum possible $n$ and calculate the corresponding resistance of one resistor.

Imagine that we accelerate a usually sized bacteria into velocity $v = 50 \mathrm{km\cdot h^{-1}}$ in the horizontal direction and we let it move freely in air. Estimate the distance traveled by the bacteria before it stops.

The result might be surprising for you. How is it possible to become infected this way with a bacterial infection? Discuss why is it possible despite the result.

We have a satellite and we want to launch it out of the Solar System. We launch it from Earth's orbit so that after some corrections of the trajectory it gets a velocity which is higher than the escape velocity from the Solar System. What is the probability that the satellite will collide with some cosmic material with higher diameter that $d=1 \mathrm{m}$ before leaving the Solar System.

Measure the length density of the catgut which arrived to you together with the tasks. You are forbidden to weigh the catgut.

Hint: You can try to vibrate the string.

1. Find all (three) real roots of the function $\exp (x)-5x^2$. Choose an appropriate method yourself and comment on the reasons behind your choice.
2. Newton’s method works even for functions of complex variable. Your task is to render so called Newton fractals, i.e. areas in complex plane in which choosing an initial guess for Newton’s method leads to converging on a specific root. Render the fractals for the functions $z^3-1$ and $z^6+z^3-1$, where $z$ is a complex number. The derivations of these functions are $3z^2$, and $6z^5+3z^2$ respectively. For calculations and rendering you may utilize the Python code attached to this task.
   Note: Complex derivation, if it exists, can be calculated the same way as normal derivation..
   Bonus: Find as beautiful or interesting Newton fractal as you can.
3. Simulate on computer (or calculate by hand) an elementary cellular automaton abiding by the rule 54 on a grid with size 20 and periodical conditions for at least 10 time steps (more certainly can’t hurt). At the beginning, one arbitrary cell has the value 1, the rest 0. Plot the result on a spacetime diagram.
4. Simulate the changes in roughness $W$ of a 1D surface using a model of random deposition. The roughness $W$ is given by the equation \[\begin{equation*} W(t,L) = \sqrt {\frac {1}{L^d}\sum _i h_i^2-\(\frac {1}{L^d}\sum _i h_i\)^2} \, . \end {equation*}\] Where $d$ is the dimension, $L = 100$ is the length of the surface and $h_i$ is the height of the i-th column. Initially, the surface is perfectly flat. Plot the roughness as a function of time for at least $10^8$ steps (one step $=$ one new particle), discuss the results.
   Note: Random deposition simply means that in each step of the simulation, the height of one randomly selected column will increase by one.