Problem Statement of Series 5, Year 31

If we once colonized the Moon, would it be appropriate to use stairs on it? Imagine the descending staircase on the Moon. The height of one stair is $h=15 \mathrm{cm}$ and it's length is $d=25 \mathrm{cm}$. Estimate the number $N$ of stairs that a person would fly over if he walked into the staircase with a velocity $v=5{,}4 \mathrm{km\cdot h^{-1}}=1{,}5 \mathrm{m\cdot s^{-1}}$. The gravitational acceleration on the Moon's surface is six times weaker than on Earth's surface.

A light ray falls on a glass plate with an absolute reflective index $n = 1,5$. Determine its angle of incidence $\alpha _1$ if the reflected ray forms an angle $60 \dg$ with the refracted ray. The board is stored in the air.

We have two wedges with the masses $m_1$, $m_2$ and the angle $\alpha $ (see figure). Calculate the acceleration of the left wedge. Assume that there is no friction anywhere.

Bonus: Consider friction with the $f$ coefficient.

At what temperature does the indoor environment of the flat in a block of flats stabilise? Consider that our flat is adjacent to other apartments (except its shorter walls), in which the temperature $22 \mathrm{\C}$ is maintained. The shorter walls adjoin the surroundings where the temperature is $ - 5 \mathrm{\C}$. The inside dimensions of the flat are height $ h = 2{,}5 \mathrm{m}$, width $ a = 6 \mathrm{m}$ and length $ b = 10 \mathrm{m} $. The coefficient of the specific thermal conductivity of the walls is $ \lambda = 0{,}75 \mathrm{W\cdot K^{-1}\cdot m^{-1}} $. The thickness of the outer walls and the ceilings are $ D\_{out} = 20 \mathrm{cm}$, and the thickness of the inner walls are $ D\_{in} = 10  \mathrm{cm}$.

How will the result be changed if we add polystyrene insulation to the building? The thickness of the polystyrene is $ d = 5 \mathrm{cm}$, and its specific heat conductivity is $ \lambda '= 0{,}04 \mathrm{W\cdot K^{-1}\cdot m^{-1}} $.

Let's take a rounded drop of radius $ r_0 $ made of water of density $ \rho \_v $ which coincidentally falls in the mist in the homogeneous gravity field $g$. Consider a suitable mist with special assumptions. It consists of air of density $\rho \_{vzd}$ and water droplets with an average density of $ rho\_r $ and we consider that the droplets are dispersed evenly. If a drop falls through some volume of such mist, it collects all the water that is in that volume. Only air is left in this place. What is the dependence of the mass of the drop on the distance traveled in such a fog?

Bonus: Solve the motion equations.

Try to invent as much „physics tricks“ as possible thanks to which mercury would float on the liquid water for at least a limited time. The more permanent solution you find, the better.

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1. Solve the two-body problem using the Verlet algorithm and the fourth-order Runge-Kutta method (RK4) over several (many) periods. Use a step size large enough for the numerical errors to become significant. Observe the way the errors manifest themselves on the shape of the trajectories.
2. Solve for the time-dependent position equation of a damped linear harmonic oscillator described by the equation $\ddot {x}+2\delta \omega \dot {x}+\omega ^2 x=0$, where $\omega $ is the angular velocity and $\delta $ is the damping ratio. Change the parameters around and observe the changes in the oscillator’s motion. For which values of the parameters is damping the fastest?
3. Model sedimentation using the method of ballistic deposition \[\begin{equation*} h_i(t+1) = max(\(h_{i-1}(t), h_i(t)+1, h_{i+1}(t)\)) \, , \end {equation*}\] where $h_i$ is the height of i-th column. And study the development of the roughness of the surface $W(t,L)$ (see this year’s series 4, problem S). Initially (for small values of $t$) the roughness is proportional to some power of $t$: $W(t,L) \sim t^{\beta }$. For large values of $t$, however, it is proportional to some (possibly different) power of the grid length $L$. $W(t,L) \sim L^{\alpha }$. Find the powers $\alpha $ and $\beta $. Choose an appropriate step size so that you could study both modes of sedimentation. The length of the surface should be at least $L = 256$. (Warning: the simulations may take several hours.)
4. Simulate on a square grid the growth of a tumor using the Eden growth model with the following variation: when a healthy and an infected cell come into contact, the probability of the healthy one being infected is $p_1$ and the probability of the infected one being healed is $p_2$. Initially, try out $p_1 \gg p_2$, the proceed with $p_1 > p_2$ and then with $p_1 < p_2$. At the beginning, let only 5 cells (arranged into the shape of a cross) be infected.
   Describe qualitatively what you observe.
5. Rewrite the attached code for the growth of a fractal (diffusion limited aggregation model) on a hexagonal grid to the growth of a fractal on a square grid and calculate the dimension of the resultant fractal.

Note: Using the codes attached to this task is not mandatory, but it is recommended.