# 5. Series 31. Year

### (3 points)1. staircase on the Moon

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.

Dodo read The Moon Is a Harsh Mistress.

### (3 points)2. death rays on the glass

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.

Danka likes solvine more problems simultaneously.

### (5 points)3. wedge

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.

Jáchym robbed the CTU scripts.

### (7 points)4. thermal losses

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 $\lambda '= 0{,}04 \mathrm{W\cdot K^{-1}\cdot m^{-1}}$.

### (8 points)5. sneaky dribblet

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.

Karal wanted to assign something with changing mass.

### (9 points)P. floating mercury

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.

### (10 points)S. Differential equations are growing well

Mirek and Lukáš have already grown their algebra, now they have different seeds.