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## thermodynamics

### (3 points)4. Series 33. Year - 2. Mach number

Planes at high flight levels are controlled using the Mach number. This unit describes velocity as a multiple of the speed of sound in the given environment. However, the speed of sound changes with height. What is the difference in the speed of a plane, flying at Mach number $0{,}85$, at two different flight levels FL 250 ($7\;600 \mathrm{m}$) and FL 430 ($13\;100 \mathrm{m}$)? At which flight level is the speed higher and by how much (in $\jd {kph}$)? The speed of sound is given by $c =\(331{,}57+0{,}607\left \lbrace t \right \rbrace \) \jd {m.s^{-1}}$, where $t$ is temperature in degrees Celsius. Assume a standard atmosphere, where temperature decreases with height from $15 \mathrm{\C }$ by $0,65 \mathrm{\C }$ per $100 \mathrm{m}$ (for heights between $0$ and $11 \mathrm{km}$) till $-56{,}5 \mathrm{\C }$, and then remains constant till $20 \mathrm{km}$ above mean sea level.

### (10 points)3. Series 33. Year - P. meteor swarm

Is it possible that droplet of rain evaporates earlier than it hits the ground? Think up suitable model of evaporating of rain droplets during their fall and show under what conditions (some of the relevant parameters are initial radius, behaviour of outdoor temperature in relation to height above sea level) the droplet can evaporate completely. You can assume that the droplet arises suddenly in particular height $h_0$ with initial radius $r_0$ and in first approximation it falls through dry atmosphere. And when is it possible that the droplet freezes?

Mirek waited for rain.

### (3 points)4. Series 32. Year - 1. cube with the air

Consider a hollow cube with edge of $a = 20 \mathrm{cm}$ filled with air. Air as well as the enviroment has a temperature of $t_0 = 20 \mathrm{\C }$. We will cool down the air inside the cube to the temperature of $t_1 = 5 \mathrm{\C }$. Find the force acting on each of the cube's side. The cube has got a fixed volume. The pressure outside of the cube equals $p_0 = 101{,}3 \mathrm{kPa}$.

Danka was irritated by the shower curtain.

### (3 points)3. Series 32. Year - 2. efficient coffee

It is 2 am and Jáchym is going to make a coffee. He places a kettle with the heat capacity of $C_k$ on a hot plate, which is made of a cast-iron cylinder of a radius $r$ and of height $h$. The kettle contains water with a volume of $V$ with an initial temperature of $T\_v$. The rest of the system has got an initial temperature of $T\_s$. What is the overall efficiency (ratio of energy absorbed by water vs energy input) of water heating from its initial temperature $T = 100 \mathrm{\C }$ $(T\_s, T\_v < T).$ Assume, that the heat transfer is very fast and therefore there is no heat loss. You can estimate the unknown values or find them in physics tables.

Jáchym run out of energy drink

### (6 points)3. Series 32. Year - 3. heat in the Dyson sphere

What would be the diameter of a Dyson sphere that would surround a star with the luminosity of the Sun, so the temperature on the outer surface of the sphere is $t= 25 \mathrm{\C }$?. Don't consider the presence of the atmosphere in the Dyson sphere. A Dyson sphere should be a relatively thin concave structure of spherical shape surrounding the star.

Karl likes Dyson spheres.

### (3 points)2. Series 32. Year - 2. Finnish sauna

Imagine that Dan has a sauna with dimensions $2,5 \mathrm{m}$ x $3 \mathrm{m}$ x $4 \mathrm{m}$ with a relative humidity of $20 \mathrm{\%}$ and temperature of $90 \mathrm{\C }$. How much water would have to evaporate, so the relative humidity inside the sauna is $35 \mathrm{\%}$? The water evaporates inside the sauna without changing the overall temperature.

Karl was thinking about corroding Dan’s swimsuit

### (10 points)2. Series 32. Year - P.

Create an accurate weather forecast for address V Holešovičkách 2, Prague 8, for Wednesday 14th of November from 12:00 to 15:00. How will the weather change throughout the whole day? You are allowed to use previous data about the weather in this area (remember you are only permitted to use data until 10th of November). It is necessary to justify your weather prediction, write down references and ideally to use as many data and resources as possible.

Karl listened to radio on a motorway

### (3 points)6. Series 31. Year - 2. hot wire

Calculate the current, that needs to pass through a metal wire of a diameter $d = 0{,}10 \mathrm{mm}$ located in a vacuum bulb, so that its temperature stays at $T = 2 600 K$. Assume the surface of the wire radiates like an ideal black body and neglect any losses by heat conduction. The resistivity of the material of the wire at the given temperature is $\rho = 2{,}5 \cdot 10^{-4} \mathrm{\Ohm \cdot cm}$. \taskhint {Hint}{Use the Stefan-Boltzmann's law.}

Danka was contemplating the light bulb efficiency

### (12 points)6. Series 31. Year - E. don't play with matches

Measure the speed with which a wooden skewer burns as a function of its tilt with respect to the vertical.

Because the gasoline that Karel suggested was a bit too much.

### (7 points)5. Series 31. Year - 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}} $.