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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.

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.

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.

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.}

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

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}} $.

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.

When is it most efficient to pour cold milk into a hot coffee for the coffee to be drinkable as soon as possible? We expect a detailed description of the cooling process rather than a precise calculation.

What happens when we close an almost empty 1.5 liter PET bottle in a well-heated office, let's say at  $t\_k = 26 \mathrm{\C }$, and then walk down the stairs with it? The bottle starts to make cracking noises. What has a bigger effect on this phenomenon, the fact the atmospheric pressure is changing as we walk down 10 floors of the building, or the fact that the temperature on the stairs is a bit lower, let's say, $t\_s = 15 \mathrm{\C }$?

Mirek wants to shoot a rat he sees at the dorm. To that end, he made a simple air gun which can be modeled as a tube with constant cross-section $S=15\;\mathrm{mm}$ and length $l=30\;\mathrm{cm}$ closed on one side and open on the other. Mirek plans to place a bullet of mass $m=2g$ into the tube so that the bullet seals to tube exactly and is fixed at a distance $d=3\;\mathrm{cm}$ from the closed end. He that pumps up the closed section to a pressure $p_{0}$ and then releases the bullet. He wants the speed of the bullet to be at least $v=90\;\mathrm{m}\cdot \mathrm{s}^{-1}$ as it exits the tube. What pressure will he need to achieve if the gas is ideal? Discuss the realism of the situation. Assume the bullet is released by a quasi-static adiabatic process where $κ=7⁄5$, as the gas is diatomic. Assume an external atmospheric pressure $p_{a}=10^5Pa$. Neglect losses due to friction, air resistance and gas compression ahead of the bullet.