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## gas mechanics

### 1. Series 32. Year - 1. baloons

How many balloons of volume $V=10 \mathrm{\ell }$ filled with helium of density $\rho _{\scriptscriptstyle \rm He} = 0{,}179 \mathrm{kg\cdot m^{-3}}$ are needed to lift Filip, whose mass is $m_{\scriptscriptstyle \rm F} =80 \mathrm{kg}$, and keep him afloat in air of density $\rho _{v} =1{,}205 \mathrm{kg\cdot m^{-3}}$? How many would be necessary to lift Danka, who weighs $m_{\scriptscriptstyle \rm D} =50 \mathrm{kg}$? Neglect the mass of the empty balloons.

Danka gave Filip a promo balloon to lift his mood.

### 1. Series 31. Year - P. model plane at the ISS

How would a plane behave in microgravity (in other words ignore the effects of gravity)? Describe what effects would the ailerons, the rudder, the elevators and thrust vectoring have of the engines? Which acrobatic maneuvers would be possible? (E.g. a flat spin probably wouldn't be).

Erik read discussions on the internet.

### 6. Series 30. Year - 4. shoot your rat

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.

Karel wanted to find out if the solvers could pass the Masters programme admissions at MFF

### 5. Series 30. Year - 5. balloon

Consider a balloon with mass $m$ (blown up) and volume $V$ filled with helium. An infinite string of length density $τ=10gm^{-1}$ is tied to the balloon. Assuming the atmosphere is isothermal, in which the pressure depends on height $z$ as

$p=p_0e^{-z/z_0}\$,,

($z_{0}$ is a parameter of the atmosphere), what is the maximum height the balloon will reach?

### 4. Series 30. Year - 4. heat engine

Consider a heat engine filled with a diatomic gas. This engine works thanks to a cycle ABCDEFA as shown in the picture. The 6 processes that make up the cycle are

- A $→$ B - isobaric heating from a state 4$p_{0}$ and $V_{0}$ (let us denote temperature at A 4$T_{0})$ to a state with volume 3$V_{0}$,
- B $→$ C - isothermic expansion to volume 4$V_{0}$,
- C $→$ D - isochoric cooling to pressure $p_{0}$,
- D $→$ E - isobaric cooling to volume 2$V_{0}$,
- E $→$ F - isothermic compression to volume $V_{0}$,
- F $→$ A - isochoric heating to pressure 4$p_{0}$. Determine the remaining state variables in B, C, D, E, and F, the maximal and the minimal temperature of the ideal gas during the process (as a multiple of $T_{0})$, heat received and lost by the gas in each process, and the overall efficiency of the engine. Compare this efficiency with that of a Carnot engine working between the same minimal and maximal temperatures. Assume for simplicity that the molar amount of the gas does not change and there are no chemical changes during the cycle. A sketch can be seen in figure.

**Bonus:** Do the same for a much simpler „square“ cycle, ABCDA, where the gas starts in a state $p_{0}$, $V_{0}$ and $T_{0}$ and izochorically heats up to 4$p_{0}$, isobarically heats up and expands to 4$V_{0}$, isochorically cools down to $p_{0}$ and isobarically cools down to $V_{0}$. Compare the efficiency of these two heat engines and suggest which one is better.

Karel was alternately warm and cold

### 2. Series 30. Year - 5. tea container problem

We have a tea container with a tap near the bottom and an airtight lid (maybe you know these from your school canteen). Determine the volume of tea we can pour from the tap before we have to open the valve to equalize the pressure in the container.

Lukáše na soustředění trápilo, kolik čaje má být ve várnici.

### 2. Series 30. Year - P. an effective machine

Guns can be considered to be heat engines. Calculate the efficiency of a gun, say a pistol or a rifle.

Michal původy svých nápadů raději nesděluje.

### 1. Series 30. Year - P. The sky is falling

Did you ever think about, why the clouds simply don't fall down, when they consist of water, which is much denser than air? The raindrops fall to the ground in minutes, so why not clouds? Try to physically explain this. Support all of your claims with calculations.

Mirek se zadíval na nebe a dostal strach.

### 6. Series 29. Year - S. A closing one

- Find, in literature or online, the change of enthalpy and Gibbs free energy in the following reaction

$$2\,\;\mathrm{H}_2 \mathrm{O}_2\longrightarrow2\,\mathrm{H}_2\mathrm{O},$$

where both the reactants and the product are gases at standard conditions. Find the change of entropy in this reaction. Give results per mole.

- Power flux in a photon gas is given by

$j=\frac{3}{4}\frac{k_\;\mathrm{B}^4\pi^2}{45\hbar^3c^3}cT^4$.

Substitute the values of the constants and compare the result with the Stefan-Boltzmann law.

- Calculate the internal energy and the Gibbs free energy of a photon gas. Use the internal energy to write the temperature of a photon gas as a function of its volume for an adiabatic expansion (a process with $δQ=0)$.

**Hint:** The law for an adiabatic process with an ideal gas was derived in the second part of this series (Czech only).

- Considering a photon gas, show that if $δQ⁄T$ is given by

$$\delta Q / T = f_{,T} \;\mathrm{d} T f_{,V} \mathrm{d} V\,,$$

then functions $f_{,T}$ and $f_{,V}$ obey the necessary condition for the existence of entropy, that is

$$\frac{\partial f_{,T}(T, V)}{\partial V} = \frac{\partial f_{,V}(T, V)}{\partial T} $$