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## gravitational field

### 4. Series 26. Year - P. Mrazík

In the fairy tale Mrazik, Ivan fought several bandits, stole their clubs, and threw them so high up into the sky that they did not fall back until half a year later. What is the altitude the clubs had to reach in order to stay in the air for so long? Make a first guess and then go on and improve it. Carefully analyze all the approximations you made and explain why are these estimates most likely wrong. Furthermore, explain why it makes no sense for the clubs to fall back at the same spot where Ivan threw them.

Lukáš was watching fairy tales.

### 2. Series 26. Year - P. messing with gravity

What if the gravitational constant suddenly doubled (without affecting the value of other physical constants)? What if it increased a hundred times? Discuss the impact the change would have on the life on the Earth and on the trajectories of bodies in the universe.

### 6. Series 25. Year - 2. space station

Estimate the minimal energy needed to put a space station on an orbit around the Earth. You can work with the values valid for the International Space Station which orbits the Earth at height approximately $h=350\;\mathrm{km}$ and has mass $m=450000\;\mathrm{kg}$. Explain why is this estimate minimal and why, in reality, much more energy is needed.

Astrokarel.

### 4. Series 25. Year - 5. gas leakage

What is the mass percentage of Earth's atmosphere that escapes to the outer space each year? Assume the atmosphere reaches 10 km above the ground, the pressure is everywhere the same (equal to the pressure at sea level) and it consists of ideal gas at tepmperature 300K whose molecular speeds obey the Maxwell-Boltzmann distribution. Also assume that the gravitational field is homogeneous.

Aleše napadlo při úniku.

### 1. Series 25. Year - P. Cubeworld

Imagine that the Earth is not a sphere but a cube. Would it be able sustain this shape? If so for how long and on what parameters would this time depend on? What about the life on such a planet? What gravitational force would you feel while walking around this planet?

Mysterious…

### 4. Series 24. Year - 2. To the Sun

Karel has decided to throw his notes into the Sun.
**Help:** him calculate the necessary initial velocity of his notebook so that it reaches the Sun. You can neglect the frictional froces acting on his notebook and you can also assume that the trajectory of the Earth around the Sun is circular. You know the masses of the Earth and the Sun as well as the distance of the Earth from the Sun.

Karel

### 6. Series 22. Year - P. flying people

Titan (satellite of Saturn) is very cold (surface temperature is 94 K) with nitrogen atmosphere, with icy surface and hydrocarbon lakes on the surface. Radium of Titan is 5150 km, mass is 1 ⁄ 45 mass of the Earth, thickness if its atmosphere is 200 km and pressure on the surface is 1,5 bar.

Based on previous facts calculate acceleration due to gravity on the surface and estimate density of atmosphere. Comparing with parameters of birds at earth conditions decide, if feathered person could fly on Titan.

létat se zachtělo Honzovi P.

### 1. Series 22. Year - S. equivalence principle

- What had to be a conditions for Galileo to fail its experiment? Leaning tower in Pisa is $h=55\;\mathrm{m}$ high. Assume that both balls have the same diameter $R=8\;\mathrm{cm}$ and that one is made from lead of density $ρ=11300\;\mathrm{kg}\cdot \mathrm{m}^{-3}$. What density must be the second ball to achieve the difference in time of impact to be bigger than $ΔT=0,3s?$

- What is precision of original Eötvös measurement of equality of gravitational and inertial mass for neutrons and protons, if in wood neutrons make 50 percent of mass and in platinum 60 percent of mass? Neglect mass of electrons and binding energy.

- Verify the assumption, that in Budapest is $g_{s}′$ negligible comparing to $g$.

Zadali autoři seriálu Jakub Benda a Pavel Motloch.

### 3. Series 21. Year - P. high and low tide

High tide and low tide are caused by tidal forces, mainly gravitational force of the Moon. High tide repeats every 12 hours and 25 minutes, however on the Earth we always see two high tides on opposite sides of the Earth. It means that one high tide circles the Earth in approximately twice the time which is 25 hours. Therefore on the equator of length 40 000 km the high tide is moving at speed approximately 40 000/25 km ⁄ h = 1 600 km ⁄ h. This is even more that the speed of sound in air.

However, from the experience we know, that water in ocean does not move at this speed, at the ships does not shipwreck regularly and bring bananas from Kostarika. Is there some mistake in calculation, or do we have to interpret it differently?

Úlohu navrhl Honza Hradil.

### 1. Series 21. Year - 3. weighting the Sun

Suggest several methods for estimating the mass of the Sun. Explain in detail each of them and estimate mass of the Sun.

K zahřátí mozků do nového ročníku FYKOSu zadal Pavel Brom.