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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?

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.

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.

• 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 $g$.

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?

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

Download program Planeta.pas, ThreeStars-TriHvezdy.pas. As an excerscice try to understand the programs, modify them and change them.

• First task is to add into at least two of programs something new. For example change origin conditions, add another planet or star. Also try to change gravitational law – calculate with force $F=A⁄R+B⁄R$, where $A$ and $B$ are constants etc.
• Assume two same mass stars, which orbit around each other on circle. In axis of this circle the third star is approaching, at the beginning with the same speed as the two stars are orbiting. It has also the same mass. Create simulation what will happen.

As solution of both problems please send us pictures with extensive commentary. State at least short explanation what is at the images. Also mention how have you proceed during calculations and which computation system have you used.

The space probe Huygens (named after the discoverer of Titan) has landed on Titan on Friday 14th of January 2005. The mother ship Cassini was travelling 7 years to reach Saturn. It is the most distant landing of man-made probe in history.

The landing module of nett weight (without fuel) $m$, equipped with reactive engine, levitated at fixed position above the surface of the moon (gravitational acceleration is $g)$. The module had available fuel of mass $M$ and energy $E_{0}$ which is used to accelerate the fuel (the speed and amount of fuel which is ejected from the engine can be regulated without any limitations). What is the maximum time the module can stay in constant height? Suggest how the speed and amount of fuel should be programmed to reach this maximum time.

<h3>Deep space, year 2224</h3>

Federation got a message: in sector 0056 is happening some collection of mysterious matter with fascinating properties. Matter is black, as it does not interact with electromagnetic radiation, its particles interacts only with gravity. Federation had sent a small two-crew space ship to explore. During a trip the space hit asteroid and was deviated out of its original direction.

The crew managed to land on small unknown planet of class „M“. During the free fall the space-ship shield was heated up due to the contact with the planet's atmosphere. From the speed of free fall the gravitation acceleration on the planet was established as 0.5 $g$.

Planet is covered by very strange trees. Observation showed that they are very similar to foliage-trees known from the Earth and they reach maximum possible height. However the branches are covered by leafs, also strobiles grows on them.

During one of exploring trip on the surface of this new planet humanoid creatures were discovered. They lived in heights of trees, which can be used to send SOS signal. The humanoids used some sort of color stones to send color-coded signals. The stones are transparent and allow to pass some colors. If the light from different stones is combined they give different (other) colors.

After sending SOS-signal from the top of trees the space ship USS Odyssey came to collect stranded crew. They then sent all the results from this adventure trip back to headquarters on the Earth.

A cloud of mysterious intergalactic matter is isotropic, homogeneous and spherical shape formed by gas. At the beginning the whole cloud is completely in rest, without any movement of matter, with total mass of $M$. Find out how the local speed of collapse will change during gravitational collapse. Comment on the speed of collapse at the moment just before collapsing into a single point.

Do you believe that average density of Saturn is lower than density of water?

You can look at Saturn in a telescope. Except the rings you will see several moon (if they are not behind Saturn in the moment – in such case you have to wait maximum 6 hours, which is the time of transit over the planet circle). You will find, that Titan orbits around Saturn once in 16 days. Can you calculate average density of Saturn from observation of Titan. If not, justify why and if yes, calculate it. You will notice one curiosity!