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Imagine you are going on a date with your girlfriend/boyfriend and you end up watching the sunset on the beach. The sun above the sea horizon looks very romantic, so to prolong this special moment, you decide to use a forklift to lift you up. The forks of the forklift move up with such speed that you can see the sun touching the horizon at any moment. Determine the speed of the forks.

Lukas has been weightlifting and he managed to make a black hole of mass 1 kg. As he isn't too fond of quantum field theory in curved spacetime, the black hole does not radiate. Lukas drops this hole and it begins oscillating within the earth. Try to estimate how long would it take for the mass of the black hole to double. Is it safe to make black holes at home?

Imagine that around the Sun on a circular orbit is a convex lens with a diameter that is equal to the diameter of the Sun, the focal point of which orbits with a sufficient precision on the orbit of Earth. Determine how the lens will burn the Earth during one of its orbit (ie. how much solar energy will be given to Earth by the lens), if it orbits at the distance of Mercury and compare it with the state where it will be as far as Venus.

Bonus: Consider the eclipse that the lens will cause during its orbit.

Can you see the Moon from Mars with a naked eye.Ground your answer in calculations.

A planet is orbiting around a star on a circular orbit and a moon is orbiting around the planet on a circular orbit in the plane of the planet's orbit. We know that, during the eclipse of the sun the angular size of the moon is the same as the angular size of the sun if observed from the surface of the planet (the moon perfectly covers the sun). Furthermore we know that the planet perfectly covers the moon during the moon's eclipse. Determine the ratio of the radius of the planet $R$ and the moon $r$, if the distance of the planet from the star is very large compared to the distance of the moon from the planet $L$ and this is in turn larger by several orders of magnitude than the dimensions $R$, $r$.

It is known that the Moon when it is full has the apparent magnitude of approximately -12 mag and the Sun during the day has the apparent magnitude of -27 mag. Try to figure out what is the apparent magnitude of the Moon directly before a solar eclipse, if you know that the albedo of the Earth is approximately 0.36 and the albedo of the Moon 0.12. Presume that light after reflection disperes the same way on the surface of both the Moon and Earth.

How fast does the boundary between regions with and without sunlight move on the surface of the Moon? Is it possible to run away from dark when you are at the equator?

A favorite topic in astrology is the relation between cosmic catastrophes and planetary conjunctions. Imagine that, precisely at noon, all the large planets and the Sun lied on a ray which originated at the center of the Earth. What was the percantege change of your weight compared to a situation with only the Earth present in the solar system?

• Assume the validity of the Newton model derived in the text. For $E=0$ solve the case that the Universe is expanding and the energy of vacuum is constant. What is the future of the Universe according to this model?

• Since the Universe is full of stars, the light from each of them should reach the Earth sooner or later. However as you know the nights are pretty dark. Explain this paradox and support your answer with quantitative arguments.

• In the text a simple derivation of the existence of dark matter in galaxy clusters was presented. Can you figure out another way to prove the existence of dark matter in galaxy clusters? Suggestion is enough, you do not have to work out any calculations.

• Active galaxies appear as a point sources, same as stars. Try to find some ways to distinguish between star and AGN. The more ways, the better.

Radio observations of quasar 3C&nbsp273 showed that there's a blob of radio emission moving away from the nucleus with angular velocity μ = 0.0008 year^{ − 1}. Assuming the radio knot is moving perpendicular to the line of sight, and using the distance $d=440/hMpc$, where $h$ is Hubble constant, compute apparent velocity $v_{zd}$.

• Derive for which value of angle $φ$ is $β_{T}$ maximal?
• Let's assume that the supermasive black hole in the galactic centre is 30 % efficient. How much energy will the hole produce when swalloving an Earth size