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(8 points)5. Series 36. Year - 4. Dark Side Time

FYKOS plans to send its own satellite into space. It will be powered by solar cells; hence, it cannot stay in the Earth's shadow for too long. What is the height above the Earth's surface for which the time of the satellite passing through the Earth's shadow is the shortest? In your calculations, assume (same as the organizers did) that the Earth is a perfect sphere, that sunrays close to Earth's surface are parallel, and that the Sun, the Earth and the satellite's trajectory are in the same plane.

Bonus: While solving the problem, you will encounter an analytically unsolvable equation. Do not use online solvers, but try to create your own solution.

Honza's batteries died in Kerbal.

(9 points)4. Series 36. Year - 5. space visit

Two aliens each live on their own space station. The stations are in free space and the distance between them is $L$. When one alien wants to visit the other, he has to board his non-relativistic rocket and fly to his neighbor. What is the shortest time an alien can spend on its way there and back? The mass of the rocket with fuel is $m$, without fuel $m_0$. The exhaust velocity is $u$. The fuel flow is arbitrary, and his neighbor won't let him load any fuel (he has little himself).

Jarda needed no one to notice that he had disappeared from the meeting for a while.

(3 points)6. Series 35. Year - 2. generational threat

Imagine there is a comet that threatens the Earth once a generation, just when it is in the perihelion. What is the distance between the Earth and such a comet when the comet is in its aphelion? What is the length of the semi-major axis and the orbital eccentricity of the comet's trajectory? Do not consider gravitational influences other than from the Sun, and assume that one generation is $g = 20 \mathrm{years}$.

Karel threatened civilization over and over again.

(3 points)5. Series 35. Year - 1. illuminated satellite

On average, what part of the day does a satellite in low orbit spend in the shadow of Earth? Assume that the satellite's orbit is circular and lies in the ecliptic plane at height $H = R/10$ above the surface of Earth, where $R$ is the mean radius of Earth.

Karel was thinking about satellites.

(10 points)5. Series 35. Year - P. hot asteroid

Come up with as many physics reasons as possible on why an asteroid might have a higher temperature than its surroundings.

Karel was thinking about the Fermi paradox.

(9 points)3. Series 35. Year - P. artificial gravitation

How could artificial gravity be implemented on a spaceship? What would be the advantages and disadvantages depending on the different characteristics of the spacecraft? Is it realistic to have gravity in different directions on different floors of the spaceship or for it to change rapidly, as we can sometimes see in sci-fi movies when „artificial gravity fails“?

Karel was day-dreaming while watching sci-fi.

(8 points)2. Series 35. Year - 5. Shkadov thruster

A long time ago in a galaxy far, far away, one civilisation decided to move its whole solar system. One of the possibilities was to build a „Dyson half-sphere“, i. e. a megastructure which would capture approximately half of the radiation output of the start and reflect it in a single direction. An ideal shape would therefore be a paraboloid of revolution. What would be the relation between the radiation output of the star, surface mass density of such a mirror and its distance from the star such that this distance is constant?

Karel watches Kurzgesagt.

(8 points)6. Series 34. Year - 4. I have seen the comet

Long-period and non-periodic comets usually begin the outgassing process when they reach the orbit of Saturn. Until that, they appear only as small rocks to an observer on Earth, and therefore they are almost unobservable. Assume a comet with the perihelion distance $q = 0,5 \mathrm{au}$. Estimate the time that it takes for the comet to reach the Earth's orbit once it passes the orbit of Saturn. The eccentricity of the trajectory of the comet is very close to one.

Dodo at a tutorial of astrophysics.

(10 points)6. Series 34. Year - P. more dangerous corona

When there is a coronal mass ejection from the Sun, the mass will start to propagate with high velocity through the space. Sometimes the mass can hit the Earth and affect its magnetic field. Estimate the magnitude of the electric currents in the electric power transmission network on Earth which could be generated by such ejection. What parameters does it depend on? Comment on what effects would such event have on the civilisation.

Karel was at a conference and then he saw a video on the same topic.

(3 points)5. Series 34. Year - 2. retarded Jupiter

The sidereal period of Jupiter is approximately $11,9 \mathrm{years}$, the speed of light is $3 \cdot 10^{8} \mathrm{m\cdot s^{-1}}$. Assume the relative distance between the Earth and the Sun to be $150 \cdot 10^{9} \mathrm{m}$. Using these values, calculate how long will the light travel from Jupiter to Earth if Jupiter is located at a point to which it will get from opposition in one quarter of the sidereal period.

Vašek remembered the observations of Ole R\o {}mer.

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