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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?

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.

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.

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.

Efchári and Goiteía are two components of a double planet around recently arisen stellar system. They orbit around a common centre of mass on circular trajectories in the distance $a = 250 \cdot 10^{3} \mathrm{km}$. Efchári has the radius $R_1 = 4~300 km$, density $\rho _1 = 4~100 kg.m^{-3}$ and siderial period $T_1 = 14 \mathrm{h}$. Goiteía is smaller – it has the radius $R_2 = 3~800 km$, but it has a higher density $\rho _2 = 4~500 kg.m^{-3}$ and a shorter period $T_2 = 11 \mathrm{h}$. Rotation axes of both planets and the system are parallel. After several hundred years, the system transfers due to tidal forces into so-called tidal locking. Find the resulting difference in the period of the system, assuming that both bodies are homogeneous and roughly spherical.

Imagine placing a large number of satellites on the geosynchronous orbit. Coincidentally, a runaway series of collisions occurs and forms a thin spherical layer homogenously scattered with ten million shards with an average size of $x = 10 \mathrm{cm}$. Assume that the velocity directions of the individual shards are oriented randomly in the plane tangent to the sphere. On average, how much time passes between two collisions?

The length of daytime and nighttime varies during the year and it may vary differently in different places on Earth. What about the average length of daytime during one year? Is it the same everywhere or does it vary in different places? A qualitative description is sufficient.

Bonus: Try to estimate the maximum difference between the average length of daytime and $12 \mathrm{h}$.

We are sorry. This type of task is not translated to English.

Consider an astronaut of weight $M$ remaining still (with respect to a space station) in zero-g state, holding a heavy tool of weight $m$. The distance between the astronaut and the wall of the space station is $l$. Suddenly, he decides to throw the tool against the wall. Find his distance from the wall when the tool hits it.

The interstellar space is not empty but contains an insignificant amount of mass. For simplicity, assume hydrogen only and look up the required density. Could we build a spaceship that would „suck in“ the hydrogen and would use energy from it? How fast/large would the spaceship have to be in order to keep up the thermonuclear fusion only from the acquired hydrogen? What reasonable obstacles in realization should we consider?