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## astrophysics

### (8 points)4. Series 34. Year - 5. Efchári-Goiteía

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.

Dodo keeps confusing Phobos and Deimos.

### (6 points)3. Series 34. Year - 3. kaboom, kaboom

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?

Dodo learned about transport phenomena in gasses for his state exams.

### (3 points)2. Series 34. Year - 1. there is light -- there is none

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 $12 \mathrm{h}$.

Dodo was discarding old problems.

### (10 points)6. Series 33. Year - S.

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

### (3 points)4. Series 33. Year - 1. tchibonaut

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.

Karel wanted to set this name for this problem.

### (10 points)4. Series 32. Year - P. V-1 in the space

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?

crypto-facism → Red Dwarf → drive → thrust → V-1 and the circle closes

### (12 points)1. Series 32. Year - E. hourly

Measure the length of one day. However, there is a limitation: one continuous measurement can't take longer than one hour. For the sake of statistical accuracy, though, do repeat your measurements multiple times.

Jachym had an hour until deadline

### (10 points)1. Series 32. Year - P. terrible cold

Some nebulas constituted of a gas from stars, e. g. Bumerang, have lower temperature than the Cosmic Microwave Background (CMB), hence are technically colder than space. How is this possible? Try to determine a condition for a gas ejected by a hot star to cool down below the temperature of the CMB.

Karel wasn't satisfied with the claim that the temperature everywhere in space is at least that of the CMB.

### (9 points)6. Series 31. Year - P. universe expansion compensation

According to the current observations and cosmological models, it seems that our Universe is expanding and the rate of expansion is accelerating. What if that wasn't the case? What if the Universe stayed the same, but the physical laws/constants were changing so that it would seem like the universe is expanding, the way we observe it? Describe as many laws that would need to change.

Karel was intrigued whether one can compensate the expansion of universe.

### (9 points)4. Series 31. Year - P. Voyager II and Voyager I live!

We have a satellite and we want to launch it out of the Solar System. We launch it from Earth's orbit so that after some corrections of the trajectory it gets a velocity which is higher than the escape velocity from the Solar System. What is the probability that the satellite will collide with some cosmic material with higher diameter that $d=1 \mathrm{m}$ before leaving the Solar System.

Karel was wondering why NASA doesn't consider this possibility…