# 1th Round of 20th year

### 1. ... mysterious matter( points)

<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.

### 2. ... collision with asteroid( points)

Calculate the angle between asteroid's velocity and space ship's velocity after collision. Before the impact the spherical asteroid had same mass as the ship. Assume that the ships has spherical shape.

### 3. ... color mixing of stones( points)

Explain, why by combining two lights from two color stones the scientists get different color to the color which would appear by mixing the colour of two stones.

### 4. ... captains diary( points)

Contribute by some interesting record to the diary of the expedition (image, artistic creature, adventure story of length of daily observation, physical observation, …).

### P. ... tree height( points)

Estimate the height of the trees on the planet. Think over all possible aspects influencing the height of the trees.

### E. ... collection of strobiles( points)

The number of spirals made from scutes of strobiles (pine-cones) coming from the center is not arbitrary, but is either 1, 2, 3, 5, 8, 13, 21, … These are numbers of Fibonacci sequence, where the next term is generated by adding two previous terms and first two terms of sequence are 1 and 1. As every rule also this one has its exceptions. Sometimes the number of spirals is equal to 1, 3, 4, 7, 11, …, which is Lucas series. Lucas series is derive in similar order as Fibonacci series, the only difference is starting numbers 1 and 3.

Your task is to find how often and at which conditions this exception occurs on the Earth. More detailed information can be found at http://artax.karlin.mff.cuni.cz/~zdebl9am/phyllot.pdf.) Explore most possible number of parameters (e.g. if the tree grows in middle of woods, is stand alone etc.).

### S. ... Bohr hypothesis( points)

In this question we will deal with hydrogen atom, which consists of heavy nucleus with electric charge $e$ and light electron of mass $m$ and charge $-~e$, which orbits around nucleus at circular trajectory.

- Calculate (using classical physics) the distance of electron from the nucleus depending on its total energy (kinetic and potential) $E$.
- If we accept Bohr hypothesis, that electron's momentum is quantised i.e. can have only discrete values $L~$=$~nh/2π$,

where $n$ is integer number. In which distance from nucleus can electron orbit around nucleus?

- Calculate frequency of emitted photon, if the atom change its energy level from $n$-th allowed to $m$-th allowed distance from nucleus.