Problem Statement of Series 6, Year 25

A small ball of radius $r$ is placed on top of a bigger ball of radius $R$. The lowest point of the bigger ball is at height $h$ above ground. Both balls are released simultaneously. Calculate the maximum height that can be achieved by the smaller ball. Assume that all collisions are perfectly elastic and that the mass of the smaller ball is negligible compared to the larger ball.

Bonus: Generalize the problem if there are $N$ balls present. You can still assume that a smaller ball on top of a bigger ball has a negligible mass compared to the larger one.

Estimate the minimal energy needed to put a space station on an orbit around the Earth. You can work with the values valid for the International Space Station which orbits the Earth at height approximately $h=350\mathrm{km}$ and has mass $m=450000\mathrm{kg}$. Explain why is this estimate minimal and why, in reality, much more energy is needed.

Imagine a bent tube of length $l$ that is full of water and whose lower end is submerged in a container (see $\text{???}$). We rotate the tube once per time $T$. Calculate the pressure that causes the water to flow out of the container. You can neglect viscosity and the pressure due to the column of water in the vertical part of the tube.

A sphere of ice is floating in the Arctic Ocean. Calculate the fraction of its surface that is above the sea level. The density of ice is 917 kg ⁄ m and the density of salty water is 1025 kg ⁄ m.

Aleš was procrastinating with his tablet when he realized that he is late for his class. The only way to make it on time was to run without stopping. Therefore he started running uphill with speed $v$. The road was inclined at an angle $\alpha$. After a while (at time $T)$ he realized that he still carries a brick that he meant to leave in his tent. He is able to throw the brick only with an initial speed $w$. Determine the angle at which Aleš should throw the brick in order to hit his friend that is sitting in the same spot he was sitting. Is it possible that Aleš will not be able to do this? You should not account for any reaction time.

When you illuminate your fingers with a very intense light in order to see through them, you should notice that you can see individual vessels but the rest seems rather homogeneous. Explain why the vessels are visible but bones are not.

Original way to get rid of an empty coke can is to pour a little bit of water into it, seal the opening and place it on a cooker. After it is hot enough throw it into a cold water and if you are lucky it will collapse so that it is ready to be recycled. Try this also without the water inside the can and explain why is the outcome different. Your goal is to crush the can into the smallest possible form. Send us a picture of your result together with the description of the conditions under which you achieved it. Warning The can will get hot, do not burn yourself!

• 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.