Problem Statement of Series 1, Year 31

When is it most efficient to pour cold milk into a hot coffee for the coffee to be drinkable as soon as possible? We expect a detailed description of the cooling process rather than a precise calculation.

Consider an optical switch (transfer speed $10\,\mathrm{Gb\!\cdot\! s^{-1}}$), whose output (after any necessary amplification) is used to illuminate the Moon. Thanks to the mirrors left behind by the Apollo mission, the signal comes back and can be used (after any necessary amplification) as an input to the switch. If we make sure the switch works reliably the transmitted data will circle in the system indefinitely. Thus we acquire a memory. What is its maximum capacity? Ignore any delays caused by the processing of the signal and any headers of the data.

We have a thin homogeneous pipe in the shape of a letter L with segments of lengths $b,c$. It is hung freely by one of the ends in a rail car so  that its bend points in the direction of travel. Find the acceleration of the rail car required for the bottom side of the L to stay parallel to the direction of travel? Ignore relativistic effects.

Bonus: Consider relativistic effects.

What happens when we close an almost empty 1.5 liter PET bottle in a well-heated office, let's say at $t_{\mathrm{k}} = 26\,\mathrm{^\circ\mskip-2mu\mathup{C}}$, and then walk down the stairs with it? The bottle starts to make cracking noises. What has a bigger effect on this phenomenon, the fact the atmospheric pressure is changing as we walk down 10 floors of the building, or the fact that the temperature on the stairs is a bit lower, let's say, $t_{\mathrm{s}} = 15\,\mathrm{^\circ\mskip-2mu\mathup{C}}$?

You have probably already thought about whether there are alien civilizations in the universe. As a general rule, the bigger a star the higher is its radiation power and the shorter its life. Let's focus on two stars, where one has twice the power output than the other. If the band where life is possible is given by the steady state temperature of a black body orbiting there, which has a lower and upper limit (that are the same for each and every system), around which star is this life-supporting band wider? How many times larger it will be in comparison to the other star?

This problem has an open solution, so be sure to cite all sources used.

How would a plane behave in microgravity (in other words ignore the effects of gravity)? Describe what effects would the ailerons, the rudder, the elevators and thrust vectoring have of the engines? Which acrobatic maneuvers would be possible? (E.g. a flat spin probably wouldn't be).

Measure the bending of a wooden skewer stick freely supported at its ends as a function of the force applied in its middle (see picture).