Problem Statement of Series 1, Year 40

In an air dome, there is an excess pressure of $\Delta p = 310\,\mathrm{Pa}$ relative to the surroundings. The atmospheric pressure outside the dome is $p_{\mathrm{out}} = 950\,\mathrm{hPa}$. If we were very sensitive, leaving the hall would give us a sensation similar to a sudden change in elevation. What altitude difference does this pressure difference correspond to?

On a rainy day, Matěj needs to get home as dry as possible. He has chosen the run-walk method, that is, he runs for half the time at speed $v_{\mathrm{b}}=15\,\mathrm{km\!\cdot\! h^{-1}}$ and walks for half the time at speed $v_{\mathrm{ch}}=5\,\mathrm{km\!\cdot\! h^{-1}}$, and he can choose when to take breaks from running. Small parts of the route pass through covered spaces (underpasses, arcades, etc.). Is it worthwhile for Matěj to plan his run-walk intervals so that he runs through or walks through these sections? The area of Matěj's effective horizontal cross-section is five times smaller than that of his vertical cross-section. Assume that the rain falls with a vertical speed $u_{\mathrm{s}}=20\,\mathrm{km\!\cdot\! h^{-1}}$ and a horizontal speed $u_{\mathrm{v}}=5\,\mathrm{km\!\cdot\! h^{-1}}$ in the direction of Matěj's motion. Neglect the relative motion of Matěj's limbs during motion.

What shape must a hand-held mirror have to view a screen such that we see an aspect ratio of $4:3$, when the screen actually has an aspect ratio of $16:9$? The mirror is held at a distance of $65\,\mathrm{cm}$ from our eyes. The line intersecting the screen and the observer is perpendicular to the surface of the mirror. Where and how far away from us must the screen be located in order for us to observe this change? What is the focal length of the mirror?

Two point masses, each of mass $m$, are located on the surface of a sphere of radius $R$ at an initial distance $d_0$ (where $R \gg d_0$). The segment connecting these points forms an arc on a great circle of the sphere, which we will refer to as the „equator“ for the purposes of this problem. Both points are given the same initial velocity $v$ directed „north“ (i.e., perpendicular to the defined equator). No external forces act on the points, other than the constraint that keeps them on the surface of the sphere. The points move along great circles (meridians), which converge toward the pole. Consequently, the mutual distance between them decreases. Imagine an observer who is unaware that the points are moving on a curved spherical surface and assumes they are moving in a flat plane. This observer attributes the fact that the points are approaching each other to the existence of an apparent attractive force acting between them.

What dependence of this attractive force on the instantaneous distance $d$ between the points would this observer derive from the experiment?

A hotplate is heated from a constant ambient temperature $T_1$ to a temperature $T_2$ in a certain time interval. After the power is switched off, it cools to the temperature $\left(T_2 + T_1\right)/2$ in the same amount of time as in the heating process. Determine the maximum temperature $T_{\mathrm{max}}$ to which the hotplate can be heated. Assume that the power supplied to the hot plate is independent of its temperature and that the rate at which it transfers heat to the surroundings is directly proportional to the temperature difference between the hotplate and the ambient air.

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

You have certainly heard that bananas are slightly radioactive. How many would be needed for them to function as a nuclear power plant (and replace, e.g., Czechia's Temelín)? Would it not be more efficient to feed bananas to the employees and have them manually spin the turbines, or perhaps convert bananas into electricity in another way?

Measure the dependence of the coefficients of static and sliding friction of wood on wood on the orientation of the wood cut.