Problem Statement of Series 4, Year 33

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.

Planes at high flight levels are controlled using the Mach number. This unit describes velocity as a multiple of the speed of sound in the given environment. However, the speed of sound changes with height. What is the difference in the speed of a plane, flying at Mach number $0.85$, at two different flight levels FL 250 ($7600\mathrm{m}$) and FL 430 ($13100\mathrm{m}$)? At which flight level is the speed higher and by how much (in $\mathrm{kph}$)? The speed of sound is given by $c=(331.57+0.607\left\{t\right\})\mathrm{m}\cdot {\mathrm{s}}^{-1}$, where $t$ is temperature in degrees Celsius. Assume a standard atmosphere, where temperature decreases with height from $15{}^{\circ }\mathrm{C}$ by $0.65{}^{\circ }\mathrm{C}$ per $100\mathrm{m}$ (for heights between $0$ and $11\mathrm{km}$) till $-56.5{}^{\circ }\mathrm{C}$, and then remains constant till $20\mathrm{km}$ above mean sea level.

The FYKOS bird found an optical bench at the Faculty of Physics. The bench allows him to place different tools along an optical axis. He started to play with it and gradually placed onto it: a point source of light, a first lens, a second lens and a screen, with the same spacing between them (so the distance between the screen and the light source is three times bigger than any distance of two neighbouring tools). A sharp image of the source was created on the screen. Then, he dipped the whole system into an unknown liquid, which he found in a strange container. To his amazement, the image on the screen stayed sharp. Figure out the refractive index of the given liquid, which is certainly different from the refractive index of air. You can assume that the refractive index of air is unitary. One of the lenses has ten times bigger focal length than the other and both are thin, manufactured from a material with refractive index $2$.

Jachym is located in a two dimensional Cartesian system at a point $J=(-2a,0)$. As fast as possible, he wants to get to a point $T=(2a,0)$, which is located (luckily) in the same system. Jachym moves exclusively with velocity $v$. This is not so easy, because there is a moving strip in the shape of a line passing through points $(-3a,0)$ and $(0,a)$. On the moving strip, Jachym is moving with total velocity $kv$. For what minimum $k\ge 1$ is it profitable for Jachym to get on the moving strip?

Travel by airplane affects the atmosphere not only by well-known carbon emissions. Discuss how the aircraft industry affects warming of the atmosphere of Earth.