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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 ($7\;600 \mathrm{m}$) and FL 430 ($13\;100 \mathrm{m}$)? At which flight level is the speed higher and by how much (in $\jd {kph}$)? The speed of sound is given by $c =\(331{,}57+0{,}607\left \lbrace t \right \rbrace \) \jd {m.s^{-1}}$, where $t$ is temperature in degrees Celsius. Assume a standard atmosphere, where temperature decreases with height from $15 \mathrm{\C }$ by $0,65 \mathrm{\C }$ per $100 \mathrm{m}$ (for heights between $0$ and $11 \mathrm{km}$) till $-56{,}5 \mathrm{\C }$, and then remains constant till $20 \mathrm{km}$ above mean sea level.

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?

Ladybird moves with velocity $4 \mathrm{cm\cdot s^{-1}}$. When we place the ladybird onto a rubber, she comes through it in $10 \mathrm{s}$. What happens when the ladybird starts moving and we start prolonging the rubber the way that its length will be increasing with velocity $5 \mathrm{cm\cdot s^{-1}}$? Is the ladybird able to come through the whole rubber to its end? If yes, how long will it take? Consider that the rubber prolongs uniformly and never breaks.

We are sorry. This type of task is not translated to English.

They say that people inside an elevator can bear with acceleration $a = 2{,}50 \mathrm{m\cdot s^{-2}}$ without any major problems. We want to get to the planned floor as soon as possible. If the elevator started running with that acceleration for a quarter of a time, a half of the time went by constant velocity, and the rest quarter of the time it was decelerating, how high it could go by total time of the ride $t = 1{,}00 \mathrm{min}$?

Let us assume a pulley in a fixed position with a rope of negligible weight. A weight $m_1$ is located on one end of the rope and a winch (mass $m_2$) on the other. Initially, the winch is in the same height as the weight $m_1$. In the first case, the winch is fixed to the ground and pulls only the weight. In the second case, the weight is firmly connected to the winch. Therefore, while the rope is being attached, both the weight and the winch are pulled up. Find out which case requires less pull force (and therefore weaker winch).

Cutting board with thickness $t=1,0 \mathrm{mm}$ and width $d =2,0 \mathrm{cm}$ is made up of two parts. The first part has density $\rho _1 =0,20 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_1 = 10 \mathrm{cm}$, the second part has density $\rho _2 =2,2 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_2 = 5,0 \mathrm{cm}$. We place the cutting board on water surface, which density is $\rho \_v = 1{,}00 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and then we wait until it is in equilibrium position. What angle will a plane of the cutting board hold with the water surface? How big the part of the cutting board which will stay above the water level will be?

We are sorry. This type of task is not translated to English.

A truck driving on a highway has a $2 \mathrm{\%}$ higher speed than a bus in front of it. The driver of the truck decides to overtake the bus, but when the truck is exactly next to the bus, a right curve begins on the highway, making the path of the truck longer. As a consequence, the two vehicles drive next to each other all the way along the curve, whilst a notable traffic jam starts to build up behind them. Determine the radius of the curve (at the middle of the inner driving lane) if the separation between the centers of the lanes is $3{,}75 \mathrm{m}$.

We are sorry. This type of task is not translated to English.