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## mechanics of a point mass

### (3 points)4. Series 33. Year - 1. tchibonaut

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.

### (3 points)4. Series 33. Year - 2. Mach number

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.

### (9 points)4. Series 33. Year - 5. a shortcut across time

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?

### (8 points)3. Series 33. Year - 4. ladybird on a rubber

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.

Matej was watching Vsauce.

### (10 points)3. Series 33. Year - S.

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

### (10 points)2. Series 33. Year - S.

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

### (3 points)1. Series 33. Year - 1. D1

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 $3{,}75 \mathrm{m}$.

Matej doesn't like trucks on highways.

### (10 points)1. Series 33. Year - S. slow start-up

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

Karel wants to have the longest problem assignment.

### (3 points)6. Series 32. Year - 2. bookworm

Vítek has been spending some time in the library. Because of his clumsiness, a book fell down from a shelf and he managed to press it with a swift move towards the wall. He pushes the book with a force $F$ applied at an angle $\alpha$ (see figure). The book's mass equals $M$ and the coefficient of friction between the wall and the book is $\mu$. Find the condition under which the force keeps the book from falling down (and at rest) and determine the critical value $\alpha _0$, below which there does not exist any force that will keep the book up.

Vítek was in a mobile library.

### (6 points)6. Series 32. Year - 3. range

A container is filled with sulfuric acid to the height $h$. We drill a very small hole perpendicularly to the side of the container. What is the maximal distance (from the container) that the acid can reach from all possible positions of the hole? Assume the container placed horizontally on the ground.

Do not leave drills where Jáchym may take them!