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## mechanics of a point mass

### 2. Series 22. Year - E. slanted PET bottle

How much of water has to be inside PET bottle placed upside down onto cap to make it the most stable (stability is defined as at small inclination bottle will fall in longest time). Do not forget to make theoretical assumption.

nad vypitou lahví se zamyslel Béda

### 1. Series 22. Year - 1. sliding and oscillations

Two weights of masses $m$ and $M$ are connected by a spring of stiffness $k$ and lies on flat smooth surface (friction can be neglected). Body of mass $m$ gets velocity $\textbf{v}$ (see picture). What will be the smallest distance between the bodies? And when it will be reached?

V ročenkách kanadské FO našel Honza Prachař.

### 6. Series 21. Year - 1. free fall of Fykos-bird

A Fykos-bird (the symbol of the seminar) is flying 1 km above the forest at speed 4 m ⁄ s. It is shooted by a men with a gun. The bullet leaves gun at speed 710 m ⁄ s. Our men saw the bird just above his head and fired. How far from him the body of bird will touch the ground? Air friction is negligible.

Vyplodil Honza Prachař.

### 6. Series 21. Year - 4. quick retreat

Fykos-bird is running through a coridor, which is turning to left. The width of coridor is $b$, bird is running at speed $v_{0}$ and the turn is in distance $d$. If the acceleration will be bigger than $a_{0}$, it will slip and fall down. What is the optimum trajectory to have smallest possible delay?

Napadlo Honzu Jelínka při dobíhání tramvaje za rohem.

### 5. Series 21. Year - 2. question of survival

The entrance and the internal surface are connected by a ladder. You have already descended one kilometer, when you slipped and are falling down. What will be the speed you will fall down on Rama surface? How long it will take? Is there a chance to survive?

Martin Formánek

### 5. Series 21. Year - 3. staircase from the sky

The ladder is only 2 km long going to the platform, from which you have to take a stairs, which goes with long arch above the countryside. The staircase has very special shape: for each step you need the same amount of mechanical work. Calculate, how the step height depends on the distance from the Rama axis, if the length of stairs is constant. What shape is the arch?

Martin Formánek

### 5. Series 21. Year - P. Rama-quake

Finally you got onto the surface of Rama. Suddenly it started to shake and you think the rotation speed has changed. How you can measure the rotation speed? Can you find more than one experiment?

Martin Formánek

### 3. Series 21. Year - 1. English and Scots

Find the change in rotation speed of the Earth, if English and Scots would start to drive on right instead on left. Make just an estimation.

Úlohu zaslechl Aleš Podolník.

### 3. Series 21. Year - 2. lift to the skies

Find the properties of material, which you need to make a rope for a lift from geostationary orbit to the Earth surface. Is such material available on Earth?

### 3. Series 21. Year - 3. jumping on inclined plane

A small ball is thrown in horizontal direction onto a inclined plane. The ball starts to jump on the plane and after $Ncontacts$ it falls at right angle onto the plane. An example of such trajectory for $N=4$ is in figure. What is the inclination angle of the plane $α?$ Assume, the ball bounce elastically, do not assume rotation.

Pavel Motloch.

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