# Series 6, Year 23

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### 1. Hussite one

The wagon fort stands on the inclined plane (elevation angle is $α)$. The wagons are of the same weight, the distance between adjacent ones is $s$ and their total number is $N$. All of them have their brakes on at the beginning. Cunning Hetman Žižka sets the brakes of the uppermost wagon off and the wagon starts to accelerate. On the way down it hits other wagons (when the impact occurs, the wagon which was hit sets its brakes off immediately and continues to run down the slope together with the one which hit it). Calculate the velocity of the wagon fort after the last impact.
**Bonus:** task: Calculate the distribution of weight of wagons in order to achieve constant velocity after each impact.

### 2. John the voyeur

John is standing on the top of the Žižkov Tower and looks into the windows of neighbouring houses. All the windows face the tower, are of same size and are at the same height. Calculate the radius of the lowest privacy in the houses with such windows. John does not have any binoculars.

### 3. atomic capacitor

Luke has bought a Uranium atom and started to take away its electrons just for fun. After taking the $n-th$ one he surprisingly discovered that the mass of the atom had increased. What has inflicted this phenomena? Determine the value of $n$.

### 4. subcritical semispheres

After Luke got bored with the ionization, he ordered more of the Uranium. The two perfect semi-spheres were delivered. Each semi-sphere has the mass $m$ ($m_{k}⁄2<m<m_{k}$, where $m_{k}$ is the critical mass). Luke has placed their flat sides opposite each other which started to bring them nearer. Calculate the distance $d$ at which the experiment was interrupted by the ignition of the chain reaction.

### P. tora tora tora

In the middle of the Battle of Midway one of the Japanese pilots got thirsty. After searching the cockpit he found out that the supply management gave him only bottles with soda. What should he do in order to satisfy his thirst?

### E. the drop

The task of this problem is to investigate optical parameters of a water drop. If you place a drop on a thin glass plate, you will get an improvised magnifying glass. Examine the focal length and maximal magnification of this $drop$ glass$ in correlation with its dimensions and compare it with the theory. Notice the optical aberrations of the drop. What will happen if you try to magnify the display of the computer?