# 3. Series 26. Year

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### 1. Conjunction

A favorite topic in astrology is the relation between cosmic catastrophes and planetary conjunctions. Imagine that, precisely at noon, all the large planets and the Sun lied on a ray which originated at the center of the Earth. What was the percantege change of your weight compared to a situation with only the Earth present in the solar system?

Ales's brain

### 2. Marble

If you throw a small marble of diameter r from a very tall building, it appears to become smaller as it falls down. What is the time dependence of the apparent radius if, initially, it is at rest and a distance $x_{0}$ away from your eyes. Also assume that you are watching the marble directly from above at all times. Feel free to neglect any friction forces acting on the marble.

Karel was bored.

### 3. Burning brakes

What is the optimal speed for a car to go downhill in order for the brakes to be warming up with the fastest rate? Assume that the temperature difference between the brakes and the air around them is directly proportional to the power dissipated in the brakes.

Lukas was on fire.

### 4. Supersonic or subsonic?

Imagine a bomb that is free-falling on a military base of your enemies. Although you threw the bomb off an airplane, you should assume that its motion is vertical (i.e. no horizontal component of its velocity relative to the surface). Because of air friction, the falling bomb emits a sound that propagates at the speed c = 340 m/s. What is the maximal impact velocity of the bomb in order for your enemies to hear it before it kills them?

Lukas was staring at ducks on a lake.

### 5. Gas crises

A Siberian gas pipeline with a liquefied natural gas needs to be closed. Váňa Vasilijevič decided to do this manually by closing a frictionless valve. What is the work he needed to do so? What is the force he acted on the valve with (choose an appropriate parameter to describe it)? You can imagine the valve as a board that is being inserted into the pipeline (the pipeline is perpendicular to this board). Initially, the pressure inside the pipeline was $p=2MPa$. Its cross-section is square-shaped with a side of length $a=1\;\mathrm{m}$. The board is $d=10\;\mathrm{cm}$ wide, the density of liquefied natural gas is $ρ=480\;\mathrm{kg}⁄m$, and its flow rate is $q=20m3\;\mathrm{s}$.

Ales wanted to know what is it like to live in Russia.

### P. Conspiracy theory?

In this problem you are asked to think about the vapor trails that sometimes form behind an airplane. What are the physical parameters that determine the length of these lines? Estimate and/or find the values of these parameters and determine the range of all possible lengths. Use your results to dismiss the myth of chemtrails that says that these lines are formed by poisonous chemicals being released from the planes.

Michael declared a war on stupidity.

### E. Fun with cans

Determine the time dependence of the velocity of a can filled with water that is rolling down an inclined plane (starting at rest). Your inclined plane should be at least 2 meters long. Also it must be a plane! I.e. check that it is not bent, and that its surface is smooth. You can perform the measurement for example by recording the motion on a video, or by partitioning the inclined plane into several intervals and measuring the average velocity on each of them.

Karel inspired by his past.

### S. tokamak

• Calculate the specific resistance of hydrogen plasma at temperature 1 keV. Compare your result with the resistance of common conductors.
• Calculate the current necessary to create a sufficiently strong poloidal magnetic field in a tokamak with a major radius of 0.5 m. The toroidal field is created using a toroidal coil with 20 windings per meter. The current inside this coil is 40 kA. The magnitude of the poloidal field should be approximately 1/10 of the magnitude of the toroidal field.
• Create a physical model of the field lines of the force field inside the tokamak, take a photo of it, and send it to us.
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