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## mechanics of a point mass

### (3 points)6. Series 37. Year - 1. ballons with Martin

A car is standing on a straight road, with a freely floating helium balloon tied inside. Suddenly, the car starts to accelerate with acceleration $a=5{,}0 \mathrm{km\cdot min^{-2}}$. By what angle will the balloon be deflected from the vertical line? What is the direction of the deflection?

Martin would like to hang on a balloon behind a car.

### (3 points)6. Series 37. Year - 2. bombarded organizer

Estimate how many antineutrinos created in Czech nuclear power plants pass through the body of an average FYKOS organizer in one meeting held for a FYKOS camp. The meeting is 4 hours long and takes place on the tenth floor of the Matfyz building at the Troja campus in Prague.

Jarda felt under pressure at the meeting.

### (7 points)6. Series 37. Year - 4. infite pulleys

Let us have an infinite system of intangible pulleys as shown in the figure, where the mass of each additional weight is one-third of the weight of the previous one. What is the acceleration of the first weight of mass $m$?

Matěj was looking for the difference between coutable and uncoutable many pulleys.

### (3 points)5. Series 37. Year - 2. basic problem of acoustics

Adam can take meaningful notes at the speed $v_1$. Unfortunately, his calculus professor speaks at the speed of $v_2$. There is an airflow in the lecture hall, moving from Adam towards the professor, with the air flowing at a velocity of $v_3$. At what velocity and in which direction along a straight line intersecting Adam and the lecturer should Adam move to transcribe everything the lecturer says into his notebook?

Adam likes the word \uv {meaningful}.

### (7 points)5. Series 37. Year - 4. centrifuge

Consider a centrifuge of length $L = 30 \mathrm{cm}$ filled with a solution in which there are homogeneously distributed small spherical particles of radius $r = 50 \mathrm{\micro m}$ and mass $m = 5,5 \cdot 10^{-10} \mathrm{kg}$. The density of the solution is $\rho \_r = 1~050 kg.m^{-3}$ and its viscosity is $\eta = 4,8 \mathrm{mPa\cdot s}$. The container with the solution is in a horizontal position and suddenly begins to rotate at an angular velocity of $\omega = 0,5 \mathrm{rad\cdot s^{-1}}$. Determine how long it will take for $90 \mathrm{\%}$ of all the particles to reach the end of the centrifuge. Do not consider interparticle collisions and movement of the particles due to diffusion.

Jarda loves to make enriched uranium.

### (3 points)4. Series 37. Year - 1. the flight over the moon

One day, the FYKOS-bird was watching the sky during a full moon. An airplane just passed over the moon in $0{.}35 \mathrm{s}$, and the perpendicular distance of its flight path from the center of the moon was $1/3$ of the full moon's radius. This plane flies typically with a speed of $800 \mathrm{km\cdot h^{-1}}$. The FYKOS-bird wondered what altitude the plane was at so he could fly with it next time. Like him, determine this altitude.

Jarda was sunbathing in the garden.

### (3 points)4. Series 37. Year - 2. they got off in Hněvice

Tomáš got into the train wagon in the shape of a rectangular cuboid and decided to take a nap. When he woke up, he found that he was alone in the wagon, which was suspended at its geometric center on a cargo crane and rotating around the hinge axis at an angular velocity of $\omega $. Tomáš didn't notice it at first since he was sitting in the wagon's centre with a width of $d$. When he realized it, he was pleased because he thought of using one of his kilogram standards, which he carries around for situations like this, to measure the length of the carriage. After a few attempts, he managed to throw the standard at an initial velocity of $\vec {v}$ so that after two revolutions of the wagon, the standard hit the far corner of the wagon and broke the window. Neglecting air resistance, what length $L$ of the wagon did he determine?

Tomáš fell asleep on the train and was thrown off by the conductor.