Series 5, Year 35

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Upload deadline: 29th March 2022 11:59:59 PM, CET (local time in Czech Republic)

(3 points)1. illuminated satellite

On average, what part of the day does a satellite in low orbit spend in the shadow of Earth? Assume that the satellite's orbit is circular and lies in the ecliptic plane at height $H = R/10$ above the surface of Earth, where $R$ is the mean radius of Earth.

Karel was thinking about satellites.

(3 points)2. cherry pit

Elon Musk plans to colonize Mars. However, he has to build supply bases on the Moon's surface to make colonization possible. Help him solve a crucial problem: how far can a $180 \mathrm{cm}$ tall person spit a cherry pit at a base on the Moon if they spit it in a horizontal direction. The same cherry pit spit on the Earth lands at a distance $4,3 \mathrm{m}$. Bonus: Determine the ratio of distances that the same person reaches by spitting the cherry pit on the Earth and the Moon if they can spit at an arbitrary angle with respect to the ground.

Katarína was looking for an excuse for a trip to the Moon.

(6 points)3. under the lid

A lid has a shape of a hollow cylinder of radius $6,00 \mathrm{cm}$. The lid under which is an air of atmospheric pressure $1~013 \mathrm{hPa}$ is placed in a horizontal washbasin. While doing the dishes, we start filling the washbasin with water at room temperature. The water also gets under the lid and compresses the air trapped inside. At a certain moment, the lid starts floating. At what height is the water level at that moment? The lid weighs $200 \mathrm{g}$, its height is $2,00 \mathrm{cm}$ and negligible volume.

Danka was doing the dishes.

(7 points)4. hit

The FYKOS bird plays with a baseball bat (homogeneous rod of linear density $\lambda $) and hits a baseball of mass $m$. Assume that the rod is attached at one of its ends and can rotate around that point freely. The FYKOS bird can either act on it by a constant torque $M$ or start rotating it by a constant power $P$. After completing a rotation of $\phi _0 = 180\dg $, the end of the rod hits yet motionless baseball, which results in an elastic collision. At what length of the rod $l$ does the baseball gain maximum speed? Compare both situations (i.e., constant $M$ vs. constant $P$).

Jáchym was playing with a baseball bat.

(8 points)5. alternating triangle

figure

Let us construct the finite Sierpiński triangle of a degree $N$ (for $N = 1$ it is a single triangle, in case of $N = 2$ it is four triangles, etc.). The bases of small triangles (that the Sierpiński triangle is made of) consist of a resistor with resistance $R = 150 \mathrm{\ohm}$, the left legs are coils of inductance $L = 0{,}4 \mathrm{H}$ and the other legs are capacitors of capacitance $C = 20 \mathrm{\micro F}$. We measure the impedance between the triangle's bottom left and right corners. The angular frequency of the source is $\omega = 50 \mathrm{s^{-1}}$. Find the recurrent relation for the measured impedance and find its value for $N = 7$. What does the recurrent formula looks like if we replace coils and capacitors with resistors $R$? Determine its numerical value for $N = 15$.

Honza likes fractals.

(10 points)P. hot asteroid

Come up with as many physics reasons as possible on why an asteroid might have a higher temperature than its surroundings.

Karel was thinking about the Fermi paradox.

(13 points)E. it's already going

Measure the moment of inertia of a cylinder (regarding its main axis) and a ball (with respect to the axis passing through its center) by rolling them on an inclined plane.

Karel imagined participants rolling.

(10 points)S. stabilizing

  1. What intensity must a laser with a wavelength of $351 \mathrm{nm}$ have in order to stabilize a Rayleigh-Taylor (RT) instability using the surface ablation of a fuel pellet? Suppose the boundary between the ablator and DT ice is corrugated with a wavelength of
  1. $0,2 \mathrm{\micro m}$,
  2. $5 \mathrm{\micro m}$.
  1. How will the intensity of the laser change if we also apply a magnetic field with magnitude $5 \mathrm{T}$?
  2. What else can help us minimize the RT instability?
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