# 4. Series 35. Year

* Post deadline: 21st February 2022*

* Upload deadline: 22nd February 2022 11:59:59 PM (local time in Czech Republic)*

### 1. planet dependent units

Units of many physical quantities on Earth are historically tied to the properties of our planet. What would be the units such as meter, knot or atmosphere, if we defined them using the same methodology as on Earth, but while living on Mars? Derive both the ratios between „terran“ and „martian“ units and their values in SI units.

Karel was pondering about non-SI units.

### 2. Freeway

Besides designing its own beer, the Faculty of Mathematics and Physics plans to build an amusement park. They intend to create a unique physics-themed bobsleigh track, where the sleigh starts with non-zero vertical velocity $v_y$ and starts moving directly downwards. The track gradually curves towards the horizontal direction, while the vertical component of the velocity remains constant. What is the dependence of sleigh's horizontal velocity component $v_x$ on the decline in height $y$? And what is the dependence of the magnitude of velocity ($v$) on time ($t$)? Assume the sleigh moves on the track without friction.

**Bonus:** What is the shape of the coaster?

Karel had a „bright“ moment.

### 3. pendular collisions

Two small marbles are attached to ends of strings of the same length ($l = 42,0 \mathrm{cm}$) and negligible mass. The other ends of both strings are attached to a single point. The marbles are of the same size, but they are made from the different materials; first one is made of steel ($\rho _1 = 7~840 kg.m^{-3}$) and second one is made of dural ($\rho _2 = 2~800 kg.m^{-3}$). Both marbles are initially at the angle $5 \mathrm{\dg }$ with respect to the equilibrium position, and after releasing them, they collide elastically. What is the maximum height the individual marbles reach after the collision? What is the result after the second collision?

Karel wanted to hypnotize others. You want to solve this problem \dots

### 4. Analogy

Assume we have two linear springs with elastic modulus $E = 2,01 \mathrm{GPa}$ and a piston with viscosity $\eta = 9,8 \mathrm{GPa\cdot s}$. The dependence of stress $\sigma $ on relative extension $\epsilon $ is characterized by formula $\sigma \_s = E\epsilon \_s$ for spring, and by formula $\sigma \_d = \eta \dot {\epsilon }\_d$ for piston, where the dot represents the time derivative (Newton's notation). We connect a spring of length $l\_s$ and a piston of length $l\_d$ into series, and then we connect the other spring of length $l\_p$ in parallel to them (see figure ). Abruptly, we stretch the entire system into the state of $\epsilon _0 = 0,2$, and we hold the extension constant. Determine, in what time (from stretching) will the stress decrease to half of the original value, if $l\_s / l\_p = 0,5$ holds.

Mirek was thinking about problems while taking an exam.

### 5. Helicopter

The FYKOS bird started to think about constructing his own helicopter because he was tired of flying using his wings. He started by creating a simple model of the main rotor and wondered what the rotor's angular velocity needed to be. Rotor blades are inclined at angle $45 \mathrm{\dg }$; thus, air molecules are pushed directly downwards, creating a momentum flux. Initially, we assume air molecules to be at rest and their collision with the rotor blades to be elastic. The effective part of the rotor blade (i.e., the part inclined at $45 \mathrm{\dg }$ angle) is blade's part that is distant $r_1 = 50 \mathrm{cm}$ to $r_2 = 6,00 \mathrm{m}$ from the centre of rotation. The projection of one blade onto the vertical plane has height $h = 10,0 \mathrm{cm}$, and the helicopter will have four such blades. What is the minimum frequency of the rotor to keep the helicopter of mass $m = 2~500 kg$ at a constant height?

Jindra was hot, so he stood under the helicopter.

### P. winter wonderland

Think about the possibilities for simplification of movement of a human through a landscape during winter. Consider different terrain inclinations, types of snow („powder“, wet snow, melted and resolidified snow, ice, \dots ), and equipment (snowshoes, skis, crampons, ice skates, \dots ). Describe the physical principle behind each type of equipment, and based on this principle, determine which type is best suited for which environment.

Dodo wishes for a proper winter at last.

### E. Useful Coin

Measure at least three physical properties of the smallest coin of legal tender in your country. We consider macroscopic dimensions as one property. We evaluate not only the accuracy of the measurement and the detail of the description but also the originality in the selection of quantities.

Karel wanted the participants to observe the money.

### S. shining

* *

- How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma (The density of plasma $n_e$ is typically expressed as a funciton $n_e = f\(\frac {x}{x_c}\)$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.) is~$50 \mathrm{\micro m}$? Next assume
- that the density of the plasma decreases exponentially with distance from the target,
- that the density of the plasma decreases linearly with distance from the target.

- What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.
- What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?
- Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?