# 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.

### 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?

### 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?

### 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.

### 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?

### 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.

### 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.