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## mechanics of a point mass

### (3 points)4. Series 35. Year - 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 on the decline in height? And what is the dependence of the magnitude of velocity on time? Assume the sleigh moves on the track without friction.

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

Karel had a „bright“ moment.

### (5 points)4. Series 35. Year - 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\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

### (9 points)4. Series 35. Year - 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\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\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.

### (3 points)3. Series 35. Year - 2. playing with keys

Vašek likes to plays with keys by swinging them around on a keychain and then letting them wrap around his hand. We will simplify this situation by a model, in which we have a point mass $m$ in weightlessness attached to an end of massless keychain of length $l_0$. The other end of the keychain is attached to a solid cylinder of radius $r$. The keychain is taut so that it is perpendicular to the surface of the cylinder at the attachment point, and the point mass is then brought to velocity $\vect {v_0}$ in the direction perpendicular both to the axis of the cylinder and to the direction of the keychain. The keychain then starts to wrap around the cylinder. What is the dependence of the velocity of the point mass on the length of the free (not wrapped around) keychain $l$?

**Hint:** Find a variable that remains constant during the wrapping process.

**Bonus:** How long does it take for the whole keychain to be wrapped around the cylinder?

Vašek was playing with keys while falling out of window.

### (3 points)2. Series 35. Year - 1. chasing the light

Jindra walks down a long, lit corridor. His eyes are at a height of $1,7 \mathrm{m}$ above the floor, the light on the ceiling is at a height of $3,4 \mathrm{m}$. Jindra is now at a distance of $10 \mathrm{m}$ (horizontally) from the nearest light and is approaching it at a speed of $3 \mathrm{km\cdot h^{-1}}$. He sees a reflection of the light on the polished floor. How fast is the reflection approaching Jindra at this point?

Jindra remembered walking down the corridor at the elementary school.

### (3 points)2. Series 35. Year - 2. fixed station

We have two carabiners anchored in a rock, both at the same height and at a distance of $d$ from each other. We snap a loop with the length $l$ into the carabiners. Then we snap another carabiner on the loop, from which we would like to abseil, while applying a downward force of $F = 2{,} \mathrm{kN}$. Calculate the tension in the loop and the force that will act on the carabiners, in cases where the abseil carabiner is slung on one and on both parts of the loop. In which case is the force acting on the loop lesser and which case is safer?

Dodo dreamed about rock climbing.

### (6 points)2. Series 35. Year - 3. model of friction

What would be the coefficient of static friction between the body and the surface if we considered a model in which there were wedges with a vertex angle $\alpha $ and a height $d$ on the surface of both bodies? Try to compare your results with real coefficients of friction.

Karel took inspiration from KorSem.

### (7 points)2. Series 35. Year - 4. tea tap

Matěj wants to pour some tea from a bevarage dispenser into a glass of mass $M$. He uses one hand to hold the glass and second hand to control the faucet, which changes the volume of the current of the tea. The speed of the outflow $v$ is constant (we can assume that the speed at the contact with the glass is identical). Since Matěj does not want to overstrain himself, he would like to hold the glass with a constant force from the start of the pouring to its end. What is the value of the volume of the current as a function of time that satisfies this requirement? How long will it take to fill the whole glass?

Matěj likes tea from a dispenser

### (8 points)2. Series 35. Year - 5. Shkadov thruster

A long time ago in a galaxy far, far away, one civilisation decided to move its whole solar system. One of the possibilities was to build a „Dyson half-sphere“, i. e. a megastructure which would capture approximately half of the radiation output of the start and reflect it in a single direction. An ideal shape would therefore be a paraboloid of revolution. What would be the relation between the radiation output of the star, surface mass density of such a mirror and its distance from the star such that this distance is constant?

Karel watches Kurzgesagt.

### (3 points)1. Series 35. Year - 1. cars

Two cars start to move from the same point at the same time with velocities $v_1 = 100 \mathrm{km\cdot h^{-1}}$ and $v_2 = 60 \mathrm{km\cdot h^{-1}}$. Is it possible for the cars to move away from each other at any of the following velocities? If so, sketch the situations. \[\begin{align*} v_a &= 160 \mathrm{km\cdot h^{-1}} , & v_b &= 40 \mathrm{km\cdot h^{-1}} , \\ v_c &= 30 \mathrm{km\cdot h^{-1}} , & v_d &= 90 \mathrm{km\cdot h^{-1}} \end {align*}\]

Karel wanted to hit Dano at a precisely defined speed.