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## mechanics of a point mass

### 4. Series 32. Year - 4. trampoline

Two point-like masses were jumping on a trampoline into height $h_0 = 2 \mathrm{m}$. While they both were in the lowest point of the trajectory (corresponding displacement of $y = 160 \mathrm{cm}$), one of them suddenly disappeared. What is the maximum height, which was the other point-like mass bounced into? A round trampoline has circumference of $o = 10 \mathrm{m}$ and is held by $N = 42$ springs with stiffness $k = 1720 \mathrm{N\cdot m^{-1}}$. Trampoline may be modelled by $N$ springs uniformly attached around the circle and connected in the middle. Mass of the disappeared mass is $M = 400 \mathrm{kg}$.

Ivo looked after his cousin.

### 3. Series 32. Year - 5. pointy

Consider a point-like particle in one-dimensional space. Initially, the particle is in rest at the origin of coordinates. It can be moved with acceleration from interval $\left (- a , a\right )$. Let $M\left (t\right )$ be a set of all possible physical states $\left (x, v\right )$ of positions $x$ and velocities $v$, which particle can achieve after time $t$ is elapsed. If we plot set $M\left (t\right )$ into $v(x)$ coordinate system we get surface $S\left (t\right )$. Find analytic expression for boundaries of $S\left (t\right )$.

**Bonus:** Find area of $S\left (t\right )$ as a funcion of time.}

Jáchym wanted to solve a certain trivial problem as a special case of this one.

### 2. Series 32. Year - 4. lunar lander

How can the electronics of the Apollo landing module control an engine thrust $T$ (and so regulate the consumption of fuel), so the rocket floats onto the surface of the Moon at a steady linear motion? The effective velocity of exhaust gases is $u$. The rocket has already slowed down its motion on an orbit and goes straight down in a homogeneous gravitational field with an acceleration $g$. The initial weight of the module is $m_0$.

**Bonus:** How can the electronics of Apollo landing module control the engine thrust during landing from a height $h$ and initial velocity $v_0$, so the landing is so-called fall from null height and the consumption of the fuel minimalizes? Maximum engine thrust is $T\_{max}$.

Michal on the Internet\footnotei {.}{\url {http://www.root.cz/clanky/historie-vyvoje-pocitacovych-her-2-cast-vek-simulaci/}}

### 2. Series 32. Year - 5. bird on the pulley

A fixed pulley is attached to the ceiling and a rope hangs over it, so the left and right end are at the same height. On one end of the rope hangs a Fykosak bird and on the other end hangs a mass, both equally heavy. Describe what happens with the system when the bird starts climbing up (on his own side of rope) with a constant force. In the beginning, assume that the rope is weightless and the pulley is ideal. Afterwards, solve this problem for a real pulley with the following parameters, its length $l$, the moment of inertia of the pulley $I$ and pulley's radius $r$. The rope's mass per unit length is $\lambda$. Assume that the rope doesn’t slip on the pulley.

Mirek rewrote an exercise from Lewis Caroll into FYKOS form

### 2. Series 32. Year - E.

Measure an average vertical velocity of falling leaves. Use leaves from several different trees and discuss what impact the shape of a leaf has on the velocity. How should an ideal leaf look like when we want it to fall as slow as it is possible?

Jachym got this idea, when he asked his friend, whether he knew any interesting experiment

### 1. Series 32. Year - 2. fireworks

Jachym was launching fireworks. We can imagine it as a beacon, that is at some point shot straight up with velocity $v$, and explodes after a certain delay. Jachym was standing a distance $x$ from the launch site when he heard the launch of the fireworks. After a delay $t_1$ he saw the explosion and after another delay $t_2$ he heard the explosion. Calculate the velocity $v$.

Jachym can't hide his pyrotechnic affinity.

### 1. Series 32. Year - 4. Skyfall

When James Bond let go of agent 006 Alec Treveljan from the top of the Arecibo radiotelescope in the final scene of the film Golden Eye, the falling agent started screaming with a frequency $f$. How does the frequency agent 007 hears at the top of the telescope change as a function of time. Neglect air resistance.

**Hint:** Ask Mr. Doppler

Matej enjoys looking outside

### 1. Series 32. Year - S. theoretical mechanics

Before we dive into the art of analytical mechanics, we should brush up on classical mechanics on the following series of problems.

- A homogenous marble with a very small radius sits on top of a crystal sphere. After being granted an arbitrarily small speed, the marble starts rolling down the sphere without slipping. Where will the marble separate and fall of the sphere?
- Instead of the sphere from the previous problem, the marble now sits on a crystal paraboloid given by the equation $y = c - ax^2$. Again, where will the marble separate from the paraboloid?
- A cyclist going at the speed $v$ takes a sharp turn to a road perpendicular to his original direction. During the turn, he traces out a part of a circle with radius $r$. How much does the cyclist have to lean into the turn? You may neglect the moment of inertia of the wheels and approximate the cyclist as a mass point.

**Bonus:**Do not neglect the moment of intertia of the wheels.

### 6. Series 31. Year - 5. jump from a plane

Filip of mass $80 \mathrm{kg}$ jumped out of air plane, that is $h_1 =500 \mathrm{m}$ above the ground. At the same time, Danka (mass $50 \mathrm{kg}$) jumps out of a different airplane but from a height of $h_2 =569 \mathrm{m}$. Assume both of them have the same drag coefficient $C = 1{,}2$, Filip's cross-sectional area is $S\_F = 2{,}2 \mathrm{m^2}$ and Danka's is $S\_D=1{,}5 \mathrm{m^2}$. The density of air $\rho =1{,}205 \mathrm{kg\cdot m^{-3}}$ is and stays the same in all heights. At what time will Danka be at the same height above the ground as Filip?

Danka contemplated the strenuous life of a physicist and wanted to break free for a moment.

### 5. Series 31. Year - 1. staircase on the Moon

If we once colonized the Moon, would it be appropriate to use stairs on it? Imagine the descending staircase on the Moon. The height of one stair is $h=15 \mathrm{cm}$ and it's length is $d=25 \mathrm{cm}$. Estimate the number $N$ of stairs that a person would fly over if he walked into the staircase with a velocity $v=5{,}4 \mathrm{km\cdot h^{-1}}=1{,}5 \mathrm{m\cdot s^{-1}}$. The gravitational acceleration on the Moon's surface is six times weaker than on Earth's surface.

Dodo read The Moon Is a Harsh Mistress.