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## mechanics of a point mass

### (3 points)5. Series 32. Year - 1. urban walk

Matěj walks across the street with constant velocity. Every 7 minutes a tram going in opposite direction passes, while every 10 minutes a tram going in his direction passes. We assume that trams ride in both directions with the same frequency. What is the frequency?

### (9 points)5. Series 32. Year - 5. bouncing ball

We spin a rigid ball in the air with angular velocity $\omega $ high enough parallel with the ground. After that we let the ball fall from height $h_0$ onto a horizontal surface. It bounces back from the surface to height $h_1$ and falls to a slightly different spot than the initial spot of fall. Determine the distance between those two spots of fall onto ground, given the coefficient of friction $f$ between the ball and the ground is small enough.

### (3 points)4. Series 32. Year - 2. it will break

Suppose a massless string of length $l$ with a point-like mass $m$ attached to its end. We know that the maximum allowed tension in the string is equal to $F = mg$, where $g$ is the gravitational acceleration. We will attach the string to the ceiling and we hold the mass in the same height with the string straight but unstrained. Then, we will release the mass and it begins to move. Find the angle (with respect to the vertical) for which the string will break.

Karel thought he won't make it.

### (7 points)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.

### (10 points)4. Series 32. Year - S. serial

* We are sorry. This task is only avaiable in Czech. *

### (8 points)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.

### (7 points)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/}}

### (9 points)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

### (12 points)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

### (10 points)2. Series 32. Year - S. theoretical mechanics

*We are sorry, this task is available only in Czech.*