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## mechanics of a point mass

### 3. Series 34. Year - 5. smuggling in space

Two spaceships move towards each other on a straight line. The initial distance between them is $d$. The first one moves with the velocity $v_1$, the second with the velocity $v_2$ (in the same reference frame). The first one can reach the maximal acceleration $a_1$, the second one $a_2$ (both regardless of the direction). Their crews want to exchange some „goods“. In order to do that, the spaceships need to meet – i. e. they must be at the same time at the same place and have the same speed. What is the minimal time for them to reach the meeting? Neglect the relativistic effects.

Jáchym insolently stole Štěpán's original idea.

### 3. Series 34. Year - S. electron in field

Consider a particle with charge $q$ and mass $m$, fixed to a spring with spring constant $k$. The other end of the spring is fixed at a single point. Assume that the particle only moves in a single plane. The whole system exists in a magnetic field of magnitude $B _ 0$, which is perpendicular to the plane of movement of the particle. We will try to describe possible modes of oscillation of the particle. Start by the determination of equations of motion - do not forget to include the influence of the magnetic field.

Next assume that the particle oscillates in both of the cartesian coordinates of the particle and carry out Fourier substitution - substitute derivatives by factors of $i \omega$, where $\omega$ is the frequency of the oscillations. Solve the resultant set of equations in order to determine the ration of the amplitudes of oscillations in both coordinates and the frequency of oscillations. The solution obtained in this way is quite complicated, and better physical insight can be gained in a simpler case. From now on, assume that the magnetic field is very strong, i.e. $\frac {q ^ 2 B _ 0 ^ 2}{m ^ 2} \gg \frac {k}{m}$. Determine the approximate value(s) of $\omega$ in this case, always up to the first non-zero order. Next, sketch the motion of the particle in the direct (i.e. real) space in this (strong field) case.

Štěpán wanted to create a classical diamagnet.

### 2. Series 34. Year - 2. land ahoy

Cathy and Catherine are watching a ship which is sailing with a constant speed towards a port. Cathy is standing on a rock above the port and her eyes are $h_1=20 \mathrm{m}$ above the surface of the water. Catherine is standing under the rock and her eyes are $h_2=1{,}7 \mathrm{m}$ above the surface of the water. If Catherine sees the top of the incoming ship $t=25 \mathrm{min}$ after Cathy sees it, what is the time of arrival of the ship to the port? Assume that the Earth is a perfect sphere with a radius $r=6378 \mathrm{km}$.

Memories of a seaside vacation.

### 1. Series 34. Year - 2. brake!

Karel's car, going at the initial speed of $v_0$, can stop at a distance $s_0$ with the constant braking force $F_0$. How many times will the braking distance increase if the initial speed doubles and the braking force stays the same? How many times must the braking force be greater for the car to stop at distance $s_0$ with the initial speed $2v_0$?

Karel and a campaign for responsible driving.

### 1. Series 34. Year - 3. cycling anemometer

Vašek rides his bicycle in windy weather. When he rides straight with the velocity $v = 10 \mathrm{km\cdot h^{-1}}$, he measures that the wind blows at an angle $25\dg$ from the direction of Vašek's direction of travel. When he accelerates to $v' = 20 \mathrm{km\cdot h^{-1}}$, the angle is only $15\dg$. Find the velocity and direction of the wind with respect to stationary observer.

Vašek thought that the wind blows on him too much while he's cycling.

### 1. Series 34. Year - 4. solar sail

A solar sail with the surface area of $S = 500 \mathrm{m^2}$ and area density $\sigma =1,4 \mathrm{kg\cdot m^{-2}}$ is located at the distance of $0,8 \mathrm{au}$ from the Sun. What force does the solar radiation act on the sail at the beginning of the sail's motion? What is the acceleration of the sail at that moment? The luminosity of the Sun is $L_{\odot } =3,826 \cdot 10^{26} \mathrm{W}$. Assume that the radiation approaches the sail from a perpendicular direction and scatters elastically. Hint: We recommend you find the acceleration for small initial velocity $v_0$ and then let $v_0 = 0$.

Danka wants to fly.

### 1. Series 34. Year - 5. how to put your beanie on sigle-handily

Let us have a ball with the radius $R$ and a circular massless rubber band with the radius $r_0$ and stiffness $k$, while $r_0 < R$. The coefficient of friction between the band and the ball is $f$. Find conditions which ensure that it is possible to stretch the band over the ball single-handily (i.e. we are allowed to touch the band in only one point.

To keep it simple assume that the band is elastic only in the tangential direction (it is planar).

### 1. Series 34. Year - E. impact-y

Measure the dependence of the diameter of a crater, created by the impact of a stone into a suitable sandpit, on the weight of the stone and the height it is released from. Does the size of the crater depend only on the energy of the impact? Dry sand is recommended for this measurement.

Dodo returned to his childhood.

### 1. Series 34. Year - S. oscillating

Let us begin this year's serial with analysis of several mechanical oscillators. We will focus on the frequency of their simple harmonic motion. We will also revise what does an oscillator look like in the phase space.

1. Assume that we have a hollow cone of negligible mass with a stone of mass $M$ located in its vertex. We will plunge it into water (of density $\rho$) so that the vertex points downwards and the cone will float on the water surface. Find the waterline depth $h$, measured from the vertex to the water surface, if the total height of the cone is $H$ and its radius is $R$. Find the angular frequency of small vertical oscillation of the cone.
2. Let us imagine a weight of mass $m$ attached to a spring of negligible mass, spring constant $k$ and free length $L$. If we attach the spring by its second end, we will get an oscillator. Find the angular frequency of its simple harmonic motion, assuming that the length of the spring does not change during the motion. Subsequently, find a small difference in angular frequency $\Delta \omega$ between this oscillator and the one in which the spring is substituted by a stiff rod of the same length. Assume $k L \gg m g$.
3. A sugar cube with mass $m$ is located in a landscape consisting of periodically repeating parabolas of height $H$ and width $L$. Describe its potential energy as a function of horizontal coordinate and outline possible trajectories of its motion in phase space, depending on the velocity $v_0$ of the cube on the top of the parabola. Mark all important distances. Use horizontal coordinate as displacement and appropriate units of horizontal momentum. Neglect kinetic energy of cube motion in the vertical direction and assume it remains in contact with the terrain.

Štěpán found a few basic oscillators.

### 6. Series 33. Year - 3. hung

What weight can be hung on the end of a coat hanger without turning it over? The hanger is made of a hook from very light wire, which is attached to the centre of the straight wooden rod, which length is $l = 30 \mathrm{cm}$ and weight $m=200 \mathrm{g}$. The hook has the shape or circular arc with radius $r=2,5 \mathrm{cm}$ and angular spread $\theta =240 \mathrm{\dg }$. The distance between the centre of the arc and the rod is $h=5 \mathrm{cm}$. Neglect every friction. 