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After we press the break pedal, the car does not start to break immediately. During the time $t_{r}$ the breaking force grows linearly up to the maximum force $F_{m}$. Coefficient of static friction between the tire and a road is $f$. What is the maximum speed of car so that the car does not skid even during emergency breaking?

The notebook of a size of A4 (297 x 210 mm) lies on a desk with an inclination of $α=5°$. The notebook weights $m$, between the desk and the notebook there acts a static friction force with coefficient $f_{0}=0.52$. Then, we hit the desk so it starts to oscillate (in the direction of the inclination of the desk) with a frequency $ν=10\;\mathrm{Hz}$ and an amplitude $A=1\;\mathrm{mm}$.

• Determine by which extra force (perpendicular to the desk) we have to act on the notebook so it does not start to move.
• Determine how long it takes the notebook to fall off the desk if at the beggining its bottom edge (the shorter one) is at the bottom edge of the desk. Dynamic friction coeficient is $f$, consider notebook as a rigid plate.

A rat is running on ice with speed $v$. Suddenly he decides to turn 90$°$ so that he keeps running with the same speed in the new direction. What is the least amount of time he needs for such a turn? Suppose that rat's feet can move independently. Coefficient of friction between rat's feet and ice is $f$.

From a spaceship on a circular orbit with height $h=2000\;\mathrm{km}$ above the surface of Earth a screwdriver is thrown with speed $v=5\;\mathrm{km}\cdot h^{-1}$ relative to the rocket towards the center of the Earth. Determine when will the screwdriver hit the surface?

In a train, that can move without friction on rails, stand 2 people, both with a mass $m$. In which of the two following situations shall the train reach a higher speed? When both jump out at the same time or when they will jumping outone after another? A person can jump out the train with a relative speed $u$ (the speed of a person jumping out the train versus the speed of the train).

$N$ people decide to play tag but not the normal variety. At the start they stand in the vertices of a regular $N-gram$ of a side $a$. The game then proceeds so that everyone chases (goes to him in a straight line)his neighbour on the right (anti-clockwise). Everyone moves with the same constant velocity $v$. Describe the progress of the game (trajectory on which the players move) and determine how quickly the game will end depending on the parameters $N$, $and$, $v$.

Autumn weather is sometimes as unstable as Spring weather and so it often happens that we can be surprised by an unforeseen torrent of rain. A happy few carried umbrellas. Approximate how large the pressure of heavy rain can be and compare the force of the rain with the gravitational force with which the umbrella is pulled down. Choose the parameters of the umbrella appropriately.

• Show that for arbitrary values of parameters $K$ and $T$ you can express the Standard map from the series express as

$$x_{n} = x_{n-1} y_{n-1},$$

$$\\ y_n = y_{n-1} K \sin(x),$$

where $x$, y$ are somehow scaled d$φ⁄dt,φ$. Show that the physical parameter $K$, x, y$$.

• Look at the model of the kicked rotor from the series and take this time the passed impuls$I(φ)=I_{0}$, after the period $T$ then $I(φ)=-I_{0}$, after another one $I_{0}$ and this way keep on kicking the rotor on and on.
• Make a map $φ_{n},dφ⁄dt_{n}$ on the basis of values $φ_{n-1},dφ⁄dt_{n-1}$ before the doublekick ± $I$ Why not?
• Solve $φ_{n},dφ⁄dt_{n}$ on the basis of some initial conditions $φ_{0},dφ⁄dt_{0}$ for an arbitrary $n$.
• *Bonus:** Try using the ingeredients from this series to design kicking which $will$ result in chaotic dynamics. Take care though because $φ$ is periodic with a period 2π and shouldn't d$φ⁄dt$ unscrew forever through kicking.

Two notebooks of the type A460 we shall insert into each other so that a page of one is always followed by the page of another and we put them on a horizontal table. What is the work we have to do to seperate them if the lists act on each other only with their own weight? Assume that we pull only in the plane of the notebooks by the back of one of them and also assume that in the beginning the pages perfectly cover each other.

Determine the acceleration (both due to gravitational and centrifugal forces) on the surface of a neutron star based on what lattitude we are. How large would the tidal forces acting on an object of height $h=1\;\mathrm{m}$ and with a mass $m=1\;\mathrm{kg}$ in the vicinity of it surface be? What would the energy of a marshmallow be if it fell to the surface from a height of $h?$ The neutron star has a radius of $R$ and rotates with a period of $T$. You can consider it spherical even though it is not precisely spherical. Find values that are typical for neutron stars and give general as well as concrete numerical answers.