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## mechanics of a point mass

### (4 points)5. Series 28. Year - 3. matfyz tag

$N$ 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 $N$, $and$, $v$.

Kuba Vosmera graduate.

### (4 points)5. Series 28. Year - 4. heavy rain

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.

Mirek was looking for excuses why not to be envious of protected passerbys.

### (6 points)5. Series 28. Year - S. mapping

- 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.

### (4 points)4. Series 28. Year - 3. unbreakable bond

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.

Mirek was unsuccesfully dividng analysis and algebra.

### (4 points)4. Series 28. Year - 4. oh the gravity

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.

Karel was dreaming of the devastating power of neutron power and their amazing non-inertiality .

### (4 points)4. Series 28. Year - 5. knife thrower

The throwing knife shall leave the hand in the moment that its center of mass is at the height $h$ and has just a purely horizontal component of velocity $v_{0}$. What must its angular velocity $ω$ be for it to hit and stick in a vertical panel at a distance$dfrom$ the point of escape? To make simplify consider the center of mass to be in the middle of its length$l$ and that the knife shall stay in the vertical if the blade shall hit it before the hilt.

Mirek's experiments with knife throwing were not following his statistical predictions.

### (8 points)3. Series 28. Year - E. sneakers on water

Measure the coefficient of static and dynamic friction between the sneaker (shoe) and a horizontal smooth surface, where the surface is dry and where it is wet. Compare the results and interpret.

Karel slipped on dry land.

### (6 points)3. Series 28. Year - S. numerical

- Look at the equations of the Lorenz model and write a script to simulate them in Octave (maybe even refresh your knowledge of the second part of series). Together with the sketching command your script should have the following form: …

function xidot = f(t,xi)

…

xdot=…;

ydot=…;

zdot= …;

xidot = [xdot;ydot;zdot];

endfunction

config = odeset('InitialStep', 0.01,'MaxStep',0.1);

initialCondition=[0.2,0.3,0.4];

solution=ode45(@f,[0,300],initialCondition,config);

plot3(solution.y(:,1),solution.y(:,2),solution.y(:,3)); </pre> Just instead of three dots fill in the rest of the code (just as in the second part of the series) and use $σ=9,5$, $b=8⁄3.Then$ figure out with a precision of at least units for what positive $r$ the system goes from asymptomatic stopping to chaotic oscillation(it is independent of the initial conditions).

- Here is the full text of the Octave script for simulating and visualising the movement of a particle in a gravitational field of a massive object in the plane $xy$, where all the constants and parameters are equal to one: clear all

pkg load odepkg

function xidot = f(t,xi)

alfa=0.1;

vx=xi(3);

vy=xi(4);

r=sqrt(xi(1)^2+xi(2)^2);

ax=-xi(1)/r^3;

ay=-xi(2)/r^3;

xidot = [vx;vy;ax;ay];

endfunction

config = odeset('InitialStep', 0.01,'MaxStep',0.1);

x0=0;

y0=1;

vx0=…;

vy0=0;

initialCondition=[x0,y0,vx0,vy0];

solution=ode45(@f,[0,100],initialCondition,config)

plot(solution.y(:,1),solution.y(:,2));

pause()</pre>

- Choose initial conditions $x0=0,y0=1,vy0=0$ and and a nonzero initial velocity in the direction $x$ such that the particle will be bound (ie. it won't escape the center.)
- Add to the gravitational force the following force $-α\textbf{r}⁄r^{4}$, where $αis$ a small positive number. Choose gradually increasing $α$ beginning with $α=10^{-3}$ and and show that they cause quasiperiodic movement.

### (5 points)2. Series 28. Year - 5. gravitational manuevers

Assume a satelite which orbits the sun on an elliptical orbit. If we lower the speed in the aphelion $v_{a}$ to 4⁄5 of the initial velocity (i.e. to 4⁄5$v_{a})$, how will the speed of the satelite change in the perihellion? Express the new velocity using the initial velocity $v_{p}$ and the parameters of the ellipse (main axis $a$ and relative eccentricity $ε)$.

Karel attended a lecture about gravitational slingshots.

### (6 points)2. Series 28. Year - S. numerical

- We give length values in metres, time values in seconds and mass values in kilograms. Angular velocity $Ω$ we give in radians per second. If you take the equations for the movement of balls from the series, there are three more parameteres included: $α$, $β$, $γ$. What are their dimensions?
- Consider a freefalling ball with $Ω=0$ and $v_{x}=0$. There then exists a terminal velocity $v_{z}^{t}$, at which the frictional force and and gavitational force are equally matched and the fall of the ball isn't accelerating anymore.
- Determine this velocity from the equations for the movement o a ball.
- Change this equation so that it will express $β$. $v_{z}^{t}$ can be easily measured and for ourfootball of mass $m=0,5\;\mathrm{kg}it$ is typically around 25 m\cdot s^{ −1}. Then what is $β?$
- Express the initial $v_{x}$ and $v_{z}$ using the angle at which it was shot out $φ$ with a fixed initial velocity $v=10\;\mathrm{m}\cdot \mathrm{s}^{-1}$. Write a program according to the series and try changing the initial conditions and the following parameters
- Choose some positive $β$, turn off the rotation $Ω=0$ and find out, if the angle under which the the ball reach the farthest is bigger or smaller than 45°. Demonstrate your finding with graphs of the trajectories.
- Choose a positive non-zero $α$ with a numerical value in the given units the same as $β$, $γ=0,01$ (in the given units) and $Ω=±5rad\cdot \;\mathrm{s}^{-1}.How$ will in these specific cases the optimal angle of the shot change?
- *Bonus:** How far would you throw with a cricket ball? Is our model good enough to make such predictions?