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• 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.

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 $ε)$.

• 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?

Explain why and how the following situations occur:

• In a cistern of a rectangular cuboid shape that is filled with water a ball is floating on the surface of the water. Describe the movement of the ball if the cistern starts moving with a constant acceleration small enough that the water shall not flow over the edge.
• In a cistern of a rectangular cuboid shape that is filled with water a ball filled with water is floating. Describe the movement of the ball if the cistern starts moving with a constant acceleration small enough that the water shall not flow over the edge.
• In a closed bus a ballon is floating near the ceiling. Describe its movement if the bus starts accelerating constantly

Calculate the power required by a bee to remain in the air and approximate how long a bee that has just eaten can remain in the air for(at a constant altitude).

• Write down the equations for a throw in a homogeneous gravitational field (you don't need to prove them but you need to know how to use them). Design a machine that will throw an item and determine the angle of approach and the velocity. You can throw with the item with a spring, determine its spring constant, mass of the object and calculate the kinetic energy and thus the velocity of the item. What do you think is the precision of the your value of the velocity and angle? Put the boundaries determined by this error into the equations and show in what boundaries we can expect the distance of the landing from the origin to be.Throw the item with your device at least five times and determine the distance of the landing and what are the boundaries within which you are certain of your distance? Show if your results fit into your predictions. (For a link to video with a throw you get a bonus point!)
• Tie a pendulum with an amplitude of $x$, which effectively oscillates harmonically but the frequency of its oscillations depends on the maximum displacement $x_{0}$

$$x(t) = x_0 \cos\left[\omega(x_0) t\right]\,, \quad \omega(x_0) = 2\pi \left(1 - \frac{x_0^2}{l_0^2}\right)\,,$$

where $l_{0}is$ some length scale. We think that are letting go of the pendulum from $x_{0}=l_{0}⁄2$ but actually it is from $x_{0}=l_{0}(1+ε)⁄2$. B By how much does the argument of the cosine differ from 2π after one predicted period? How many periods will it take for the pendulum to displaced to the other side than which we expect? Tip Argument of the cosine will in that moment differ from the expected one by more than π ⁄ 2.

• Take a pen into your hand and let it stand on its tip on the table. Why does it fall? And what will determine if it will fall to the right or to the left? Why can't you predict a die throw even though the laws of physics should predict it? When you play billiard is the inability to finish the game only due to being incapable of doing all the neccessary calculations? Write down your answers and try to enumerate physics phenomenons that occur in daily life which are unpredictable even if we know the situation well.

More than a hundred years ago the measurements of surveyors confirmed that when we sail west, gravimeters show higher values of gravitational acceleration than when travelling east. Determine the difference that we measure on the equator between the measurements we make when still (relative to the earth) and when we are travelling at 20 knots per hour westwards.

We put a roll with paper into a bearing (without friction) and we let the paper unroll itself (we neglect the sticking of layers to eac other, friction in the bearing and the weight of the bearing). What is the angular velocity of the roll after all paper is removed? We know the radiusand mass of the roll, the longitudal density of paper, its overall mass and its length. Consider that the paper shall be able to unroll into an infinite pit.

Bonus: Now consider that the paper will fall to the ground before it all unrolls.

An object of mass $m$ on a piece of rubber of length $l_{0}is$ hung at a rigid point, the coordinates of which are $x=0$ and $y=0$. From the $xaxis$, which is horizontal, we slowly release the mass. What will be the relation between the lowest point reached and its position on the axis $x?$

Two ships are sailing against each other, the first one with a velocity $u_{1}=4\;\mathrm{m}\cdot \mathrm{s}^{-1}$ and the second with a velocity of $u_{2}=6\;\mathrm{m}\cdot \mathrm{s}^{-1}$. When they are seperated by $s_{0}=50\;\mathrm{km}$, a seagull launches from the first ship and flies towards the second one. He is flying against the wind, his speed is $v_{1}=20\;\mathrm{m}\cdot \mathrm{s}^{-1}$. When he arrives to the second ship he turns around and flies back now with the wind behind his back with a velocity $v_{2}=30\;\mathrm{m}\cdot \mathrm{s}^{-1}.He$ keps on flying back and forth until the two ships meet. How long is the path that he has undertaken?