Series 3, Year 31

Let's suppose a camera with a frame rate of 24 frames per second (consider evenly spaced and perfectly sharp shots). We record a flight of a helicopter with the rotor rotation velocity of $2 900 \mathrm {cycles/min}$. Then the record is played. What is the apparent rotational velocity of the rotor in the record?

In the figure, there is an ellipse with two focal points $F_1$ $F_2$ and several marked points on the ellipse. The ellipse represents a trajectory of one material point. Plot the accelerations the point experiences in given points of its trajectory. Show it in a figure. The direction and ratio of accelerations are important.

1. There is a massive body in the focal point $F_1$. The material point is orbiting it, and Kepler's $2^{\rm nd}$ law applies.
2. The absolute value of velocity of the material point is constant. It only moves along the ellipse.

You fired at Imp from your plasma gun which shoots a cluster of particles with uniform velocity distribution in interval $\langle v_0, \; v_0+\delta v\rangle$ (all of the particles are moving in one line, there is no transverse velocity). The total kinetic energy of the cluster is $E_0$. The barrel rifle has a cross cestion of $S$ and the pulse takes an infinitely short time. How far does Imp need to stand to be safe. Assume that his skin is able to cool the heat flow of $q$.

We drop a pen (rigid stick) on a table so that it makes an angle $\alpha $ with horizontal plane during its fall. Calculate the velocity of the higher end during its impact. When we dropped the pen, its center of mass was at height $h$. All collisions are inelastic and friction between the table and the end of the pen large enough.

Bonus: Calculate the angle $\alpha $ so that the velocity (of the second end that touches the table) is maximal. For which height $h$ is it worth to tilt the pen?

We have $A_0$ particles which decay into $B$ particles with decay constant $\lambda \_A$. $B$ particles decay into $A$ particles with decay constant $\lambda \_B$. The number of $B$ particles at the beginning is $B_0$. Find a ratio of the numbers of particles $A$ and $B$ as a function of $t$.

Everyone has certainly heard and surely tried it: „Sheet of paper can not be folded in a half more than seven times.“ Is it really true? Find boundary conditions.

You got a planar magnet (magnetic foil) together with the tasks of these series. This magnet is a bit different than a rod magnet. The south and north poles are alternating parallel lines. When approaching the ferromagnetic surface, a magnetic circuit is created which holds the magnet (for example, on the fridge) and can carry even a picture on itself. Your tasks are:

• Measure the area and thickness of the film which you be used for your experiments.
• Measure the mean distance between the two closest same magnetic poles (twice the distance of opposite poles).
• Measure the maximum payload (ie. weight without magnet weight) which can be carried by a $1 \mathrm{cm^2}$ of a magnet if the magnet load is even if the magnet is attached to the bottom of the horizontal plate. The plate should be approx.  $1 \mathrm{mm}$ thick sheet made of magnetically soft steel.

1. Propose three different examples of Markov chains, at least one of which is related to physics. Is a random walk without backtracking (a step cannot be time reversed previous step) an example of Markov chain? What about a random walk without a crossing (it can lead to each point at most once)?
2. Consider a 2D random walk without backtracking on a square grid beginning at the point $(x,y) = (0,0)$. It is constrained by absorbing states $b_1\colon y = -5$, $b_2\colon y = 10$. Find the probability of the walk ending in $b_1$ rather than in $b_2$.
3. Simulate the motion of a brownian particle in 2D and plot the mean distance from the origin as a function of time. Assume a discrete time and a constant step size. (One step takes $\Delta t = \textrm{const} $, and the step size is $\Delta l = \textrm{const} $). A step in any arbitrary direction is possible, i.e. every step is described by it’s length and an angle $\theta \in [0,2\pi )$, while all directions are equally probable. Focus especially on the asymptotic behavior, i.e. the mean distance for $t \gg \Delta t$.
4. Error function is defined as \[\begin{equation*} {erf}(x)=\frac {2}{\sqrt {\pi }}\int _0^x \eu ^{-t^2} \d t . \end {equation*}\] Calculate the integral for many different values of $x$ and plot it’s value as a function of $x$. What do you get by numerically deriving this function?
5. Look up the definition of Maxwell-Boltzmann probability distribution $f(v)$, i.e. the probability distribution of speeds of particles in an idealized gas. Utilizing MC integration calculate the mean value of speed defined as \[\begin{equation*} \langle v\rangle = \int _0^{\infty } vf(v) \d v , \end {equation*}\] Use the Metropolis-Hastings algorithm for sampling the Maxwell-Boltzmann distribution. Compare the values of particular parameters with the values from literature.