Series 6, Year 31

We have two point masses with the same mass $m$ at a distance $d$ from each other. They are located freely in space with no external gravitational forces. What's the minimum velocity we need to impart on one of the points in the direction away from the other point, so that they keep flying away from each other indefinitely?

Calculate the current, that needs to pass through a metal wire of a diameter $d = 0{,}10 \mathrm{mm}$ located in a vacuum bulb, so that its temperature stays at $T = 2 600 K$. Assume the surface of the wire radiates like an ideal black body and neglect any losses by heat conduction. The resistivity of the material of the wire at the given temperature is $\rho = 2{,}5 \cdot 10^{-4} \mathrm{\Ohm \cdot cm}$. \taskhint {Hint}{Use the Stefan-Boltzmann's law.}

Imagine a pole of length $b = 5 \mathrm{cm}$ and mass $m = 1 \mathrm{kg}$ and a spring of initial length $c = 10 \mathrm{cm}$, spring constant $k = 200 \mathrm{N\cdot m^{-1}}$ and negligible mass, that are connected at one of their ends. The other ends of the spring and the pole are affixed at the same height $a = 10 \mathrm{cm}$ from each other. The spring and the pole can both freely rotate about the fixed points and their joint. Label $\phi $  the angle of the pole to the horizontal. Find all angles $\phi $, for which the system is in an equilibrium. Which of these are stable and which unstable?

Matej was making a gun and wanted to measure what is the speed of the projectiles leaving the barrel. Unfortunately, he doesn't have any other measuring device, than a ruler. However, he found a block that is made half from steel half from wood. He lays it down at the edge of the table (of height $100 \mathrm{cm}$ and length $200 \mathrm{cm}$), and shoots at it horizontally. With the steel part of the block facing the gun, the bullet bounces off perfectly elastically and lands $50 \mathrm{cm}$ from the edge of the table. The block slides $5 \mathrm{cm}$ on the table. Then Matej turns around the block and shoots into the wooden side. This time the bullet stays in the block and the block slides only $4 \mathrm{cm}$. Help Matej with calculating the speed of the bullet. It might be also helpful to know, that when Matej lifts one edge of the table by at least $20 \mathrm{cm}$, the moving block won't stop sliding.

Filip of mass $80 \mathrm{kg}$ jumped out of air plane, that is $h_1 =500 \mathrm{m}$ above the ground. At the same time, Danka (mass $50 \mathrm{kg}$) jumps out of a different airplane but from a height of $h_2 =569 \mathrm{m}$. Assume both of them have the same drag coefficient $C = 1{,}2$, Filip's cross-sectional area is $S\_F = 2{,}2 \mathrm{m^2}$ and Danka's is $S\_D=1{,}5 \mathrm{m^2}$. The density of air $\rho =1{,}205 \mathrm{kg\cdot m^{-3}}$ is and stays the same in all heights. At what time will Danka be at the same height above the ground as Filip?

According to the current observations and cosmological models, it seems that our Universe is expanding and the rate of expansion is accelerating. What if that wasn't the case? What if the Universe stayed the same, but the physical laws/constants were changing so that it would seem like the universe is expanding, the way we observe it? Describe as many laws that would need to change.

Measure the speed with which a wooden skewer burns as a function of its tilt with respect to the vertical.

1. Simulate the dynamics of a predator-prey system using Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} &= r\_x x - D\_x xy ,\\ \frac{\d y}{\d t} &= r\_y xy - D\_y y . \end {align*}\] where $x$ and $y$ are the population sizes of prey and predator respectively, the parameters $r\_x$ and $r\_y$ represent the populations’ growth and the parameters $D\_x$ and $D\_y$ represent the shrinking of the populations. Set the parameters to be $r\_x = 0{.}8$, $D\_x0= 1{.}0$, $r\_y = 0{.}75$, $D\_y = 1{.}5$. Run the simulations for several different value pairs for initial population sizes $x = 0{.}5$ and $y = 2{.}0$; $x = 1{.}5$ and $y = 0{.}5$; $x = 1{.}95$ and $y = 0{.}75$. Plot the predator population size as a function of the prey population size. Discuss the results.
   Bonus: Find the solutions for the same situations analytically (by integrating the differential equations).
2. Using the competitive Lotka–Volterra equations \[\begin{align*} \frac{\d x}{\d t} = r\_x x \(1 - \(\frac {x + I\_{xy} y}{k\_x}\)\)  ,
   \frac{\d y}{\d t} = r\_y y \(1 - \(\frac {y + I\_{yx} x}{k\_y}\)\) . \end {align*}\] simulate the dynamics of two competing populations (e.g. hawks and eagles) for the following values of parameters: $r\_h = 0{.}8$, $I\_{he} = 0{.}2$, $k\_h = 2{.}0$, $r\_e = 0{.}6$, $I\_{eh} = 0{.}3$, $k\_e = 1{.}0$. Set the initial population sizes to be $h = 0{.}01$, $e = 1{.}0$. Then, simulate the same situation, but change the interaction coefficients to $I\_{he} = 1{.}5$ a $I\_{eh} = 0{.}6$. Plot the results in one graph - the sizes of populations vs time. Discuss the results.
3. Verify the importance of pivoting.
   Solve the system of linear equations \[\begin{equation*} \begin{pmatrix} 10^{-20} & 1\\ 1 & 1 \end{pmatrix} \begin{pmatrix} x_1\\ x_2 \end{pmatrix} = \begin{pmatrix} 1\\ 0 \end{pmatrix} \end {equation*}\] at first exactly (on paper), then using LU factorization with partial pivoting (you may utilize some Python module, e.g. scipy.linalg.lu()), and finally, solve the system using LU factorization without pivoting. Compare the resultant  $\vect {x}$ obtained from the three methods and the results of matrix multiplication $L^{-1}\cdot U$ ($P\cdot L^{-1}\cdot U$ in the case of pivoting).
4. Consider an infinite parallel-plate capacitor. The gap between plates has a thickness $L=10 \mathrm{cm}$ and the voltage between the plates is $U=5 \mathrm{V}$. Between the plates of the capacitor grounded electrode in the shape of an infinitely long prism with square base of side length $a=2 \mathrm{cm}$, whose center lies $l=6{,}5 \mathrm{cm}$ away from the grounded plane of the original capacitor. The prism is oriented such that one of its short sides is perpendicular to the capacitor plates. Find the distribution of electric potential in the condensator. Since the problem has a translational symmetry in the direction of the infinite side of the prism, it is sufficient to solve it only in the plane parallel to the plates, i.e. it is a 2D problem. Render the potential distribution in this plane. You may utilize the code attached to this task.
   Bonus: Calculate and render the distribution of the electric field strength $\vect {E}$.