Problem Statement of Series 4, Year 39

Jarda is driving a car at a speed of $70\,\mathrm{km\cdot h^{-1}}$ and is approaching a car that has just entered a town and slowed down to $50\,\mathrm{km\cdot h^{-1}}$. Jarda sees the trees behind him in the reflection on the rear window of this car. What is the speed and direction of the trees' motion in the reflection relative to Jarda?

Consider two containers filled with an ideal gas of equal volume and temperature, consisting of a mixture of oxygen and nitrogen. Each container is sealed at the top by a piston of identical thickness and material, positioned at the same height. The pistons exert pressure on the gas only through their own weight and do not interact with the container walls.

The ratio of the partial pressures of oxygen in the first and second containers is $3:5$. The partial pressure of nitrogen in the first container exceeds that in the second by $40\,\mathrm{Pa}$, and the sum of the oxygen partial pressures in both containers equals the sum of the nitrogen partial pressures. Determine the resulting displacement of the pistons and the total pressure after the containers will be interconnected.

What is the minimum initial velocity required for the Little Prince to jump from the asteroid B-612, of density $\rho_1=3~000\,\mathrm{kg\cdot m^{-3}}$ and radius $R=5\,\mathrm{km}$, in order to land on the surface of the King's asteroid, which has the density of $\rho_2=2~500\,\mathrm{kg\cdot m^{-3}}$ and radius $r=3\,\mathrm{km}$? The distance between the centers of the asteroids is fixed at $d = 50\,\mathrm{km}$. The asteroids do not move toward each other. Atmospheric effects, orbital motion, and rotation of both homogeneous asteroids may be neglected.

Consider an ideal impulsive change in momentum. Two rockets are in orbit about the Earth, both moving on the same circular orbit of radius $r_0$ with identical speeds. The angle $\theta \ll 1$ between their position (radius) vectors is constant. The trailing rocket, of mass $m$, applies a radial momentum impulse $\Delta \vec{p}$ ) such that the semi-major axis of its orbit remains unchanged, while its eccentricity is altered. That is, the orbital trajectories before and after the impulse are ellipses with the same semi-major axis.

As a result of this maneuver, the trailing rocket collides with the leading rocket in less than one half of an orbital period. Determine the time of flight $\tau$ from the instant the impulse is applied to the moment of collision, and determine both the direction and the magnitude of the impulse $\Delta \vec{p}$. Assume that both rockets are sufficiently far from the Earth that there are no constraints on their ability to maneuver.

Two weights are suspended from a pulley. One weight is of mass $m_1$ and the other of mass $m_2$. The mass $m_1$ carries an electric charge $Q_1$, while the mass $m_2$ is not charged. A point electric charge $Q$ is located above the pulley or below the pulley at a distance $h$. Is there an equilibrium position for the weights on the pulley? Is it stable? Consider the radius of the pulley to be negligible. Solve the problem for the following cases:

1. The charge $Q = -1.00\cdot 10^{-5}\,\mathrm{C}$ is located at a height $h = 0.15\,\mathrm{m}$ above the pulley. The values of the weights are $m_1 = 1.00\,\mathrm{kg}$, $m_2 = 2.00\,\mathrm{kg}$, $Q_1 = -1.00\cdot 10^{-5}\,\mathrm{C}$. The length of the rope connecting the two weights via the pulley is $l = 1.00\,\mathrm{m}$.
2. The charge $Q = 5.00\cdot 10^{-6}\,\mathrm{C}$ is located at a height $h = 0.30\,\mathrm{m}$ above the pulley. The values of the weights are $m_1 = 1.50\,\mathrm{kg}$, $m_2 = 2.00\,\mathrm{kg}$, $Q_1 = 2.00\cdot 10^{-4}\,\mathrm{C}$. The length of the rope connecting the two weights via the pulley is $l = 0.75\,\mathrm{m}$.
3. The charge $Q = 4.00\cdot 10^{-5}\,\mathrm{C}$ is located at a depth of $h = 0.50\,\mathrm{m}$ below the pulley. The values of the weights are $m_1 = 1.20\,\mathrm{kg}$, $m_2 = 1.90\,\mathrm{kg}$, $Q_1 = -2.00\cdot 10^{-5}\,\mathrm{C}$. The length of the rope connecting the two weights is $l = 2.00\,\mathrm{m}$.

This problem has an open solution, so be sure to cite all sources used.

Choose any plain chocolate bar (without fillings such as nuts, biscuits, etc.) and estimate as accurately as possible how much energy is required for a single bar to reach you, the consumer (that is, from its production through distribution and sale). Compare this value with the energy content of the chosen chocolate bar.

Determine the period of small oscillations of a standard clothes hanger suspended from a thin horizontal rod. Try to estimate the period theoretically through calculation, and compare the calculated value with the experimentally measured period.

1. In the attached file¹ you will find the measured lifetimes of atoms of the radioactive isotope $\ce{^214Po}$. The time is in nanoseconds. Determine the half-life of this isotope. – 3 points

2. An electron with kinetic energy $150\,\mathrm{keV}$ flew into the detector and transferred all of its energy to the detector. Determine the uncertainty of the measured energy if the detector was

   • a gas detector with argon;

   • a silicon semiconductor detector;

   • a scintillator $\ce{LaBr_3 (Ce)}$.

   Consider that the source of uncertainty is only the fluctuation in the number of free charge carriers or photons created. Also assume that the detector collects all charge carriers (or all photons in the case of a scintillator). For the scintillator, consider the Fano factor $F=1$. – 2 points

3. Consider the AMS-02 experiment aboard the International Space Station (ISS). In addition to other detectors, it includes a transition radiation detector, a RICH detector, and a magnetic spectrometer with 7 layers of silicon detectors. The RICH detector contains $\ce{NaF}$ with a refractive index of $1.33$ and aerogel with a refractive index of $1.05$.

   In addition to measuring position with an accuracy of $10\,\mathrm{\upmu{}m}$ , the silicon detectors in the magnetic spectrometer can also measure energy loss per unit length $\mathrm{d} E/\mathrm{d} x$. Describe qualitatively how these three detectors can be used to distinguish between the following four particles: electron, proton, antiproton, and helium nucleus $\ce{^4He}$. Each particle has the same kinetic energy $3\,\mathrm{GeV}$. – 2 points

4. From the Bethe-Bloch equation, derive the $\beta$ factor at which the energy loss per unit length $\mathrm{d} E/\mathrm{d} x$ is the lowest (minimum ionizing particle). Neglect the correction terms $\delta\!\left(\beta\right)$ and $C\!\left(\beta\right)$ and also assume that the equation in this form also applies to electrons. Do not be afraid of numerical solutions. You can use the average excitation energy $I = 92.2\,\mathrm{eV}$, which we calculated for nitrogen using the equation from the series text. In this way, we can approximate the passage through air. What are the kinetic energies of a MIP electron, a MIP muon and a MIP proton? – 3 points