Problem Statement of Series 2, Year 39

Jarda is $110\,\mathrm{m}$ away from a traffic light showing red and is approaching it at a speed of $50\,\mathrm{km/h}$. He intends to decelerate with an acceleration of no more than $3.0\,\mathrm{m\cdot s^{-2}}$.

What is the maximum speed at which he can pass the traffic light while it is green, assuming he never presses the accelerator during the entire approach? Assume that when Jarda is not actively braking, the car moves at a constant speed. The green light will turn on in $21\,\mathrm{s}$.

A mass point is attached to a thin horizontal rod by a massless, inextensible, flexible string of length $L$. The mass is initially placed so that the string is horizontal and perpendicular to the rod. The string is then taut and the mass is released downwards. At a distance $d

A young soccer player finds that after an unfortunate shot, his football becomes wedged between two vertical parallel walls separated by a distance of $21 cm$. The boy can exert a maximum force of $450\,\mathrm{N}$, while the coefficient of friction between the ball and the walls is $0.6$.

However, being a participant of VYFUK physics competition, he knows quite a bit about physics. He therefore begins to spray the ball with cold water at a temperature of $10\,\mathrm{^\circ\mskip-2mu\mathup{C}}$.

Will he be able to pull the ball out afterwards? The ball has a radius of $11\,\mathrm{cm}$, and the contact areas with the walls are circular. Assume that the volume of the ball remains constant, both during its deformation between the walls and during cooling. Before the shot, the ball was inflated to $120\,\mathrm{kPa}$ and had a temperature of $44\,\mathrm{^\circ\mskip-2mu\mathup{C}}$.

Consider an immovable, homogeneous, uncharged, conducting sphere of radius $R$. A particle carrying charge $q$ is launched from infinity with speed $v$ toward the sphere, along a trajectory characterized by an impact parameter $b$ (i.e., the perpendicular distance from the sphere's center to the asymptotic path of the particle).

Determine the conditions under which the particle does not collide with the uncharged sphere. Neglect the effect of any induced magnetic field.

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

There are plans to develop a probe made up of two identical material sections equipped with instruments, connected by a fiber between them. Each section would be black on one side and highly reflective on the other, allowing the probe to spin up to a very high angular velocity under the influence of solar radiation pressure. At the point of maximum rotational speed, the fiber would break, causing the two sections—or at least one of them—to escape the Solar System. What is the maximum speed that one part of the probe could reach using the strongest materials available today? Assume each instrument section has a mass of $M = 1.0\,\mathrm{kg}$ and the fiber has a mass of $m = 0.50\,\mathrm{kg}$.

What maximum velocity could one section achieve when leaving the Solar System, assuming that before separation the probe orbited the Sun along an elliptical orbit with perihelion $a_{\mathrm{pe}} = 0.50\,\mathrm{au}$ and aphelion $a_{\mathrm{af}} = 1.0\,\mathrm{AU}$?

Bonus: How long would the acceleration of the probe to that speed take? After how long could we consider that the probe has left the Solar System?

Using any method, measure speed by means of the Doppler effect. Repeat the measurement several times for different speeds, using appropriate software to generate and to analyze the frequencies. Discuss the values obtained and compare them with estimated or otherwise determined values. Also evaluate how good the measurements were with the method you chose for the various speeds.

1. Attempt to interpret the simple spectrum of hydronium, or the hydroxonium $\ce{H3O+}$. Based on the information from the series text, assign four specific peaks in the infrared and Raman spectra to the molecular vibrations determined by the quantum-chemical calculation program.

   

   The wavenumbers in the spectrum have intensity maxima estimated at $872\,\mathrm{cm^{-1}}$, $1~686\,\mathrm{cm^{-1}}$, $3~611\,\mathrm{cm^{-1}}$, and $3~700\,\mathrm{cm^{-1}}$. The second and fourth modes in this sequence are actually doubled due to the symmetry of the cation; however, this does not affect this problem. Assign the depicted vibrational modes in the figure below to the individual wavenumbers. Justify your answer as well as possible in terms of vibrational symmetry, charge displacement, i.e., changes in the dipole moment vector, and molecular orbitals. – 5 points

   Hint: You can take inspiration from the spectrum of water, which has the same number of electrons as hydronium, and its interpretation can also be found on many online resources.

   

2. Using subproblem 1, determine the stiffness of the hypothetical “spring” corresponding to the vibration of the hydronium at the wavenumber $3~611\,\mathrm{cm^{-1}}$. The equilibrium $\ce{O-H}$ bond length in the cation is $98\,\mathrm{pm}$. Neglect more complex effects, such as possible changes in the effective mass during the vibration. – 3 points.

3. Derive how the maximum mentioned in the previous subproblem shifts when the light hydrogen atoms $(\ce{^1 H})$ are replaced with deuterium $(\ce{ {}^2 H})$. Is this estimate reasonable if molecules of ordinary water are present in an acidic deuterated solution? Discuss what would actually be observed in aqueous solutions. – 2 points.

   Hint: Consider that solvation occurs in the solution, i.e., the substance is surrounded by the solvent. This is often a subject of modeling in quantum chemistry. The whole process is dynamic, involving clustering through hydrogen bonding and (de)protonation of water clusters.