Problem Statement of Series 1, Year 35

Two cars start to move from the same point at the same time with velocities $v_1 = 100\,\mathrm{km\cdot h^{-1}}$ and $v_2 = 60\,\mathrm{km\cdot h^{-1}}$. Is it possible for the cars to move away from each other at any of the following velocities? If so, sketch the situations. \begin{alignat*}{2} v_a &= 160\,\mathrm{km\cdot h^{-1}} \,, \quad & v_b &= 40\,\mathrm{km\cdot h^{-1}} \,, \\ v_c &= 30\,\mathrm{km\cdot h^{-1}} \,, \quad & v_d &= 90\,\mathrm{km\cdot h^{-1}} \end{alignat*}

The two-second rule is a driving principle which states that a safe time distance between two vehicles is at least two seconds long. Suppose a traffic junction where a $n_1$-lane road changes into a $n_2$-lane one. The maximum allowed speed in the first section is $v_1$. What is the lowest possible maximum speed $v_2$ that can be allowed in the second section so that there is no traffic jam and everyone can follow the two-second rule? The average length of a car is $l$ and it can change its speed in leaps.

Skaters can stop using the “parallel slide” method, in which they turn the blades of both skates perpendicular to the direction of movement, which significantly increases the friction with the ice. During this, the skater must tilt by the angle $\phi = 35^\circ$ from the vertical direction, so he doesn't fall. Assume that he weighs $m = 70\,\mathrm{kg}$ and that he is $H = 1.8\,\mathrm{m}$ high (including the skates), with the center of gravity at a height of $h = 1.1\,\mathrm{m}$ above the ice. Calculate the distance in which he stops from the initial speed $v_0 = 15\,\mathrm{km\cdot h^{-1}}$.

A cylindrical capsule (Puddle Jumper) with a diameter $d = 4\,\mathrm{m}$, a length $l = 10\,\mathrm{m}$ and with a watertight partition in the middle of its length is submerged below the ocean surface and falls to the seabed at a speed of $v = 20\,\mathrm{ft\cdot min^{-1}}$. At the depth $h = 1~200\,\mathrm{ft}$, the glass on the front base breaks and the corresponding half of the capsule is filled with water. At what speed will it fall now? How long will it take for the capsule to sink to the bottom at the depth $H = 3~000\,\mathrm{ft}$? Assume that the walls of the capsule are very thin against its dimensions.

Assume a charged parallel-plate capacitor in a horizontal position. One of its plates is fixed and the other levitates directly below it in an equilibrium position. The lower plate is not mechanically fixed in its place. What is the capacitance of the capacitor depending on the voltage applied? Is the capacitor mechanically stable?

You may have noticed that not all volcanos on Earth have the same “universal” shape – they differ from each other. For example, compare the photos of the Hawaiian volcano Mauna Loa and the Italian Vesuvio. They differ not only in the steepness of their walls but also in the style of eruptions. Both of these properties are related to the viscosity of magma. Discuss the effect of the viscosity of magma on the style and dangerousness of eruptions. Is is related to the geographic location of the volcanoes?

Measure the dependency of the time it takes for water to start boiling on its volume. Repeat the measurement several times for at least five different volumes. Pay attention to the consistency of the external conditions, especially the criterion you use for assessing when the water starts boiling and the initial temperature of the water, vessel and stove. Try to explain the resulting relation.

1. Determine the energy gain of the following reactions and the kinetic energy of their products \begin{align*} {}^{2}\mathrm{D} + {}^{3}\mathrm{T} &\rightarrow {}^{4}\mathrm{He} + \mathrm{n} \,,\\ {}^{2}\mathrm{D} + {}^{2}\mathrm{D} &\rightarrow {}^{3}\mathrm{T} + \mathrm{p} \,,\\ {}^{2}\mathrm{D} + {}^{2}\mathrm{D} &\rightarrow {}^{3}\mathrm{He} + \mathrm{n} \,,\\ {}^{3}\mathrm{T} + {}^{3}\mathrm{T} &\rightarrow {}^{4}\mathrm{He} + 2\mathrm{n} \,,\\ {}^{3}\mathrm{He} + {}^{3}\mathrm{He} &\rightarrow {}^{4}\mathrm{He} + 2\mathrm{p} \,,\\ {}^{3}\mathrm{T} + {}^{3}\mathrm{He} &\rightarrow {}^{4}\mathrm{He} + \mathrm{n} + \mathrm{p} \,,\\ {}^{3}\mathrm{T} + {}^{3}\mathrm{He} &\rightarrow {}^{4}\mathrm{He} + {}^{2}\mathrm{D} \,,\\ \mathrm{p} + {}^{11}\mathrm{B} &\rightarrow 3\;{}^{4}\mathrm{He} \,,\\ {}^{2}\mathrm{D} + {}^{3}\mathrm{He} &\rightarrow {}^{4}\mathrm{He} + \mathrm{p}\,. \end{align*}
2. By using the graph of fusion reaction rate (sometimes called volume rate) as a function of temperature in the Serial study text, derive the Lawson criterion for the inertial-confinement-fusion time for a temperature of your choosing, while considering the following reactions:
   1. deuterium - deuterium,
   2. proton - boron,
   3. deuterium - helium-3.
   Determine the product of the size of a fuel pellet, and the density of a compressed fuel for each case. Are there any advantages of these reactions compared to the traditional DT fusion?
3. What form would the Lawson criterion take for the non-Maxwellian velocity distribution, considering the case with the following kinetic energy of a particle
   1. $E_{\mathrm{k}} = k_{\mathrm{B}} T^\alpha$,
   2. $E_{\mathrm{k}} = a T^3 + b T^2 + c T$.
   Could such a fusion be even possible? If so, what (the fuel) should drive the fusion reaction, what is the ideal size of the fuel pellet and what density should it be compressed to?