Problem Statement of Series 6, Year 35

Lex Luthor kidnapped Lois Lane and threw her off the plane at altitude $h$. Superman follows her and catches her at some unknown altitude. Suppose that the maximum acceleration Lois can survive is $10\,g$. What is the lowest altitude at which can Superman catch Lois to save her?

Imagine there is a comet that threatens the Earth once a generation, just when it is in the perihelion. What is the distance between the Earth and such a comet when the comet is in its aphelion? What is the length of the semi-major axis and the orbital eccentricity of the comet's trajectory? Do not consider gravitational influences other than from the Sun, and assume that one generation is $g = 20\,\mathrm{years}$.

Imagine we create a small soap bubble with a bubble blower. How fast does it fall to the ground? The bubble has an outer radius $R$ and an areal density $s$.

What is the probability that three-quarters of the initial one mole of atoms decay during one half-life? Commonly it happens only after two half-lives. What could cause such a situation?

We have constructed a small rocket weighing $m_0 = 3\,\mathrm{kg}$, from which $70\%$ is fuel. The exhaust velocity is $u = 200\,\mathrm{m\cdot s^{-1}}$ and the initial flow of the exhaust fumes is $R = 0.1\,\mathrm{kg\cdot s^{-1}}$ and both these values remain constant during the flight. The rocket is equipped with stabilization elements, so it does not deviate from its desired trajectory. It has been launched from the rest position vertically. Assume that the friction force of the air is proportional to the velocity $F_{\mathrm{o}} = -bv$, where $b = 0.05\,\mathrm{kg\cdot s^{-1}}$, $v$ is the velocity of the rocket and the sign minus means that the force exerts against the direction of the motion. What height above the ground level does the rocket fly in time $T = 25\,\mathrm{s}$ from the engine startup?

Is it convenient to hide from the rain in the woods? Create a suitable model describing this issue. Consider, for example, foliage density, and the intensity and duration of the rain. Describe how long after the rain starts, the drops from the leaves start to fall to the ground, as well as how long after the rain ends, it stops raining in the woods, and so on.

Create a device that can measure one minute as accurately as possible. You are not allowed to use any time measuring devices for calibration when designing your own. After you finalize your device, use a stopwatch to determine "your minute" accuracy.

Bonus Measure ten minutes.

1. How big must an aperture in a spatial filter be if we created it from a lens with a diameter of $40\,\mathrm{cm}$ and its focal length is $4\,\mathrm{m}$? Our Gaussian laser beam has an input diameter $30\,\mathrm{cm}$ and a wavelength $1~053\,\mathrm{nm}$. The radius of the focus (parameter $\sigma$) of the Gaussian beam can be obtained using \begin{equation*} r = \frac{2}{\pi}\lambda \frac{f}{D} \end{equation*} where $D$ is the diameter of the beam, $f$ is the focal length of the lens and $\lambda$ is the wavelength of the laser.

2. The laser beam is focused on a surface of a nuclear fuel pellet of a $1\,\mathrm{mm}$ diameter. What energy should it have in order for the intensity in its focus to reach $10^{14}\,\mathrm{W\cdot cm^{-2}}$? The radius of the focus is $25\,\mathrm{\upmu{}m}$ and a pulse lasts $10\,\mathrm{ns}$. How many beams do we need to equally cover the surface of a pellet? What is their total energy?

   What energy must the laser beam have if it is not focused on a surface of a nuclear fuel pellet, but the beam diameter matches exactly the diameter of the pellet and the density is its focus reaches $10^{14}\,\mathrm{W\cdot cm^{-2}}$? Assume that we have one such beam and it shines homogenously on the pellet “from all directions”.