Assignment of Series 6 of Year 33

What energy (in electronvolts) will a proton gain during a fall from an infinite distance to the surface of Earth?

The water level in a bathtub is at a height $15.0\,\mathrm{cm}$. The plug is a conical frustum which perfectly fits into the hole at the base of the bathtub. The radii of its bases are $16.0\,\mathrm{mm}$ and $15.0\,\mathrm{mm}$ and the mass of the plug is $11.0\,\mathrm{g}$. What force does the bottom of the bath exert on the plug? Assume that the drain pipe below contains air at atmospheric pressure.

Out of joy over the end of an exam period, Danka's hair count began to increase at a constant rate. Later, she noticed that she lost one hair, which scared her. The more hair she lost, the more she felt stressed, which increased her hair loss rate. More precisely, the rate of hair loss is proportional to the amount of already lost hair. The rate of growth of new hair remains the same. Again, we are interested in finding out when all her hair falls out.

The magic field of Discworld is so strong that the speed of light no longer has its common meaning. This applies only close to the surface, where the refractive index of the magic field has magnitude $n_0 = 2.00\cdot 10^{6}$. The refractive index decreases with height $h$ as $n(h) = n_0\mathrm{e}^{-kh}$, where $k = 1.00\cdot 10^{-7}\,\mathrm{m^{-1}}$. Calculate the optimal angle (with respect to the vertical direction) under which a light signal should be emitted from one end of the Discworld to reach the opposite end in the shortest possible time. The diameter of the Discworld is $d = 15~000\,\mathrm{km}$ and the speed of light in vacuum is $c = 3.00\cdot 10^{8}\,\mathrm{m\cdot s^{-1}}$.

As you have probably heard, planets and any other bodies in a central gravitational field move along conic sections (in the case of the Solar system, ellipses with small eccentricity). Find out how trajectories would look in a universe where the gravitational force was proportional to the multiplicative inverse of distance raised to the third power (instead of the second power).

Measure the viscosity (in $Pa.s$) of two different oils using Stokes' method.