Series 2, Year 31

How big would the storage facilities of the Tooth Fairy need to be, to store all of the primary teeth of all of the children of the world? Or, in other words, how rapidly would they need to grow? How long would it take for the whole Earth supply of phosphorous to be contained in those storage facilities?

The solar constant, or more accurately the solar irradiance, is the influx of energy coming from the Sun at the distance where Earth is. It technically doesn't have a constant value, but let's suppose it is approximately $P = 1{,}370\,\mathrm{W\cdot m^{-2}}$. Also, suppose that Earth's orbit is circular and its axis of rotation is tilted with respect to the normal of the orbital plane by $23.5\dg $. What would be the maximum power captured by a solar panel of area $S= 1\,\mathrm{m^2}$ at the summer and winter solstice, if the panel lies flat on the ground in Prague (latitude $50\dg $ N)? Ignore the effects of any obstructions or the atmosphere.

What fraction of a spherical planet's surface cannot be seen from the stationary orbit above the planet? (A stationary orbit is one where the satellite stays fixed above a certain point on the planet.) The density of the planet is $\rho $ and its rotation period is $T$.

Imagine we have a thing (e.g. a nuclear waste container) and we want to get rid of it. We transfer the object to a circular orbit around the Sun at the same distance as Earth, but far enough from Earth to ignore its gravitational influence. Which of these methods of the objects disposal would require the least amount of energy and thus would be the most efficient?

• Throw it into the Sun. Getting it to the solar surface would be sufficient to burn the object.
• Transfer it to a circular orbit in the Asteroid belt (located between the orbits of Mars and Jupiter).
• Get it out of the Solar System completely.

A worker brought a bag of marbles to a skyscraper construction, to show off in front of his colleagues. But, what an unlucky accident – the marbles pour out and start falling through the scaffolding towards the ground. The scaffolding consists of different levels separated by height $h$. The floor of each level is made out of identical metal grid in which the holes constitute $k  \%$ out of the whole grid area. Consider a simplified model of marbles falling through the scaffolding, in which if marble lands in the hole of the grid it goes through unobstructed and if it lands on the solid part of the grid its velocity drops to $0$ and starts to fall down again immediately (i.e. the size of the marbles is insignificant with respect to the size of the holes in the scaffolding and the marbles don't bounce upon landing, instead they stop and immediately roll down into a hole and continue with their fall). Ignore any potential collisions between marbles themselves. If we assume the marbles pour out of the bag with a constant mass flow of $Q$, what is the force on each level of the scaffolding, when the situation comes to a steady state?

What properties does the $118^{\rm th}$ element in the Periodic table have? Alternatively, what sort of properties would it have, had it been stable? Discuss at least three physical qualities.

Measure the angle of repose for at least two granular materials commonly found in the kitchen (e.g flour, sugar, salt, etc.).

1. Plot the error as a function of step size for the method \[\begin{equation*} f'(x)\approx \frac {-f(x+2h)+f(x-2h)+8f(x+h)-8f(x-h)}{12h} \end {equation*}\] derived using Richardson extrapolation. What are the optimal step size and minimum error? Compare with forward and central differences. Use $\exp (\sin (x))$ at $x=1$ as the function you are differentiating.
   Bonus: Use error estimate to determine the theoretical optimal step size.
2. There is a file with experimentally determined $t$, $x$ and $y$ coordinates of a point mass on the website. Using numerical differentiation, find the time dependence of components of speed and acceleration and plot both functions. What is the most likely physical process behind this movement? Choose your own numerical method but justify your choice.
   Bonus: Is there a better method for obtaining velocity and acceleration, then direct application of numerical differentiation?
3. We have an integral $\int _0^{\pi } \sin ^2 x \d x$.
   1. Find the value of the integral from a geometrical construction using Pythagoras theorem.
   2. Find the value of the integral using a Monte Carlo simulation. Determine the standard deviation.
      Bonus: Solve the Buffon's needle problem (an estimate of the value of $\pi $) using MC simulation.
4. Find the formula for the volume of a six-dimensional sphere using Monte Carlo method.
   Hint: You can use the Pythagoras theorem to measure distances even in higher dimensions.