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(3 points)5. Series 32. Year - 2. warm reachability into the ball

Imagine you have subcooled homogeneous metal ball that you have just taken out of a freezer set to very low temperature. You want to find out how fast the ball temperature will increase if you put it in a warm room. It would be a university-level problem. Because of that, we made it easier for you. We ask about how deep into the ball will the „warm area“ reach. You can estimate it using dimensional analysis. We know relevant parameters of the ball - its density $\left [ \rho \right ] = \jd {kg.m^{-3}}$, specific heat capacity $\left [c\right ] = \jd {J.kg^{-1}.K^{-1}}$, thermal conductivity of the ball $\left [ \lambda \right ] = \jd {W.m^{-1}.K^{-1}}$ and we are interested in dependence on time $\left [t\right ] = \jd {s}$.

Karel inspired himself by a problem from Eötvös Competition.

(10 points)6. Series 31. Year - S. Matrices and populations

Mirek and Lukáš fill matrices with atto-foxes.

(9 points)5. Series 31. Year - P. floating mercury

Try to invent as much „physics tricks“ as possible thanks to which mercury would float on the liquid water for at least a limited time. The more permanent solution you find, the better.

(10 points)5. Series 31. Year - S. Differential equations are growing well

Mirek and Lukáš have already grown their algebra, now they have different seeds.

(8 points)3. Series 31. Year - P. folded paper

Everyone has certainly heard and surely tried it: „Sheet of paper can not be folded in a half more than seven times.“ Is it really true? Find boundary conditions.

Kuba was bored and folded a paper.

(10 points)3. Series 31. Year - S. going for a walk with integrals

We are sorry, this task is not yet translated…

Mirek and Lukáš random-walk to school.

(3 points)2. Series 31. Year - 1. Tooth Fairy

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?

Karel's mind wandered to the Discworld

(12 points)2. Series 31. Year - E. grainy

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

Michal almost slumped.

(10 points)2. Series 31. Year - S. derivatives and Monte Carlo integration

 

  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.

Mirek and Lukáš read the Python documentation.

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