3th Round of 23th year

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1. ... indistinguishable people on Earth( points)

What is the maximum distance for two people to be indistinguishable to others whenever they are visible? Do not forget that people are point light sources at 2 meters height and the Earth is an ideal sphere.

2. ... noisy people in front of my house( points)

Consider the double doors of a building, between which is in the middle even in closed position small gap, where air can flow. Each wing has a spring that returns it to the starting (closed) position. Wing is displaced and then released. What will happen be the other wing?

If you are not quite sure, try it first (e.g. Ke Karlovu 5, Faculty of Science, Prague, …).

3. ... Hospodine, pomiluj ny! (medieval Czech song)( points)

How grows the volume (define yourself) of choir with the number of its members? What is the conclusion? Members of the choir can be approximated as point sound sources of the same amplitude and frequency, but shifted by a random phase. All point singers are in one place.

4. ... Mr. Wurfl, but on Moon(0 points)


Mr. Broucek (from children fairy tale) was followed up by Etherea, which is closest to the mass point. After having ordered pork and cabbage, Mr. Broucek tried to deprive her by imprisonement between two fixed table tennis bats. The distance between bats was $l$ and both were turned at an angle, and Etherea was jumping between them as table tennis ball so that it always re-bounced the same height. To make torturing more realistic, he put a net of the height of $h$ at the centre. Mr. Broucek is a clever wag, so he wanted to (as in ping-pong) on each ball (=Etherea) crossing that the ball would touch the ground Calculate the frequency with which, depending on all sorts of parameters (rotation bats, initial ball speed, angle, …) Etherea flies, and when this frequency is highest. Assume that the motion happens in plane and the reflections from obstacles (from the Moon or from the bat) is just the opposite as the speed, all movements happen take place in vacuum.

P. ... the rotten fibers(0 points)


Investigate the double knot, which connects two fibers of radius $r$ and static friction coefficient  $f$. What is the minimum force, we have to pull the fibers to slide the fibers through the knot? Insert typical values and make sure, that the fiber will not break.

E. ... steps( points)

Build a long domino line and measure the speed of falling wave. For known dimensions of domino bricks and variable distance between them. Will the speed stabilise?

S. ... game with shadows( points)

* In serial we discussed discrete distribution of point sources on a line and its projection onto another line. Now assume, that the points are distributed on a plane and the screen is the plane parallel to it. Describe the distribution of intensity on the screen in case, that light sources:

  • lie at one line with equal spacing $d$.
  • are as two parallel lines, where the distance between two of them is $d$.
  • lie in the corners of rectangle network, where rectangles have sides $a$, $b$.
  • Assume following situation: before the screen, presented by a plane $xy$ is a disc of radius $R$, parallel to the plane. The screen is illuminated from the side of the disc by a beam of parallel beams perpendicular to the screen $xy$. Explain, why this situation can be described using point light sources located continuously in the same plane as disc is located, excluding the disc itself. Find the intensity distribution in plane $xy$ as function of  $x$ and $y$ (you can supply just integral, not it solution) and show, that point opposite the center of disc shows strange behaviour, which we would not expect from ray optics
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