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(9 points)2. Series 34. Year - P. costly ice hockey
Estimate how much the complete glaciation of an ice hockey rink costs.
Danka doesn't like ice hockey, but she likes figure skating.
(10 points)6. Series 33. Year - S.
We are sorry. This type of task is not translated to English.
(10 points)4. Series 33. Year - P. climate changes feat. airplanes
Travel by airplane affects the atmosphere not only by well-known carbon emissions. Discuss how the aircraft industry affects warming of the atmosphere of Earth.
(13 points)2. Series 33. Year - E. I need a\protect \unhbox \voidb@x \penalty \@M \ {}hug
Measure your volume using several different methods.
Matej was having a bath.
(10 points)2. Series 33. Year - P. Earth fired up
Estimate about how much would a content of $\ce {CO2}$ in the atmosphere rise, if all flora on Earth was burnt down?
Karel is pyromaniac.
(10 points)2. Series 33. Year - S.
We are sorry. This type of task is not translated to English.
(10 points)1. Series 33. Year - P. planet destroyer
How small could a weapon capable of destroying a planet be? We are interested in the smallest and the lightest such weapons. The process should be reasonably fast, at least shorter than a human lifetime, and the faster it is, the better.
Karel watches sci-fi too much, this time the intro of Men in Black II.
(10 points)1. Series 33. Year - S. slow start-up
We are sorry. This type of task is not translated to English.
Karel wants to have the longest problem assignment.
(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)4. Series 32. Year - S.
- Show that in an arbitrary central-force field, i.e. a force field where the potential only depends on distance (not on angular position), a particle will always move in a plane. Instructions:: Set up Lagrangian equations of the second kind for this situation using appropriate generalized coordinates.. Then, set the coordinate $\theta = \pi /2$ and initial velocity in the direction of this coordinate equal to zero. Think about and explain why this choice of coordinates does not cause a loss of generality.
- Set up the Lagrangian for a mass point moving in a plane in a central-force field. Find all the integrals of motion for this Lagrangian and use them to find the first orded differential equation for the variable $r$.
- Think about how to find the angular distance between two points on a sphere, given their spherical coordinates. Check your solution on the stars Betelgeuse and Sirius. Hint:: This problem can be easily solved even without the knowledge of spherical trigonometry.