# Problems in 2nd round

A booklet with all the problems and the actual chapter of the series (in Czech):

## Problem II . 1 … *beach date* (3 points)

Imagine you are going on a date with your girlfriend/boyfriend and you end up watching the sunset on the beach. The sun above the sea horizon looks very romantic, so to prolong this special moment, you decide to use a forklift to lift you up. The forks of the forklift move up with such speed that you can see the sun touching the horizon at any moment. Determine the speed of the forks.

## Problem II . 2 … *ultra high temperature superconducticvity* (3 points)

Many types of materials, mostly metals,
have increasing dependence of resistivity on temperature.
However, there are semiconductors or graphite which show a decreasing dependence.
And you have also probably heard about superconductivity,
the natural phenomenon when a cooled material shows almost no electrical resistance
and becomes a perfect conductor. Our current state of knowledge says
that the temperature of a superconductor must be well below room temperature,
but let's assume that the equation defining the resistance is *R* = *R*_{0} ( 1 + *αΔt* ) , where *R*_{0} is the resistance at room temperature, *α* is the temperature coefficient of resistance and *Δt* is the temperature difference with respect to room temperature,
and the equation holds for any temperature.
Using this equation and coefficients *α*_{C} = -0.5 · 10^{-3} K^{ − 1} for graphite and *α*_{Si} = -75 · 10^{-3} K^{ − 1} for silicon,
we obtain zero resistance for high temperatures.
Determine these two temperatures and explain why the superconducting
phenomenon does not work this way, i.e. neither carbon nor silicon
are superconductors at high temperatures.

## Problem II . 3 … *looping* (6 points)

We have a plane inclined at an angle *α* which is smoothly connected to a loop of radius *R*. What is the minimal initial height *h* where we have to place
a ball of radius *r* (comparable to *R*, but smaller), so that
the ball will roll through the loop? The ball must be always in
contact with the plane or the loop and we assume that the ball does not slip.

## Problem II . 4 … *tiny ball* (6 points)

Imagine the motion of a homogeneous ball: it starts with translation (without rotation) and gradually transitions to rolling (without slipping). Determine the time of the transition from pure translation to rolling without slipping. Consider different radius, mass, initial speed and coefficient of friction of the ball.

## Problem II . 5 … *tea container problem* (7 points)

We have a tea container with a tap near the bottom and an airtight lid (maybe you know these from your school canteen). Determine the volume of tea we can pour from the tap before we have to open the valve to equalize the pressure in the container.

## Problem II . P … *an effective machine* (9 points)

Guns can be considered to be heat engines. Calculate the efficiency of a gun, say a pistol or a rifle.

## Problem II . Exp … *one full fat milk, please* (12 points)

Milk with higher fat content should be „whiter“ – more light is scattered and less is transmitted. Conduct a measurement of the fat content of milk with the help of a color scale (contact us at fykos@fykos.czto get the pdf with the scale -- you have to print it yourself). The difference in whiteness is most apparent when you add a dye to each glass of milk. You can use e.g. black ink or any other dye, but with different colors you have to create your own color scale which you have to add to your solution. Use different types of milk and mixture of milk with water. Discuss the reliability of this method of measurement.