Problem Statement of Series 1, Year 25

Pepička has bought a light bulb, two switches and a wire. Help: her to design a circuit such that if you change the state of any of the switches the light bulb will also change its state (from on to off or reverse). After you find the solution try to generalize it to any number of switches.

A man wants to swim across a river which flows at a speed of 2 km/h. He is able to swim at a speed of 1 km/h. What is the optimal trajectory and direction he should take so that his trip is the least exhausting? Where and at what time will he reach the other bank? How would the situation change if his aim was the shortest possible trajectory? The width of the river is $d=10\;\mathrm{m}$.

What is the temperature of the air leaving a bicycle pump when we want to inflate the tube to a pressure of 3 atm? Assume that the air entering the pump has temperature 20°C.

A small conductive ball of negligible size is bouncing between two charged plates of opposite polarities. What is the frequency of the resulting periodic motion of the ball? Voltage between the plates is $U$. When the ball touches a charged plate it charges to a charge of magnitude $Q$ whose polarity is the same as polarity of the plate. The ratio of kinetic energies of the ball before and after an impact is $k$.

Bonus: Does the output power of this resistor correspond to the energy losses during impacts?

When shooting from a gun the resulting backward impact causes the bullet to shoot out in a different direction than was originally meant. What is the angle difference between these two directions? Assume that hand muscles compensate for any influence that gravity can have. Also assume that the gun is rotating only around some point in a wrist. You know the moment of inertia of the gun-hand system (with respect to the previously mentioned point) as well as the mass of the projectile, its speed when leaving the gun and the dimensions described in the picture. After you solve this problem qualitatively make a quantitative guess of the necessary quantities and find a numerical value for the angle.

Imagine that the Earth is not a sphere but a cube. Would it be able sustain this shape? If so for how long and on what parameters would this time depend on? What about the life on such a planet? What gravitational force would you feel while walking around this planet?

Find out experimentally at least three different physical properties of gummy bears candy. You should explore even how do these properties depend on the candy's color. Some possible properties you can measure are melting temperature, Young's modulus, maximal tension, absorbency (change in volume or mass after being in water for certain amount of time), density, conductivity, index of refraction, solubility, dependence of any of the preceding properties on temperature or anything else you can think of.

• Some stars are considered circumpolar. Does it mean that they can be seen the whole year? What stars are visible throughout the whole year in Czech Republic? What coordinate describes circumpolar stars? What is the situation in Czech Republic, at the pole and at the equator? We recommend that you download the program <a href=http://www.stellarium.org/>Stellarium</a> (GNU GPL license) where you can enter your location and look at these different cases.

• Compare the absolute magnitudes of Alpha Lyrae (7.79 pc far, apparent magnitude 0.01 mag) and Betelgeuze (Alpha Ori, approximately 200 pc far, apparent magnitude 0.42 mag). How would we see these stars if they exchanged their distances from the Earth? Discuss visibilities.

• Transformations and some more transformations. Find the transformation between galactic and equatorial coordinates or the second kind. Do not worry if the resulting equations do not look exactly like those found in literature.

• Janap likes to get lost once in a while. It is not always desired but it happens anyway. This time however she brought a theodolite – a magic box that can measure how high above a horizon a star is. She found out that the star Arcturus was located at 23.20 gon at 18:46:30 and the star Capella at 13.60 gon at 19:18:30. The scale of theodolite was in grad (gon), where 100 gon = 90°. What was her location?