# 6. Series 33. Year

* Post deadline: 27th April 2020*

* Upload deadline: 28th April 2020 11:59:59 PM (local time in Czech Republic)*

### (3 points)1. gravitational accelerator

What energy (in electronvolts) will a proton gain by a fall from infinite distance to the surface of Earth?

Kačka saw a vertical accelerator.

### (3 points)2. under pressure

The water level in bath reaches height $15{,} \mathrm{cm}$. The plug has a shape of a conical frustum which perfectly fits the hole in the base of the bath. Its radii of bases are $16,0 \mathrm{mm}$ and $15,0 \mathrm{mm}$ and its mass is $11,0 \mathrm{g}$. What force does the bath bottom act on the plug? Assume that the drain pipe below contains air of atmospheric pressure.

Jindra felt pressure to think simple problems up.

### (5 points)3. hung

What weight can be hung on the end of a coat hanger without turning it over? The hanger is made of a hook from very light wire, which is attached to the centre of the straight wooden rod, which length is $l = 30 \mathrm{cm}$ and weight $m=200 \mathrm{g}$. The hook has the shape or circular arc with radius $r=2,5 \mathrm{cm}$ and angular spread $\theta =240 \mathrm{\dg }$. The distance between the centre of the arc and the rod is $h=5 \mathrm{cm}$. Neglect every friction.

Dodo is seeking for a scarce.

### (7 points)4. frightened hair

Thanks to joy from the end of an exam period, Danka's hair count begun to increase by a constant rate. Later she noticed that she lost one hair, which scared her. The more hair she lost the more she feels stressed, which increases hair loss rate. More precisely, the rate of hair loss is proportional to the number of already lost hair. The rate of new hair growth remains the same. Again, we are interested, when will her last hair fall out.

Jáchym desired to calculate this for a long time.

### (10 points)5. golden nectar

Magic field of Discworld is so strong that the speed of light does no longer have its common meaning. This applies only close to the surface, where the refractive index of the magic field has magnitude $n_0 = 2,00 \cdot 10^{6}$. The refractive index decreases with height $h$ as $n(h) = n_0\eu ^{-kh}$, where $k = 1,00 \cdot 10^{-7} \mathrm{m^{-1}}$. Calculate the optimal angle (measured from vertical direction) under which a light signal shall be emitted from one end of the Discworld to reach the opposite end in the shortest time possible. Diameter of the Discworld is $d = 15\;000 \mathrm{km}$ and speed of light in vacuum is $c = 3,00 \cdot 10^{8} \mathrm{m\cdot s^{-1}}$.

Mirek waited for the light from the traffic light to reach him.

### (10 points)P. 4D universe

As you have probably heard, planets and any other bodies in the central gravitational field move on conic sections (in case of the Solar system ellipses with small eccentricity). Find out, how would trajectories look like in a universe, where gravitational force was proportional to multiplicative inverse of distance raised to the third power (instead of second power).

Matěj likes higher dimensions.

### (12 points)E. viscosity

Measure viscosity (in $\textrm{Pa}\cdot\textrm{s}$) of two different oils using Stokes' method.

Jáchym stole Jirka's idea to steal this problem from labs.

### (10 points)S.

*We are sorry. This type of task is not translated to English.*