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## nuclear physics

### (7 points)6. Series 33. Year - 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.

### (8 points)3. Series 31. Year - 5. decay here, decay there

We have $A_0$ particles which decay into $B$ particles with decay constant $\lambda \_A$. $B$ particles decay into $A$ particles with decay constant $\lambda \_B$. The number of $B$ particles at the beginning is $B_0$. Find a ratio of the numbers of particles $A$ and $B$ as a function of $t$.

### (12 points)1. Series 30. Year - E. Pechschnitte

Does bread always falls on the side that has the spread on it? Explore this Murphy's law experimentally with emphasis on statistics! Does it depend on the dimensions of the slice, or the composition and the thickness of the spread? Try to explain the experimental results with a theory. Use a sandwich bread.

Terka má stůl ve špatné výšce.

### (2 points)6. Series 29. Year - 1. It's about what's inside of us

In the year 2015, a Nobel prize for Physics was given for an experimental confirmation of the oscillation of neutrinos. You have probably already heard about neutrinos and maybe you know that they interact with matter very weakly so they can pass without any deceleration through Earth and similar large objects. Try to find out, using available literature and Internet sources, how many neutrinos are at any instant moment in an average person. Don't forget to reference the sources.

### (5 points)6. Series 29. Year - 5. Particle race

Two particles, an electron with mass $m_{e}=9,1\cdot 10^{-31}\;\mathrm{kg}$ and charge $-e=-1,6\cdot 10^{-19}C$ and an alpha particle with mass $m_{He}=6,6\cdot 10^{-27}\;\mathrm{kg}$ and charge 2$e$, are following a circular trajectory in the $xy$ plane in a homogeneous magnetic field $\textbf{B}=(0,0,B_{0})$, $B_{0}=5\cdot 10^{-5}T$. The radius of the orbit of the electron is $r_{e}=2\;\mathrm{cm}$ and the radius of the orbit of the alpha particle is $r_{He}=200\;\mathrm{m}$. Suddenly, a small homogeneous electric field $\textbf{E}=(0,0,E_{0})$, $E_{0}=5\cdot 10^{-5}V\cdot \;\mathrm{m}^{-1}$ is introduced. Determine the length of trajectories of these particles during in the time $t=1\;\mathrm{s}$ after the electric field comes into action. Assume that the particles are far enough from each other and that they don't emit any radiation.

### (4 points)1. Series 29. Year - 5. chernobyl

If someone would eat 5 µg of the isotope of cesium ^{137}Cs, how long would it take for them to have only0,04 % of the original amount of this isotope? Assume that cesium ^{137}Cs has a half-life of 30,42 let and a biological half-life (the time it takes for half of the original amount of the material to leave the body) is approximately 5 days. Determine also, how many of the particles will have decayed in the body up till then.

Kiki was hungry after her toxikology exam.

### (4 points)2. Series 28. Year - 3. impatient core

The core of Bismuth ^{209}Bi sits impatiently at peace on the same spot. Suddenly it can't hold it any lonher and it falls apart. A thalium core ^{205}Tl remains and from it one can see an$αparticle$ shoot away. What is the speed of the $αparticle$, if the energy released during the decay becomes its kinetic energy? What is the velocity of the $αparticle$ in reality? Compare the results. The rest masses of the atoms are $M=m_{^{209}Bi}=208,980399u$, $M′=m_{^{205}Tl}=204,974428u$, $m=m_{^{4}He}=4,002602u$. Don't forget to check if one should use relativistic relations.

Jakub was sad that Bismuth must wait whole eons to decay.

### (2 points)1. Series 28. Year - 1. consumption of antimatter

How much antimatter would we need to generate enough electricity for the Czech Republic for a year? We have enough matter and we assume that there would be no losses.

Karel was watching Angels and Demons from Dan Brown.

### (2 points)5. Series 27. Year - 2. uranium star

Imagine that no thermonuclear fusion occurs in stars and instead they run on nuclear fission. Estimate how long such a star would be able to shine if at the beginning of its life cycle it is composed of uranium 235, its mass and luminosity are both aproximately constant and are equal to the current values of the sun.

Mirek was reading through his new textbooks.

### (5 points)6. Series 26. Year - P. turn it of I, can't!

How many people per second can be killed by a nuclear reactor without any protective walls?