Consider a planet with the same total mass and radius as the Earth. How much uranium $^{238}\mathrm{U}$ would it have to contain, so that its surface temperature is $15 \mathrm{\C }$, assuming it is not lit by any nearby star.

(9 points)6. Series 36. Year - 5. gadolinium sphere

What is the smallest amount of gadolinium $148$ needed to put together to cause local melting from the heat generated by its nuclear decay? Assume that only $\alpha $ decays take place and the material is at room temperature in the air.

Karel was thinking about elements, but Matěj Rz. changed that.

(7 points)6. Series 35. Year - 4. short half-life

What is the probability that three-quarters of the initial one mole of atoms decay during one half-life? Commonly it happens only after two half-lives. What could cause such a situation?

How far from the surface of the target (suppose it is made of carbon and the laser has wavelength of $351 \mathrm{nm}$) is critical surface situated and how far does two-plasmon decay occur, if the characteristic length of plasma^{1)}

^{1)}

The density of plasma $n_e$ is typically expressed as a funciton $n_e = f\(\frac {x}{x_c}\)$, where $x$ is the distance from the target and $x_c$ is so called characteristic length of plasma, which represents scale parameter for the distance from the target.))is~$50 \mathrm{\micro m}$? Next assume

that the density of the plasma decreases exponentially with distance from the target,

that the density of the plasma decreases linearly with distance from the target.

What energy must electorns have in order to go through the critical surface to the real surface of the target? To calculate the distance electron travels in carbon plasma use an empirical relationship $R = 0{,}933~4 E^{1{,}756~7}$, where $E$ has units of \jd {MeV} and $R$ has units of \jd {g.cm^{-2}}.

What is the distance that an electron has to travel in the electric field of the plasma wave in order to reach the energies determined in second exercise?

Which wavelengths of scattered light are present in the case of stimulated Raman scaterring for laser with wavelength of $351 \mathrm{nm}$?

Determine the reach of helium nuclei in central hot spot (using the figure ).

What energy must be released in the fusion reactions in order for the fusion to spread to the closest layer of the pellet? How thick is the layer?

Estimate the most probable amount of energy transferred from helium nucleus to deuterium. How many collisions on average does the helium nucleus undergo in the central hot spot before it stops?

By using the graph of fusion reaction rate (sometimes called volume rate) as a function of temperature in the Serial study text, derive the Lawson criterion for the inertial-confinement-fusion time for a temperature of your choosing, while considering the following reactions:

deuterium - deuterium,

proton - boron,

deuterium - helium-3.

Determine the product of the size of a fuel pellet, and the density of a compressed fuel for each case. Are there any advantages of these reactions compared to the traditional DT fusion?

What form would the Lawson criterion take for the non-Maxwellian velocity distribution, considering the case with the following kinetic energy of a particle

$E\_k = k\_B T^\alpha $,

$E\_k = a T^3 + b T^2 + c T$.

Could such a fusion be even possible? If so, what (the fuel) should drive the fusion reaction, what is the ideal size of the fuel pellet and what density should it be compressed to?

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.

(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 $t$.

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.