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In 2000 a new nuclear waste repository was built. The first waste arrived and the government decided that every year the amount of newly delivered waste must be reduced by five percent. Assume for simplicity that the half-life of nuclear waste is 100 years (in reality it is much longer). Find out what year are the people in nearby villages going to receive the highest amount of radiation.

Luke has bought a Uranium atom and started to take away its electrons just for fun. After taking the $n-th$ one he surprisingly discovered that the mass of the atom had increased. What has inflicted this phenomena? Determine the value of $n$.

After Luke got bored with the ionization, he ordered more of the Uranium. The two perfect semi-spheres were delivered. Each semi-sphere has the mass $m$ ($m_{k}⁄2&ltm&ltm_{k}$, where $m_{k}$ is the critical mass). Luke has placed their flat sides opposite each other which started to bring them nearer. Calculate the distance $d$ at which the experiment was interrupted by the ignition of the chain reaction.

• Derive Taylor expansion of exponential and for $x=1$ graphically show sequence of partial sums of series \sum_{$k=1}^{∞}1⁄k!$ with series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$.

Using the same method compare series { ( 1 − 1 ⁄ $n)^{n}}_{n=1,2,\ldots}$ and series of partial sums of series \sum_{$k=1}^{∞}x^{k}⁄k!$, therefore series {\sum_{$k=1}^{n}x^{k}⁄k!}_{n=1,2,\ldots}$, now for $x=-1$.

• The second task is to find concentration of electrons and positrons on temperature with total charge $Q=0$ in empty and closed cavity (you can choose value of $Q.)$ Further calculate dependence of ration of internal energy $U_{e}$ of electrons and positrons to the total internal energy of the system $U$ (e.g. the sum of energy of electromagnetic radiation and particles) on temperature and find value of temperature related to some prominent temperature and ratios (e.g. 3 ⁄ 4, 1 ⁄ 2, 1 ⁄ 4, …; can this ratio be of all values?).

Put your results into a graph – you can try also in 3-dimensions.

To get the calculation simplified, it could help to take some unit-less entity (e.g. $βE_{0}$ instead of $β$ etc.).

• Solve the system of differential equations for $M(r)$ and $ρ(r)$ in model of white dwarf for several well chosen values of $ρ(0)$ and for every value find the value which it get close $M(r)$ at

$r→∞$. This is probably equal to the mass of the whole star. Try to find the dependence of total weight on $ρ(0)$ and find its upper limit. Compare the result with the upper limit of mass for white dwarf (you will find it in literature or internet). Assume, that the star consists from helium.

Calculate time dependence of wave function of particle, which is located in potential $V(x)=\frac{1}{2}kx$ and which is at time $τ=0$ described by wave function

$ψ_{R}(X,0)=\exp(-((X-X_{0}))⁄4)$,

ψ$_{I}(X,0)=0$.

It is wave packet with the center not in the origin. We can tell you, that this is so called coherent stat of harmonic oscillator and wave packet should oscillate around origin with angular frequency √( $k⁄m)$ same as classical particle.

If you can calculate the previous, then you can try what will be the behaviour of wave packet of different width (e.g. denominator in exponential is different from 4) of how the behaviour will look like with different potential.

Uranium storage

Heating wire

Capacity of a cube

Estimate thickness of water needed to shield radiation from nuclear reactor of power 980 MW in proposed new block of nuclear power station Temelin. The total energy released by fission from Uranium is split following: 82% goes to kinetic energy of fragments, 6% goes to neutrinos, 6% goes to neutrons and remaining 6% goes to energy of gamma photons.

Hint: Probability of particle passing through a material into the depth $d$ is probably equal to e^{$-σnd}$, where $n=N/V$ is density of molecules of material (in our case number of water molecules in 1 m3 ) and $σ$ is effective cross section for absorption of particle on molecule of water. Unit of cross section is surface, therefore m^{2} but often is used unit barn = 100 fm2) and depends on energy of particles. The values of cross section can be found on Internets or in tables.

The typical composition of natural uranium is 0.72 % of isotope ^{235}U of halftime of 704 million years and the rest is isotope ^{238}U with half life of 4 468 million years.

While mining uranium in 1970's in Okla in equatorial Gabon, the uranium ore with relative content of isotope ^{235}U 0.44 % . This discrepancy can be explained by assuming existence of 'natural nuclear reactor' at the uranium deposits million years ago.

Calculate, how long the nuclear reaction was going on if the decay of ^{235}U was started by slow neutrons. The collision of slow neutron with the nucleus happens every 352 thousand years.

In Jet Propulsion Laboratory in California, U.S.A. in NASA laboratory the new rocket engine is under development. It uses momentum of $α-particles$ created during radioactive decay of fermium $^{257}_{100}Fm_{157}$, which mass is $m_{Fm}$ and half-life $T$. The second product is californium $^{253}_{98}Cf_{155}$. The mass of $α-particle$ is $m_{α}$, the mass of californium is $m_{Cf}$, and during the decay the energy $E$ is released. Assume, that each $α-particle$ leaves rocket in the same direction.

The space probe with above engine is in rest at the beginning and its mass is $M$, the mass of 'fuel' is also $M$. Calculate the speed of the probe $v$ after half of the fermium decays. Resulting speed calculate also for the following numerical values $E=1,106\cdot 10^{-12}J$, $M=4\;\mathrm{kg}$ a $T=100,5days$, for other values consult your table-book.