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Adam can take meaningful notes at the speed $v_1$. Unfortunately, his calculus professor speaks at the speed of $v_2$. There is an airflow in the lecture hall, moving from Adam towards the professor, with the air flowing at a velocity of $v_3$. At what velocity and in which direction along a straight line intersecting Adam and the lecturer should Adam move to transcribe everything the lecturer says into his notebook?

Consider a centrifuge of length $L = 30 \mathrm{cm}$ filled with a solution in which there are homogeneously distributed small spherical particles of radius $r = 50 \mathrm{\micro m}$ and mass $m = 5,5 \cdot 10^{-10} \mathrm{kg}$. The density of the solution is $\rho \_r = 1~050 kg.m^{-3}$ and its viscosity is $\eta = 4,8 \mathrm{mPa\cdot s}$. The container with the solution is in a horizontal position and suddenly begins to rotate at an angular velocity of $\omega = 0,5 \mathrm{rad\cdot s^{-1}}$. Determine how long it will take for $90 \mathrm{\%}$ of all the particles to reach the end of the centrifuge. Do not consider interparticle collisions and movement of the particles due to diffusion.

One day, the FYKOS-bird was watching the sky during a full moon. An airplane just passed over the moon in $0{.}35 \mathrm{s}$, and the perpendicular distance of its flight path from the center of the moon was $1/3$ of the full moon's radius. This plane flies typically with a speed of $800 \mathrm{km\cdot h^{-1}}$. The FYKOS-bird wondered what altitude the plane was at so he could fly with it next time. Like him, determine this altitude.

Tomáš got into the train wagon in the shape of a rectangular cuboid and decided to take a nap. When he woke up, he found that he was alone in the wagon, which was suspended at its geometric center on a cargo crane and rotating around the hinge axis at an angular velocity of $\omega $. Tomáš didn't notice it at first since he was sitting in the wagon's centre with a width of $d$. When he realized it, he was pleased because he thought of using one of his kilogram standards, which he carries around for situations like this, to measure the length of the carriage. After a few attempts, he managed to throw the standard at an initial velocity of $\vec {v}$ so that after two revolutions of the wagon, the standard hit the far corner of the wagon and broke the window. Neglecting air resistance, what length $L$ of the wagon did he determine?

Consider a homogeneous magnetic field of induction $B_1$, which spans a half-space bounded by the plane of interface $y=0$, beyond which is an equally oriented, also homogeneous magnetic field of induction $B_2$. An electron flies out of the plane perpendicularly to it and the field lines (as in the figure) with velocity $v$. Determine the size and the direction of its average velocity parallel to the plane of the interface.

Bonus: Consider now that the magnitude of the field changes linearly as $B = B_0 \(1+\alpha y\)$ and its direction is in the positive direction of the $z$-axis. Again, determine the magnitude and direction of the average velocity of the electron parallel to the interface plane. The electron is initially emitted as in the previous case.

In the microworld of cells, there are two types of transport: transport by free diffusion, also known as Brownian motion where the motion uses the energy of the environment directly. The second type, so-called active transport, requires, among other things, a motor protein moving at a constant speed along the cytoskeletal filament. Consider the typical value of the diffusion constant $D \approx 10^{-9} cm^2.s^{-1}$ and the rate of active transport speed $u\approx 10^{-6} m.s^{-1}$. For which distances is the Brownian motion more time efficient than the active transport? Assume that the transport is happening in only one direction.

1. According to definitions by International System of Units, convert these into base units
   • pressure $1 \mathrm{psi}$,
   • energy $1 \mathrm{foot-pound}$,
   • force $1 \mathrm{dyn}$.
2. In the diffraction experiment, table salt's grating constant (edge length of the elementary cell) was measured as $563 \mathrm{pm}$. We also know its density as $2{,}16 \mathrm{g\cdot cm^{-3}}$, and that it crystallizes in a face-centered cubic lattice. Determine the value of the atomic mass unit.
3. A thin rod with a length $l$ and a linear density $\lambda $ lies on a cylinder with a radius $R$ perpendicular to its axis of symmetry. A weight with mass $m$ is placed at each end of the rod so that the rod is horizontal. We carefully increase the mass of one of the weights to $M$. What will be the angle between the rod and the horizontal direction? Assume that the rod does not slide off the cylinder.
4. How would you measure the mass of:
   • an astronaut on ISS,
   • a loaded oil tanker,
   • a small asteroid heading towards Earth?

When working out, we often come across workout machines that contain pulleys. Consider the machine in the following figure. What force must be applied on the rope if the velocity of the end of the rope at point A is $v = 0,4 \mathrm{m\cdot s^{-1}}$ and its direction is downwards? Each pulley has a radius $r = 15 \mathrm{cm}$ and mass $m = 15 \mathrm{kg}$. A weight of mass $M = 25 \mathrm{kg}$ hangs over the free pulley.

Jarda is standing at the end of the platform, waiting for his train to arrive. When the train's first carriage passes him, he discovers that this is the carriage where he has his seat ticket. At this point, the speed of the train is $8{,}5 \mathrm{m\cdot s^{-1}}$, and the train begins to slow down steadily until it stops in $28 \mathrm{s}$. Jarda immediately starts walking in the direction of his carriage, but because he has to push through the crowds of passengers, his speed is only $1 \mathrm{m\cdot s^{-1}}$. What is the shortest time the train must stay in the station for Jarda to board his carriage?

Measure the coefficient of static friction between two sheets of office paper.