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## mechanics of a point mass

### (5 points)3. Series 37. Year - 3. randomly you get further

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.

### (3 points)2. Series 37. Year - 1. workout

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.

Dodo went clibming to Smíchoff.

### (3 points)1. Series 37. Year - 2. train shifting

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?

Jarda is going to Prague again.

### (12 points)1. Series 37. Year - E. wipe the paper

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

Karel browsed through the recipes.

### (3 points)6. Series 36. Year - 2. rotten apple

Jarda found an apple in his backpack after the FYKOS camp, which was no longer in good condition. He threw it into a low kitchen trash can $1{,}0 \mathrm{m}$ away, and of course, he scored a hit. He threw the apple horizontally from a height $0{,}5 \mathrm{m}$, and it landed on the spot where the wall and the base of the trash can meets, where it smashed. The basket with a mass $910 \mathrm{g}$, was displaced by a distance of $5 \mathrm{cm}$ after the apple hit. What is the coefficient of friction between the floor and the basket? The apple has a mass of $230 \mathrm{g}$.

Jarda forgot to eat his snack again.

### (3 points)5. Series 36. Year - 2. conveyor belt

Every second, a material of mass $\mu $ falls vertically onto a moving horizontal conveyor belt and falls away at its end. A resistive force $F\_{odp}=kv$, which is directly proportional to the belt speed $v$ through the constant $k$, acts on the belt. At what speed does thevbelt move if

- a constant driving force $F$ acts on it?
- it is driven by a motor of constant output power $P$?

Karel hoped it could be solved.

### (6 points)5. Series 36. Year - 3. waiting for an elevator

Karel uses an elevator in a building with a ground floor and $12$ floors above it, while the height between floors is $h=3{.}0 \mathrm{m}$. Consider that the elevator accelerates half the time and decelerates half the time at a constant acceleration of $a=1{.}0 \mathrm{m\cdot s^{-2}}$ and that there is a $50\mathrm{\%}$ probability that the elevator is stationary on the ground floor. The rest of the probability is evenly distributed among the other floors. What is the expected waiting time for the elevator on each floor of the building? Neglect the time needed for opening doors.

**Bonus:** Let us have $2$ elevators in a twelve-story building. One elevator is always recalled to the ground floor. To which floor should we send the second one to minimize the average waiting time? Similarly, assume that half of the rides will start on the ground floor and the other half, with equally distributed probability, will start on any other floor.

Karel often waits for an elevator.

### (8 points)5. Series 36. Year - 5. xenon was wandering

A once positively ionized xenon atom flew out from the center of a large cylindrical coil with velocity $v=7 \mathrm{m\cdot s^{-1}}$ and began to move through a homogeneous magnetic field, which is in a plane perpendicular to the magnetic lines of force. At a certain point the coil is disconnect from the source, thus its induction begins to decrease exponentially according to the following equation $\f {B}{t}=B_0\eu ^{-\Omega t}$, in which $B_0=1,1 \cdot 10^{-4} \mathrm{T}$ and $\Omega =600 \mathrm{s^{-1}}$. What is the deviation from the initial direction after the atom is stabilized?

Vojta spent several hours thinking about a reasonable problem assignment with a clever solution, but ultimately, it ended horrendously. And he has yet to see the solution.

### (6 points)4. Series 36. Year - 3. road closure

We all know it – road closures and endless standing at traffic lights. The light is green for $60 \mathrm{s}$, but by the time everyone gets going, it is red again. Consider the $0{,}5 \mathrm{s}$ reaction time for a driver to get moving after the car in front of him has done so. By what percentage would the number of cars that pass through the closure increase if everyone in line started moving simultaneously? The first car stands at the traffic light level, the distance of the front bumpers of all cars is estimated to be $5 \mathrm{m}$, and they all accelerate uniformly for $5 \mathrm{s}$ to a speed of $30 \mathrm{km\cdot h^{-1}}$, with which they proceed further into the closure.

They have been digging sewers in Jarda's village for three years now.

### (9 points)4. Series 36. Year - 5. space visit

Two aliens each live on their own space station. The stations are in free space and the distance between them is $L$. When one alien wants to visit the other, he has to board his non-relativistic rocket and fly to his neighbor. What is the shortest time an alien can spend on its way there and back? The mass of the rocket with fuel is $m$, without fuel $m_0$. The exhaust velocity is $u$. The fuel flow is arbitrary, and his neighbor won't let him load any fuel (he has little himself).

Jarda needed no one to notice that he had disappeared from the meeting for a while.