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## mechanics of a point mass

### (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.

### (3 points)3. Series 36. Year - 1. creative problem-solving

Danka attached a garden hose with an inner diameter of $1{,}5 \mathrm{cm}$ to a tap in her dorm room and placed the other end on the edge of a window on the eighth floor, $23 \mathrm{m}$ above the ground. What is the necessary volumetric flow rate of the water tap so that Danka can spray a stream of water on the people disturbing the night's silence? They are standing below the window at a horizontal distance $9 \mathrm{m}$ from the building. Is Danka able to achieve this if water is being sprayed horizontally from the hose and there is no wind?

**Bonus:** Where is the farthest these people can stand so Danka can still spray them if the volumetric flow rate of the tap is $0{,}4 \mathrm{l\cdot s^{-1}}$? Danka can now set the end of the hose so that water sprays at an arbitrary angle to the horizontal plane.

Danka is annoyed by the noise below the windows at night.

### (5 points)3. Series 36. Year - 3. bobsled

Matěj and David are sliding on bobsleds down the hill. The hill with a slope of $\alpha =29 \mathrm{\dg }$ turns into the horizontal ground at the bottom of it. Both of them started from rest from the same height. Matěj's bobsled always travels the same distance $l$ on an inclined plane as well as in a horizontal part. Since the bobsled digs deeper into the snow at higher loads, assume the coefficient of friction to be proportional to the normal force as $f(F)=kF$, where $k$ is a positive constant. Determine how many times Matěj will travel farther from the bottom of the hill than David if David's mass (including the bobsled) is $12 \mathrm{\%}$ greater than Matěj's. Also, assume that bobsledders don't lose any energy at the bottom of the hill.

Matej likes to talk about bobsled.

### (6 points)2. Series 36. Year - 4. parallel collision

The FYKOS-bird watches in their inertial frame of reference as two point masses move around them on parallel trajectories with constant non-relativistic velocities. They think whether these trajectories could intersect for some other inertial observer. If so, is it possible that the two point masses in question could collide at this intersection given the right initial conditions? Is this consistent with the fact that they are moving in parallel according to the FYKOS-bird?

Marek J. loves collisions.

### (3 points)1. Series 36. Year - 1. useful butter

Jarda decided to bake a cake but he found out that the battery in his kitchen scale was dead, so he can't weigh $300 \mathrm{g}$ of flour. However, he had the idea that he could use a block of butter instead. The packaging said its weight is $m = 250 \mathrm{g}$. Fortunately, he found a suitable spring and a stopwatch. He put a heap of flour in a very light bowl, attached it to the spring, perturbed it and measured the period of oscillations $T_1=2,8 \mathrm{s}$. He repeated the same process with the cube of butter and measured $T_2 = 2,3 \mathrm{s}$. How much flour does Jarda need to add or remove?

When Jarda gets kicked out of Matfyz, he will open a bakery.