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## mechanics of a point mass

### (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.

### (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).

### (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.

### (3 points)6. Series 35. Year - 1. Superman in action

Lex Luthor kidnapped Lois Lane and threw her off the plane at altitude $h$. Superman follows her and catches her at some unknown altitude. Suppose that the maximum acceleration Lois can survive is $10 g$. What is the lowest altitude at which can Superman catch Lois to save her?

Martin reminisced about his youth.

### (5 points)6. Series 35. Year - 3. wind bubble

Imagine we create a small soap bubble with a bubble blower. How fast does it fall to the ground? The bubble has an outer radius $R$ and an areal density $s$.

Karel was making bubbles in the bathtub.

### (10 points)6. Series 35. Year - 5. fly rocket, fly

We have constructed a small rocket weighing $m_0 = 3 \mathrm{kg}$, from which $70\%$ is fuel. The exhaust velocity is $u = 200 \mathrm{m\cdot s^{-1}}$ and the initial flow of the exhaust fumes is $R = 0,1 \mathrm{kg\cdot s^{-1}}$ and both these values remain constant during the flight. The rocket is equipped with stabilization elements, so it does not deviate from its desired trajectory. It has been launched from the rest position vertically. Assume that the friction force of the air is proportional to the velocity $F\_o = -bv$, where $b = 0,05 \mathrm{kg\cdot s^{-1}}$, $v$ is the velocity of the rocket and the sign minus means that the force exerts against the direction of the motion. What height above the ground level does the rocket fly in time $T = 25 \mathrm{s}$ from the engine startup?

Jindra got a homework to deliver a satellite onto the Low Earth orbit.