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mechanics of a point mass

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?

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.

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.

5. Series 35. Year - 1. illuminated satellite

On average, what part of the day does a satellite in low orbit spend in the shadow of Earth? Assume that the satellite's orbit is circular and lies in the ecliptic plane at height $H = R/10$ above the surface of Earth, where $R$ is the mean radius of Earth.

5. Series 35. Year - 2. cherry pit

Elon Musk plans to colonize Mars. However, he has to build supply bases on the Moon's surface to make colonization possible. Help him solve a crucial problem: how far can a $180 \mathrm{cm}$ tall person spit a cherry pit at a base on the Moon if they spit it in a horizontal direction. The same cherry pit spit on the Earth lands at a distance $4,3 \mathrm{m}$. Bonus: Determine the ratio of distances that the same person reaches by spitting the cherry pit on the Earth and the Moon if they can spit at an arbitrary angle with respect to the ground.

Katarína was looking for an excuse for a trip to the Moon.

5. Series 35. Year - 4. hit

The FYKOS bird plays with a baseball bat (homogeneous rod of linear density $\lambda$) and hits a baseball of mass $m$. Assume that the rod is attached at one of its ends and can rotate around that point freely. The FYKOS bird can either act on it by a constant torque $M$ or start rotating it by a constant power $P$. After completing a rotation of $\phi _0 = 180\dg$, the end of the rod hits yet motionless baseball, which results in an elastic collision. At what length of the rod $l$ does the baseball gain maximum speed? Compare both situations (i.e., constant $M$ vs. constant $P$).

Jáchym was playing with a baseball bat.

4. Series 35. Year - 3. pendular collisions

Two small marbles are attached to ends of strings of the same length ($l = 42,0 \mathrm{cm}$) and negligible mass. The other ends of both strings are attached to a single point. The marbles are of the same size, but they are made from the different materials; first one is made of steel ($\rho _1 = 7~840 kg.m^{-3}$) and second one is made of dural ($\rho _2 = 2~800 kg.m^{-3}$). Both marbles are initially at the angle $5 \mathrm{\dg }$ with respect to the equilibrium position, and after releasing them, they collide elastically. What is the maximum height the individual marbles reach after the collision? What is the result after the second collision?

Karel wanted to hypnotize others. You want to solve this problem \dots

4. Series 35. Year - 5. Helicopter

The FYKOS bird started to think about constructing his own helicopter because he was tired of flying using his wings. He started by creating a simple model of the main rotor and wondered what the rotor's angular velocity needed to be. Rotor blades are inclined at angle $45 \mathrm{\dg }$; thus, air molecules are pushed directly downwards, creating a momentum flux. Initially, we assume air molecules to be at rest and their collision with the rotor blades to be elastic. The effective part of the rotor blade (i.e., the part inclined at $45 \mathrm{\dg }$ angle) is blade's part that is distant $r_1 = 50 \mathrm{cm}$ to $r_2 = 6,00 \mathrm{m}$ from the centre of rotation. The projection of one blade onto the vertical plane has height $h = 10,0 \mathrm{cm}$, and the helicopter will have four such blades. What is the minimum frequency of the rotor to keep the helicopter of mass $m = 2~500 kg$ at a constant height?

Jindra was hot, so he stood under the helicopter.

3. Series 35. Year - 2. playing with keys

Vašek likes to plays with keys by swinging them around on a keychain and then letting them wrap around his hand. We will simplify this situation by a model, in which we have a point mass $m$ in weightlessness attached to an end of massless keychain of length $l_0$. The other end of the keychain is attached to a solid cylinder of radius $r$. The keychain is taut so that it is perpendicular to the surface of the cylinder at the attachment point, and the point mass is then brought to velocity $\vect {v_0}$ in the direction perpendicular both to the axis of the cylinder and to the direction of the keychain. The keychain then starts to wrap around the cylinder. What is the dependence of the velocity of the point mass on the length of the free (not wrapped around) keychain $l$?

Hint: Find a variable that remains constant during the wrapping process.

Bonus: How long does it take for the whole keychain to be wrapped around the cylinder?

Vašek was playing with keys while falling out of window.

2. Series 35. Year - 1. chasing the light

Jindra walks down a long, lit corridor. His eyes are at a height of $1,7 \mathrm{m}$ above the floor, the light on the ceiling is at a height of $3,4 \mathrm{m}$. Jindra is now at a distance of $10 \mathrm{m}$ (horizontally) from the nearest light and is approaching it at a speed of $3 \mathrm{km\cdot h^{-1}}$. He sees a reflection of the light on the polished floor. How fast is the reflection approaching Jindra at this point?

Jindra remembered walking down the corridor at the elementary school.