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## mechanics of a point mass

### 1. Series 30. Year - 2. Breaking

Petr likes to ride a bike on a flat road with a speed $v=10\;\mathrm{m}\cdot \mathrm{s}^{-1}$, and his smart bike tells him that his average power is $P=100W$. After an accident, his breaks are bent and they now persistently act on a wheel with a friction force $F_{t}=20N$ near the circumference. For how long ($t′)$ he needs to cycle now (with the same speed $v)$, to do the same amount of work as before, in time $t?$

Petr si uvědomil výhody zaseknuté brzdy.

### 1. Series 30. Year - 5. On a walk

Katka decided to go for a walk with her pet rat. They arrived on a flat meadow and when the rat was at a distance $x_{1}=50\;\mathrm{m}$ from Katka, she threw him a ball with the speed $v_{0}=25\;\mathrm{m}\cdot \mathrm{s}^{-1}$ and an angle of elevation $α_{0}$. In that moment, he started running towards her with the speed $v_{1}=5\;\mathrm{m}\cdot \mathrm{s}^{-1}$. Find a general formula for an angle $φ$ as a function of time, where the angle $φ(t)$ is the angle between the horizontal plane and the line between the rat and the ball. Draw this function into a graph and, based on the graph, determine, whether it's possible for the ball to obscure the Sun for the rat, when the Sun is situated $φ_{0}=50°$ above the horizon in the direction of the running rat. Use the acceleration due to gravity $g=9.81\;\mathrm{m}\cdot \mathrm{s}^{-2}$ and for simplicity imagine we are throwing the ball from a zero height.

Mirek pozoroval, co se děje v trávě.

### 6. Series 29. Year - 3. Going downhill

We are going up and down the same hill with the slope $α$, driving at the same speed $v$ and having the same gear (and therefore the same RPM of the engine), in a car with mass $M$. What is the difference between the power of the engine up the hill (propulsive power) and down the hill (breaking power)?

### 6. Series 29. Year - 4. Fire in the hole

Neutral particle beams are used in various fusion devices to heat up plasma. In a device like that, ions of deuterium are accelerated to high energy before they are neutralized, keeping almost the initial speed. Particles coming out of the neutralizer of the COMPASS tokamak have energy 40 keV and the current in the beam just before the neutralization is 12 A. What is the force acting on the beam generator? What is its power?

### 5. Series 29. Year - 4. safe ride

A car is approaching a wall with a trajectory that is perpendicular to the wall. The driver, however, wishes to approach the wall safely. Find the car's speed as a function of time, so that the distance between the car and the wall is, at every moment, the same as the distance the car would travel with its instantaneous speed in $T=2\;\mathrm{s}$.

### 3. Series 29. Year - 4. break, break, break!

After we press the break pedal, the car does not start to break immediately. During the time $t_{r}$ the breaking force grows linearly up to the maximum force $F_{m}$. Coefficient of static friction between the tire and a road is $f$. What is the maximum speed of car so that the car does not skid even during emergency breaking?

### 3. Series 29. Year - 5. running notebook

The notebook of a size of A4 (297 x 210 mm) lies on a desk with an inclination of $α=5°$. The notebook weights $m$, between the desk and the notebook there acts a static friction force with coefficient $f_{0}=0.52$. Then, we hit the desk so it starts to oscillate (in the direction of the inclination of the desk) with a frequency $ν=10\;\mathrm{Hz}$ and an amplitude $A=1\;\mathrm{mm}$.

• Determine by which extra force (perpendicular to the desk) we have to act on the notebook so it does not start to move.
• Determine how long it takes the notebook to fall off the desk if at the beggining its bottom edge (the shorter one) is at the bottom edge of the desk. Dynamic friction coeficient is $f$, consider notebook as a rigid plate.

### 2. Series 29. Year - 1. rat on ice

A rat is running on ice with speed $v$. Suddenly he decides to turn 90$°$ so that he keeps running with the same speed in the new direction. What is the least amount of time he needs for such a turn? Suppose that rat's feet can move independently. Coefficient of friction between rat's feet and ice is $f$.

Xellos dostal smyk.

### 2. Series 29. Year - 3. fatal fall

From a spaceship on a circular orbit with height $h=2000\;\mathrm{km}$ above the surface of Earth a screwdriver is thrown with speed $v=5\;\mathrm{km}\cdot h^{-1}$ relative to the rocket towards the center of the Earth. Determine when will the screwdriver hit the surface?

Karel nemá rád šroubováky.

### 1. Series 29. Year - 2. jumping out of a train

In a train, that can move without friction on rails, stand 2 people, both with a mass $m$. In which of the two following situations shall the train reach a higher speed? When both jump out at the same time or when they will jumping outone after another? A person can jump out the train with a relative speed $u$ (the speed of a person jumping out the train versus the speed of the train).

Radomir was jumping out of a train.