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## mechanics of a point mass

### (3 points)1. Series 34. Year - 2.

Carl's car, going at the initial speed of $v_0$, can stop at a distance $s_0$ with the constant braking force $F_0$. How many times will the braking distance increase if the initial speed doubles if the braking force stays the same? How many times must the braking force be for the car to stop at distance $s_0$ with the speed $2v_0$?

### (5 points)1. Series 34. Year - 3. cycling anemometer

Vašek rides his bicycle in windy weather. When he rides straight with the velocity $v = 10 \mathrm{km\cdot h^{-1}}$, he measures that the wind blows at an angle $25\dg $ from the direction of Vašek's direction of travel. When he accelerates to $v' = 20 \mathrm{km\cdot h^{-1}}$, the angle is only $15\dg $. Find the velocity and direction of the wind with respect to stationary observer.

### (8 points)1. Series 34. Year - 4. solar sail

A solar sail with the surface $S = 500 \mathrm{m^2}$ and area density $\sigma =1,4 \mathrm{kg\cdot m^{-2}}$ is located in the distance of $0,8 \mathrm{au}$ from the Sun. What force does the solar radiation act on the sail at the beginning of the sail's motion? What is the acceleration of the sail at that moment? The luminosity of the Sun is $L_{\odot } =3,826 \cdot 10^{26} \mathrm{W}$. Assume that the radiation approaches the sail from perpendicular direction and scatters elastically. **Hint:** We recommend to find acceleration for small initial velocity $v_0$ and then let $v_0 = 0$.

### (8 points)1. Series 34. Year - 5. how to put your beanie on sigle-handily

Let us have a ball with the radius $R$ and a circular massless rubber band with the radius $r_0$ and stiffness $k$, while $r_0 < R$. Coefficient of friction between the band and the ball is $f$. Find conditions which ensure that it is possible to stretch the band over the ball single-handily (i.e. we are allowed to touch the band in only one point.

To keep it simple assume that the band is elastic only in the tangential direction (so it is planar).

### (13 points)1. Series 34. Year - E. impact-y

Measure the dependence of the diameter of a crater, created by the impact of a stone into a suitable sandpit, on the weight of the stone and the height it is released from. Does the size of the crater depend only on the energy of the impact? Dry sand is recommended for this measurement.

### (5 points)6. Series 33. Year - 3. hung

What weight can be hung on the end of a coat hanger without turning it over? The hanger is made of a hook from very light wire, which is attached to the centre of the straight wooden rod, which length is $l = 30 \mathrm{cm}$ and weight $m=200 \mathrm{g}$. The hook has the shape or circular arc with radius $r=2,5 \mathrm{cm}$ and angular spread $\theta =240 \mathrm{\dg }$. The distance between the centre of the arc and the rod is $h=5 \mathrm{cm}$. Neglect every friction.

Dodo is seeking for a scarce.

### (3 points)5. Series 33. Year - 1. train on a bridge

There is a freight train standing on a $300 \mathrm{m}$ long bridge. The mass of the train is evenly distributed onto area of all nine steel pillars of the bridge. Every pillar has a base in a shape of a square with a side $a = 2,0 \mathrm{m}$ and a height $h=10 \mathrm{m}$. How much do the steel pillars shrink under the weight of the train? Young modulus of steel is $E = 200 \mathrm{GPa}$. Overall mass of the train is $m = 574 \mathrm{t}$.

Danka watched trains from her dormitory.

### (3 points)5. Series 33. Year - 2. will it move?

Jachym wants to pickle cabbage at home, so he buys a cylindrical barrel. He carries it from the shop to the home using underground. We can consider the barrel and its lid as a hollow cylinder with outer dimensions: radius $r$, height $h$ and width of the walls, the base, and the lid is $t$. The barrel is made of a material with density $\rho $. What is the maximum acceleration that the underground can go with, so the free standing barrel does not move in respect to the underground? Coefficient of friction between underground's floor and the barrel is $f$.

Dodo is listening to Jachym's excuses again.

### (10 points)5. Series 33. Year - S. min and max

*We are sorry. This type of task is not translated to English.*

They had to wait a lot for Karel.

### (3 points)4. Series 33. Year - 1. tchibonaut

Consider an astronaut of weight $M$ remaining still (with respect to a space station) in zero-g state, holding a heavy tool of weight $m$. The distance between the astronaut and the wall of the space station is $l$. Suddenly, he decides to throw the tool against the wall. Find his distance from the wall when the tool hits it.

Karel wanted to set this name for this problem.