# 2th Round of 28th year

Series

### 1. ... Saint Anne cold from the morning (this is unpunifiable)(2 points)

In the cold morning mist you are leaving the house and the garden gate works in such a way that it will open after the handle is pushed down and after closing it and letting go of the handle it stays closed shut. You return in the afternoon and ask yourself who didn't close the goddamn gate? Then you try to close the gate but can't. The steel latch won't recede far enough to pass around the aluminum frame, even after pushing down the handle. The gate is also made out of aluminum. What seems to be the problem here? What did the manufacturer forget to think about? Determine the gate's parameters (width, length, breadth) when it is 20 °C, if we know that temperatures don't fall under −30 °C and doesn't rise over 50 °C.

### 2. ... hedonistic cell(2 points)

Estimate on the basis of macroscopically measureable quantities the number of cells in the human body and the number of particles in one mole , how many molecules of oxygen&#8222;are used&#8220; daily by a human body cell. Find the relevant information needed for the calculation and don't forget to cite your sources properly.

### 3. ... impatient core(4 points)

The core of Bismuth ^{209}Bi sits impatiently at peace on the same spot. Suddenly it can't hold it any lonher and it falls apart. A thalium core ^{205}Tl remains and from it one can see an$α$ particle shoot away. What is the speed of the $α$ particle, if the energy released during the decay becomes its kinetic energy? What is the velocity of the $α$ particle in reality? Compare the results. The rest masses of the atoms are  $M$ = $m$_{^{209}Bi} = 208,980 399 u, $M′$ = $m$_{^{205}Tl} = 204,974 428 u, $m$ = $m$_{^{4}He} = 4,002 602 u. Don't forget to check if one should use relativistic relations.

### 4. ... Boeing(4 points)

Consider a tyre of a cylindrical shape and of a radius $R$ s an inner radius $r$ width $d$ filled up to a pressure of $p$. We push down on the tyre with a force $F$. With this encumbrance the shape of the tyre changes from a cylinder to a cylindrical segment with the same inner and outer radius. Assume that the temperature of the tyre will not change. Determine the contact area of the tyre and road.

### 5. ... gravitational manuevers(5 points)

Assume a satelite which orbits the sun on an elliptical orbit. If we lower the speed in the aphelion $v$_{a} to 4⁄5 of the initial velocity (i.e. to 4⁄5$v$_{a}), how will the speed of the satelite change in the perihellion? Express the new velocity using the initial velocity $v$_{p} and the parameters of the ellipse (main axis $a$ and relative eccentricity $ε$).

### P. ... problem of the travelling salesman(5 points)

When digital mobile phones started to be more common there was often an issue with accepting calls in a car. Nowadays this issue is mostly connected with trains. What factors influence the transmission of data in the GSM network and how can they influence the availability of the signal of the provider? How can one combat this problem?

### E. ... waterfall(apart)(8 points)

What is the depth under the tap where the stream of water divides into droplets? How does it depend on the the flow of water?

### S. ... numerical(6 points)

* We give length values in metres, time values in seconds and mass values in kilograms. Angular velocity $Ω$ we give in radians per second. If you take the equations for the movement of balls from the series, there are three more parameteres included: $α$, $β$, $γ$. What are their dimensions?

• Consider a freefalling ball with $Ω$ = 0 and $v$_{$x$} = 0. There then exists a terminal velocity $v$_{$z$}^{t}, at which the frictional force and  and gavitational force are equally matched and the fall of the ball isn't accelerating anymore.
• Determine this velocity from the equations for the movement o a ball.
• Change this equation so that it will express  $β$. $v$_{$z$}^{t} can be easily measured and for ourfootball of mass $m$ = 0,5 kg it is typically around 25 m·s^{ −1}. Then what is $β$?
• Express the initial  $v$_{$x$} and $v$_{$z$} using the angle at which it was shot out $φ$ with a fixed initial velocity $v$ = 10 m·s^{ −1}. Write a program according to the series and try changing the initial conditions and the following parameters
• Choose some positive $β$, turn off the rotation $Ω$ = 0 and find out, if the angle under which the the ball reach the farthest is bigger or smaller than 45°. Demonstrate your finding with graphs of the trajectories.
• Choose a positive non-zero $α$ with a numerical value in the given units the same as $β$, $γ$ = 0,01 (in the given units) and $Ω$ = ± 5 rad·s^{ −1}.How will in these specific cases the optimal angle of the shot change?
• Bonus    How far would you throw with a cricket ball? Is our model good enough to make such predictions?
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