# 6. Series 31. Year

### (3 points)1. they came apart

We have two point masses with the same mass $m$ at a distance $d$ from each other. They are located freely in space with no external gravitational forces. What's the minimum velocity we need to impart on one of the points in the direction away from the other point, so that they keep flying away from each other indefinitely?

Matej played with the universe

### (3 points)2. hot wire

Calculate the current, that needs to pass through a metal wire of a diameter $d = 0{,}10 \mathrm{mm}$ located in a vacuum bulb, so that its temperature stays at $T = 2 600 K$. Assume the surface of the wire radiates like an ideal black body and neglect any losses by heat conduction. The resistivity of the material of the wire at the given temperature is $\rho = 2{,}5 \cdot 10^{-4} \mathrm{\Ohm \cdot cm}$. \taskhint {Hint}{Use the Stefan-Boltzmann's law.}

Danka was contemplating the light bulb efficiency

### (6 points)3. non-analytic spring

Imagine a pole of length $b = 5 \mathrm{cm}$ and mass $m = 1 \mathrm{kg}$ and a spring of initial length $c = 10 \mathrm{cm}$, spring constant $k = 200 \mathrm{N\cdot m^{-1}}$ and negligible mass, that are connected at one of their ends. The other ends of the spring and the pole are affixed at the same height $a = 10 \mathrm{cm}$ from each other. The spring and the pole can both freely rotate about the fixed points and their joint. Label $\phi$  the angle of the pole to the horizontal. Find all angles $\phi$, for which the system is in an equilibrium. Which of these are stable and which unstable?

Jachym was supposed to come up with an easy problem.

### (7 points)4. dimensional analysis

Matej was making a gun and wanted to measure what is the speed of the projectiles leaving the barrel. Unfortunately, he doesn't have any other measuring device, than a ruler. However, he found a block that is made half from steel half from wood. He lays it down at the edge of the table (of height $100 \mathrm{cm}$ and length $200 \mathrm{cm}$), and shoots at it horizontally. With the steel part of the block facing the gun, the bullet bounces off perfectly elastically and lands $50 \mathrm{cm}$ from the edge of the table. The block slides $5 \mathrm{cm}$ on the table. Then Matej turns around the block and shoots into the wooden side. This time the bullet stays in the block and the block slides only $4 \mathrm{cm}$. Help Matej with calculating the speed of the bullet. It might be also helpful to know, that when Matej lifts one edge of the table by at least $20 \mathrm{cm}$, the moving block won't stop sliding.

Matej wanted all the variables to have the same unit.

### (8 points)5. jump from a plane

Filip of mass $80 \mathrm{kg}$ jumped out of air plane, that is $h_1 =500 \mathrm{m}$ above the ground. At the same time, Danka (mass $50 \mathrm{kg}$) jumps out of a different airplane but from a height of $h_2 =569 \mathrm{m}$. Assume both of them have the same drag coefficient $C = 1{,}2$, Filip's cross-sectional area is $S\_F = 2{,}2 \mathrm{m^2}$ and Danka's is $S\_D=1{,}5 \mathrm{m^2}$. The density of air $\rho =1{,}205 \mathrm{kg\cdot m^{-3}}$ is and stays the same in all heights. At what time will Danka be at the same height above the ground as Filip?

Danka contemplated the strenuous life of a physicist and wanted to break free for a moment.

### (9 points)P. universe expansion compensation

According to the current observations and cosmological models, it seems that our Universe is expanding and the rate of expansion is accelerating. What if that wasn't the case? What if the Universe stayed the same, but the physical laws/constants were changing so that it would seem like the universe is expanding, the way we observe it? Describe as many laws that would need to change.

Karel was intrigued whether one can compensate the expansion of universe.

### (12 points)E. don't play with matches

Measure the speed with which a wooden skewer burns as a function of its tilt with respect to the vertical.

Because the gasoline that Karel suggested was a bit too much.

### (10 points)S. Matrices and populations

Mirek and Lukáš fill matrices with atto-foxes.