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## gravitational field

### (9 points)3. Series 33. Year - 5. probability density of water

Imagine a container from which continually and horizontally flows out water stream with constant cross-section area. Velocity of the stream randomly fluctuate with uniform distribution from $v_1$ to $v_2$. Water from the container continually freely falls onto a horizontal floor below. Figure out arbitrary area of the floor to which falls exactly $90 \mathrm{\%}$ of water.

Another from a list of problems, which crossed Jachym's mind while being on a toilet.

### (10 points)5. Series 32. Year - S. heavenly-mechanic

- Consider a cosmic body with the mass of five Suns surrounded by a spherically symmetrical homogenous gas cloud with the mass of two Suns and radius $1 \mathrm{ly}$. The cloud starts to collapse into the central cosmic body. Neglect the mutual interactions of particles in the cloud (excluding gravity). Find how long it will take for the whole cloud to collapse into the central body. Do not solve this problem numerically.
- Show that the Binet equation solves following the differential equation, which describes the motion of a mass point of mass $m$ in a spherically symmetrical central-force field. \[\begin{equation*} \dot {r}^2 = \frac {2}{m} \(E - V(r) - \frac {l^2}{2mr^2}\) \end {equation*}\] Where $r$ is the length of the radius vector, $E$ is the total energy, $l$ is angular momentum, and $V(r)$ is the potential energy of the mass point.
- Set up the Lagrangian for the Sun-Earth-Moon system. Assume the Sun to be motionless. The Earth and the Moon move under the influence of both the Sun and each other. While setting up the Lagrangian, think about whether you are using an appropriate number of generalized coordinates.

### (3 points)2. Series 32. Year - 1. moonmen

Your weight would be lower when the Moon is in zenith than when it is in nadir. About how much?

Matej hopes that he can build something easier

### (3 points)6. Series 31. Year - 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)5. Series 31. Year - 1. staircase on the Moon

If we once colonized the Moon, would it be appropriate to use stairs on it? Imagine the descending staircase on the Moon. The height of one stair is $h=15 \mathrm{cm}$ and it's length is $d=25 \mathrm{cm}$. Estimate the number $N$ of stairs that a person would fly over if he walked into the staircase with a velocity $v=5{,}4 \mathrm{km\cdot h^{-1}}=1{,}5 \mathrm{m\cdot s^{-1}}$. The gravitational acceleration on the Moon's surface is six times weaker than on Earth's surface.

Dodo read The Moon Is a Harsh Mistress.

### (9 points)4. Series 31. Year - P. Voyager II and Voyager I live!

We have a satellite and we want to launch it out of the Solar System. We launch it from Earth's orbit so that after some corrections of the trajectory it gets a velocity which is higher than the escape velocity from the Solar System. What is the probability that the satellite will collide with some cosmic material with higher diameter that $d=1 \mathrm{m}$ before leaving the Solar System.

Karel was wondering why NASA doesn't consider this possibility…

### (3 points)3. Series 31. Year - 2. small acceleration, large acceleration

In the figure, there is an ellipse with two focal points $F_1$ $F_2$ and several marked points on the ellipse. The ellipse represents a trajectory of one material point. Plot the accelerations the point experiences in given points of its trajectory. Show it in a figure. The direction and ratio of accelerations are important.

- There is a massive body in the focal point $F_1$. The material point is orbiting it, and Kepler's $2^{\rm nd}$ law applies.
- The absolute value of velocity of the material point is constant. It only moves along the ellipse.

### (3 points)6. Series 30. Year - 2. accidental drop

From what height would we need to „drop“ an object on a neutron star to make it land with a speed 0,1 $c$ (0,1 of speed of light). Our neutron star is 1.5 times heavier than our Sun and has diameter $d=10\;\mathrm{km}$. Ignore both the atmosphere of the star and its rotation. You can also ignore the correction for special relativity. However, do compare the results for a homogenous gravitational field (with the same strength as is on the star surface) and for a radial gravitational field.
**Bonus:** Do not ignore the special relativity correction.

Karel was thinking about neutron stars (yet again)

### (3 points)5. Series 30. Year - 1. space snowman

Consider a snowman consisting of 3 homogeneous spheres of density $ρ$ with centres on a line, floating in free space. The smallest sphere (the head) has radius $r$ and each consecutive sphere has twice the radius of the previous one. Our snowman is the only thing in the universe and it does not rotate in any way. Find the force holding the head to the rest of the snowman.

**Bonus:** Generalise the problem for $N\ge3$ spheres. Will the force converge to a finite value for $N→∞$ or will it go to infinity?

Karel came up with a problem for Fyziklani and realized he wouldn't want to be checking result.

### (12 points)1. Series 30. Year - E. Pechschnitte

Does bread always falls on the side that has the spread on it? Explore this Murphy's law experimentally with emphasis on statistics! Does it depend on the dimensions of the slice, or the composition and the thickness of the spread? Try to explain the experimental results with a theory. Use a sandwich bread.

Terka má stůl ve špatné výšce.