# Series 1, Year 21

### 1. movement behind the window

Investigate following phenomenon: while in a moving train looking out of window, the objects at horizons do not move too fast. But objects close to window (e.g. telegraph poles) move extremely fast behind the window. How is this apparent speed related to the distance from the window?

Cestou domů napadla úloha Tomáše Jirotku.

### 2. save the bubble!

Batyscaphe Trieste dived into deep sea and released a bubble which started to come to the surface. What is the vertical speed of the bubble? How will this speed change? How long it will take to reach surface? What size of bubbles will be fastest?

Úlohu vymyslel Jano Lalinský.

### 3. weighting the Sun

Suggest several methods for estimating the mass of the Sun. Explain in detail each of them and estimate mass of the Sun.

K zahřátí mozků do nového ročníku FYKOSu zadal Pavel Brom.

### 4. save the beer

A truck traveling at constant speed $v$ is carrying bottles of beer. The driver suddenly discovers a dangerous curve on the road in distance $d$ in front of him. The radius of curvature is $R$. What is the ideal tactic for slowing down, if the number of broken bottles is proportional to the maximum acceleration and you want this number minimised?

Vymyslel Marek Pechal při jízdě narvanou 112kou.

### P. catch the donkey

Lets assume that you own naughty donkey, which likes to jump over fence to visit your neighbours. To stop him you have bought higher fence and are planning to erect higher fence around the perimeter of your land. However, the place for fence is on a inclined plane and therefore the situation is little bit more complicated. What would be the best angle to erect the fence to make it most difficult for your donkey to jump over?

Napadlo Honzu Prachaře na návštěvě příbuzných majících podobný problém.

Measure the smallest movement which can be registered by human eye. To be precise, measure minimal angular velocity of an object relatively to the background, which your constantly open eye can detect within 5 seconds.

Several tips for slow motion: snail movement, movement of Sun against horizon during sunset and sunrise, movement of hands on your watch, growth of flowers, growth of animals, movement of stars…

### S. gravitation

Download program Planeta.pas, ThreeStars-TriHvezdy.pas. As an excerscice try to understand the programs, modify them and change them.

• First task is to add into at least two of programs something new. For example change origin conditions, add another planet or star. Also try to change gravitational law – calculate with force $F=A⁄R+B⁄R$, where $A$ and $B$ are constants etc.
• Assume two same mass stars, which orbit around each other on circle. In axis of this circle the third star is approaching, at the beginning with the same speed as the two stars are orbiting. It has also the same mass. Create simulation what will happen.

As solution of both problems please send us pictures with extensive commentary. State at least short explanation what is at the images. Also mention how have you proceed during calculations and which computation system have you used.

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