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## geometrical optics

### 1. Series 22. Year - 4. something for motorists

At dangerous curves and crossings you often see a convex mirror. It is distorting real picture, such as distances and velocities are unreal. Assume our distance from the mirror is $d$, distance of incoming car from the mirror is $L$, its real velocity is $v$ and mirror diameter is $R$.

Based on the image in mirror, how far do we see the car? What is its apparent speed? How different is time it enters the crossing, calculated based on data from image, compared to reality? Assume some concrete values for all parameters and evaluate, if this difference can be significant.

Při cestě na soustředění zažil Marek Scholz.

### 4. Series 21. Year - 4. save the kidney

Interpol just found that mafia has mobile laser weapons, which are guided from control room located in mountain village Obernieredorf. Control room is in maximum distance of 50 km from the weapons (at longer distance, the signal is unreliable). From control room they observe situation in Carlsbad, where all the weapons are aimed at.

Help: the innocent inhabitants of Carlsbad to find a shape and location of continuous mirror surface, which will reflect all laser rays in direction of control room. Solve problem in 2D (in plane), or in 3D, if there is a solution. Obviously, we require a proof of functionality, as we we do not want to invest money without a reason.

K oprášení znalostí a dovedností z geometrie zadal Pavel Brom.

### 6. Series 20. Year - 2. strange atmosphere

The refractive index of atmosphere of a planet of radius $R$ is can be describe by following equation: $n=n_{0}-αh$. Calculate, how high above the surface of the planet $h$ the laser beam will follow circular trajectory around the planet.

Nepoužitá úloha z archivu.

### 4. Series 20. Year - 2. plum wine in china

In our popular Chinese restaurant in Prague they give to each guest with the receipt small glass of plum wine. The wine is served in small ceramic bowl with double bottom (figure 1). The top bottom is glass and you can see picture of sitting girl (figure 2). When the glass is emptied the girl disappears (figure 3). Explain what happens. The empty bowl is in figure 4.

Vymyslel Honza po několikáté návštěvě zmíněné restaurace.

### 6. Series 19. Year - 4. Sun reflection

During sunny days we often experiment with square mirror giving reflections of Sun. Sometimes the reflection has LICHOBEZNIK shape, sometimes it is elliptical shape. What are the conditions for each case to happen? If possible, formulate your condition quantitatively.

Našel Matouš v sovětské sbírce.

### 3. Series 19. Year - 2. raid at a lens

Lets have a lens of focal length $f$. The light source is at optical axis in distance $a>f$ from the lens. The light source starts moving at constant speed towards the lens. Calculate the speed of movement of the image of the light source. Decide, if this speed can be bigger than the speed of light. Would it contradict special theory of relativity?

Vymyslel Jarda Trnka, když psal studijní text z optiky.

### 1. Series 18. Year - 4. water sprite

The water sprite (unterwasermann for German speakers) is sitting 1.5 meters under water level of his pond. How he sees the world above the water level? Assume, that refractive index of eye is same as refractive index of water.

Úloha ze sbírky prof. Vybírala.