# 2. Series 32. Year

* Post deadline: 12th November 2018*

* Upload deadline: 13th November 2018 11:59:59 PM*

### (3 points)1. moonmen

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

### (3 points)2. Finnish sauna

Imagine that Dan has a sauna with dimensions $2,5 \mathrm{m}$ x $3 \mathrm{m}$ x $4 \mathrm{m}$ with a relative humidity of $20 \mathrm{\%}$ and temperature of $90 \mathrm{\C }$. How much water would have to evaporate, so the relative humidity inside the sauna is $35 \mathrm{\%}$? The water evaporates inside the sauna without changing the overall temperature.

### (6 points)3. physics trophy

Danka won the annual Derivative Bee and she obtained a statuette made of transparent material as a reward. This statuette is made in shape of a cube prism with an edge of $a = 5$ cm and height of $h \leq a$. No matter what angle she looks at the prism, she can only see the reflection on the side walls but not through it. What is the index of refraction of the material? The prism is placed in air.

### (7 points)4. lunar lander

How can the electronics of the Apollo landing module control an engine thrust $T$ (and so regulate the consumption of fuel), so the rocket floats onto the surface of the Moon at a steady linear motion? The effective velocity of exhaust gases is $u$. The rocket has already slowed down its motion on an orbit and goes straight down in a homogeneous gravitational field with an acceleration $g$. The initial weight of the module is $m_0$.

**Bonus:** How can the electronics of Apollo landing module control the engine thrust during landing from a height $h$ and initial velocity $v_0$, so the landing is so-called fall from null height and the consumption of the fuel minimalizes? Maximum engine thrust is $T\_{max}$.

### (9 points)5. bird on the pulley

A fixed pulley is attached to the ceiling and a rope hangs over it, so the left and right end are at the same height. On one end of the rope hangs a Fykosak bird and on the other end hangs a mass, both equally heavy. Describe what happens with the system when the bird starts climbing up (on his own side of rope) with a constant force. In the beginning, assume that the rope is weightless and the pulley is ideal. Afterwards, solve this problem for a real pulley with the following parameters, its length $l$, the moment of inertia of the pulley $I$ and pulley's radius $r$. The rope's mass per unit length is $\lambda$. Assume that the rope doesn’t slip on the pulley.

### (10 points)P.

Create an accurate weather forecast for address V Holešovičkách 2, Prague 8, for Wednesday 14th of November from 12:00 to 15:00. How will the weather change throughout the whole day? You are allowed to use previous data about the weather in this area (remember you are only permitted to use data until 10th of November). It is necessary to justify your weather prediction, write down references and ideally to use as many data and resources as possible.

### (12 points)E.

Measure an average vertical velocity of falling leaves. Use leaves from several different trees and discuss what impact the shape of a leaf has on the velocity. How should an ideal leaf look like when we want it to fall as slow as it is possible?

### (10 points)S. theoretical mechanics

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