# Series 2, Year 18

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### 1. Moses's miracle

Moses came to the Red Sea and said: „Lets the water open and let us go by dry foot to the land of promise.“ Then he entered into waves and they opened. What was the Moses's force, if he moved Jews over Red Sea. Assuming the sea to be 1 wide and 20 deep.

Vymyslel Jarda Trnka při čtení Bible.

### 2. how many wires at the poles?

How many phases there have to be to allow the effective voltage between phase and ground will be the same as between two neighbouring phases?

Úlohu navrhl Pavel Augustinský.

### 3. helicopter

For helicopter to levitate it needs motor of power P. What is the power P' of the helicopter, which is scale copy of the previous one in scale 1:2? Assume the effectiveness of the rotor to be 100%.

Úloha byla převzata z MFO v Kanadě.

### 4. desperate shipwrecked people

Shipwrecked people at the north pole are trying to make a cup of coffee. Advice them, how to boil the water, to get as much as possible if there are only 3 ways how to boil it:

- Re-chargeable battery of internal resistance $2R$ is connected directly to the heating element of resistance $R$.
- The same battery is connected in series in with heating element and capacitor. Each time, when the capacitor is fully charged, it is disconnected and connected into the circuit in reverse polarity.
- The same battery will be used to charge capacitor and then the capacitor powers up by the heating element.

Vymyslel Matouš Ringel, když si na výletě vařil kávu.

### P. unexpected obstacle

The driver of the car moving at the speed $v$ suddenly recognise, that is heading to the middle of the concrete wall of the width of $2d$ and is in distance of $l$ from the wall. The coefficient of the friction between the tyres and the surface of road is $f$. What is the best way to do to avoid the inevitable accident. Decide, what is the maximum velocity to avoid the crash.

Napadlo Pavla Augustinského při cestě autem.

### E. a mass is not a mass

Check experimentally the equivalence of inertial mass (the mass in second Newton's law) and gravitational mass (the one in Newton's gravitational law).

Vymyslel Jarda Trnka na přednášce z relativity.

### S. Newton's kinematics equations

- Write down and solve the kinematics equation for mass point in gravitation field of the Earth. The orientation of the coordinate system make that $x$ and $y$ are horizontal and z is vertical, pointing upwards. The starting position is $\textbf{r}_{0} = (0,0$,$h)$, starting velocity is $\textbf{v}_{0} =(v_{0}\cosα,0,v_{0}\sinα)$.
- The man with the gun sits in the chair rotating alongside vertical axe at frequency $f=1\;\mathrm{Hz}$. With the chair also the target is rotating (it is fixed to the chair). Then the man shoots the bullet at the speed of $v=300\;\mathrm{km} \cdot \mathrm{h}^{-1}$ from the rotational axes directly to the middle of the target. In what place the bullet is going to go through the target. Solve in non-inertial system and from the inertial system. The distance to the middle of the target from the centre of rotation is $l=3\;\mathrm{m}$, the air friction is negligible.

- State the dependence of the speed of the mass point at its position in gravitational field of the Sun.

Zadal Honza Prachař.