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## mechanics of a point mass

### 3. Series 21. Year - 3. jumping on inclined plane

A small ball is thrown in horizontal direction onto a inclined plane. The ball starts to jump on the plane and after $Ncontacts$ it falls at right angle onto the plane. An example of such trajectory for $N=4$ is in figure. What is the inclination angle of the plane $α?$ Assume, the ball bounce elastically, do not assume rotation.

Pavel Motloch.

### 2. Series 21. Year - 1. a spit

You travel in fast train and are looking outside of the open window. Three windows in front of you someone spits a chewing-gum. How long do you have to get back to coupe to avoid contact with chewing-gum? The chewing-gum is spherical shape and was not thrown out, but just laid in air flow.

Roman Fiala.

### 2. Series 21. Year - 2. a car in the rain

Calculate a slope of front glass of a car, so that water drops at speed 80 km ⁄ h do not run off, but to the sides. Verify that you results is compatible with reality. What else influence the slope of front window?

Nad problémem se zamýšlel Honza Prachař při jízdě autem během průtrže.

### 1. Series 21. Year - 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.

### 1. Series 21. Year - 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.

### 6. Series 19. Year - 1. climbing a bump

A carriage of mass $m$ travels in horizontal direction at speed $\textbf{v}$. In front of him is a „wooden bump“ of mass $M$ and height $h$, which slides on the horizontal plane without friction (figure 1). The carriage goes onto the bump. What conditions must be fulfilled for carriage to go over the bump? What will be the final velocity of the bump?

Našel Matouš v sovětské sbírce.

### 6. Series 19. Year - 2. cuckoo-clock on the string

Cuckoo-clock of mass $M$ are fixed at two long parallel stings (figure 2). Pendulum consists of small mass $m$ and light rod of length $l$. Calculate, how much will such clock speed up comparing to the clock fixed on the wall.

Našel Matouš v sovětské sbírce.

### 6. Series 19. Year - 3. spinning the electrical motor

On the axle of motor the sting is winded up. At the end of the string a small mass $m$ is fixed. If the motor is connected to the ideal voltage source of voltage $U$, small mass travels up at speed $v_{1}$. What speed will the small mass reach, if the voltage source is disconnected and the motor contacts are connected (shorted)? Assume friction-less devices.

Našel Matouš v sovětské sbírce.

### 4. Series 19. Year - 1. competition of balloons

*This series of questions is dedicated to the research on the „planet of the balloons“*.

This year balloons are competing in the 'The higher, the better' competition. Each balloon has a piece of string attached to measure his height. All balloons have the same parameters and noone of them have won yet.

The length density of string is 11 lufts per sprungl, density of atmosphere is 110101 lufts per cubic sprungl and the radius of each balloon is 10 sprungls and weight of balloon is 10 lufts. Every object in gravitation field of the planet increases its speed by 111 sprungls per temp. Calculate the maximum height which the referees will measure and how the balloon will move after reaching this height. Unlifted part of the string is laying freely on the ground. The competition happens at low altitudes, where density of atmosphere is approximately constant.

**Hint:** Sprungl, luft and temp are units used on planet of the balloons. Each balloon has maximum of 1 string attached.

Úlohu navrhl Petr Sýkora od Havránka.

### 2. Series 19. Year - 1. pen suspended on string

In tram at rest the pen is suspended on a sting of length $l$. The mass of the pen is $m$. The tram accelerates with constant acceleration $a$. Calculate the maximum angle of displacement of the pendulum (maximum angle between the string and the vertical direction) and the time when it crosses the starting position.

1. ročník 3.kolo