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## mechanics of a point mass

### 1. Series 19. Year - 2. train carriage

Imagine special anti-hooligan railway carriage equipped with a water cannon. The carriage weight is 30 t. Police (on the left) sprinkled hooligans with 1000 litres of water over 1 minute. How far had the carriage moved, if the wagon length is 30 m?

Assume that the carriage has brake off and that the water can escape only in vertical direction. The change in mass of carriage caused by escaped water can be neglected.

Zážitek Honzy Prachaře, když se vracel vlakem domů.

### 6. Series 18. Year - 3. space probe from NASA

In Jet Propulsion Laboratory in California, U.S.A. in NASA laboratory the new rocket engine is under development. It uses momentum of $α-particles$ created during radioactive decay of fermium $^{257}_{100}Fm_{157}$, which mass is $m_{Fm}$ and half-life $T$. The second product is californium $^{253}_{98}Cf_{155}$. The mass of $α-particle$ is $m_{α}$, the mass of californium is $m_{Cf}$, and during the decay the energy $E$ is released. Assume, that each $α-particle$ leaves rocket in the same direction.

The space probe with above engine is in rest at the beginning and its mass is $M$, the mass of 'fuel' is also $M$. Calculate the speed of the probe $v$ after half of the fermium decays. Resulting speed calculate also for the following numerical values $E=1,106\cdot 10^{-12}J$, $M=4\;\mathrm{kg}$ a $T=100,5days$, for other values consult your table-book.

SR olympiáda.

### 6. Series 18. Year - S. Hamilton formalism

Lagrangian of a particle in electromagnetic field is

$L=\frac{1}{2}mv-qφ+q\textbf{v}\cdot \textbf{A}=\frac{1}{2}\;\mathrm{m}\cdot \sum_{i=1}^{3}v_{i}-qφ+q\cdot \sum_{i=1}^{3}v_{i}A_{i}$,

where $φ$ is electrical potential and $\textbf{A}$ is magnetic vector potential.

- Calculate generalized momentum of the particle $p_{i}$ belonging to the speed $v_{i}$.
- Write Hamiltonian function (in variables ($x_{i}$, p$_{i})!)$.
- Solve Hamiltonian equation, if when $\textbf{A}=**0**$ and $φ=-Ex_{1}$.

Zadal Honza Prachař.

### 5. Series 18. Year - 2. falling from the stairs

Karel is playing with a ball. While rolling it on the floor is comes to he inclined plane, which serves as a staircase, and starts to slide down. The ball is moving in such direction, that the vector of its velocity $\textbf{v}$ and the top edge of the inclined plane shows and angle $φ$. Calculate a vector of the velocity $\textbf{v}′$ of the ball under the inclined plane (its magnitude and the direction), if the height of the plane is $h$. The friction is negligible, assume that the top edge is smooth so the ball will always follow the surface.

As a bonus: what is the difference of the direction of the ball falling into a cylindrical hole of radius $R$ and the depth $h$ with inclined sides (see figure 1). The length of the inclined wall can be neglected with respect to the overall size of the hole.

Napadlo Matouše.

### 5. Series 18. Year - S. Mercury, the pit and the pendulum

The following questions will test the knowledge from all presented chapters about mechanics – Newtons formalism, D'Alembert's principle and Lagrange's formalism.

- Imagine planet Mercury orbiting around Sun. It is know, that its elliptic trajectory is rotating, the position of perihelion is moving, which cannot be explained by gravitation force.

$\textbf{F}=κ(mM\textbf{r})⁄r^{3}$.

Proof, that adding an additional central force

$\textbf{F}=C(\textbf{r})⁄r^{4}$,

where $C$ is suitable constant, full trajectory (ellipse) will rotate at constant angular speed. In other words, that exists a frame rotating at constant speed, where the trajectory is an ellipse. Knowing this angular speed $Ω$, calculate the constant $C$. Is such correction for gravitation enough?

- Calculate equilibrium position of homogeneous rod of length $l$ supported by inner wall of excavation in the V-shape (see figure 12) as a function of the angle of V-shape $α$.
- Using Lagrange's equations calculate period of small oscillations of double-reverse pendulum in image 13. The weight are at the ends of weightless rod of the length $l$ and have masses $m_{1}$ and $m_{2}$, the distance from the joint from the weight $m_{1}$ is $l_{0}$.

a)Na úlohu narazil Matouš v jedné pěkné ruské knize. b), c) Zadal Honza Prachař a Jarda Trnka.

### 4. Series 18. Year - 1. nuclear attack in 1985

One winter afternoon the soviet generals were out of patience while watching imperialistic West and pressed The Red Button to fire nuclear bomb. Immediately after that the young lieutenant entered the room admitting the error in calculation of the rocket's trajectory. Instead to the New York the racket was aiming to the friendly Cuba.

Luckily enough, another rocket is ready and can be sent to shoot down the first one and avoid disturbance in between socialistic countries. The original rocket was fired at the speed $v$ under the angle $α$. What angle $β$ should be set for the second rocket to shoot down the first rocket if the time delay between the launches is $T$.

Discuss when the peace between socialistic countries can be saved and when not. And of course everyone knows, that the Earth is flat and the gravitation field is homogeneous.

Navrhl Jarda Trnka.

### 3. Series 18. Year - 2. bay watch

The lifeguard (plavcik) is standing in distance $D$ from the beach. He suddenly sees drowning blond girl (blondynka) which is in distance $D$ in sea (see fig. 1). The lifeguard can run at maximum speed $v$ and swim at maximum speed $v/2$. The distance from the lifeguard to the beach end is defined by following equation

$$d(\phi) = \frac{D}{3}( 8\cos{\phi}- {2\sqrt{16\cos^2{\phi}-12\cos{\phi} -3}}-3)\,,$$

where φ is angle blond-lifeguard-beach. What is the optimum trajectory for the lifeguard to safe her?

### 3. Series 18. Year - S. Langrange's equations first type

- Lets have a mass point suspended on massless string. Introduce Cartesian coordinates and write down equation for the mass point.

- Write Lagrange's equations of first type for mass point from part a). Show, that they are equivalent with the equation for mathematical pendulum

d^{2}$\varphi/dt^{2}$ + $g/l$ \cdot sin $\varphi$ = 0,

where $\varphi$ is angular displacement from equilibrium.

- Small body is in rest at the top of the hemisphere and starts to slide down. Using Lagrange's equations for first type calculate the height when the body take-off the hemisphere. (

**Hint:** The body takes-off when the $λ$ = 0.)

Autoři seriálu.

### 2. Series 18. Year - 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.

### 2. Series 18. Year - 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.