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Two point-like masses were jumping on a trampoline into height $h_0 = 2 \mathrm{m}$. While they both were in the lowest point of the trajectory (corresponding displacement of $y = 160 \mathrm{cm}$), one of them suddenly disappeared. What is the maximum height, which was the other point-like mass bounced into? A round trampoline has circumference of $o = 10 \mathrm{m}$ and is held by $N = 42$ springs with stiffness $k = 1720 \mathrm{N\cdot m^{-1}}$. Trampoline may be modelled by $N$ springs uniformly attached around the circle and connected in the middle. Mass of the disappeared mass is $M = 400 \mathrm{kg}$.

1. Show that in an arbitrary central-force field, i.e. a force field where the potential only depends on distance (not on angular position), a particle will always move in a plane. Instructions:: Set up Lagrangian equations of the second kind for this situation using appropriate generalized coordinates.. Then, set the coordinate $\theta = \pi /2$ and initial velocity in the direction of this coordinate equal to zero. Think about and explain why this choice of coordinates does not cause a loss of generality.
2. Set up the Lagrangian for a mass point moving in a plane in a central-force field. Find all the integrals of motion for this Lagrangian and use them to find the first orded differential equation for the variable $r$.
3. Think about how to find the angular distance between two points on a sphere, given their spherical coordinates. Check your solution on the stars Betelgeuse and Sirius. Hint:: This problem can be easily solved even without the knowledge of spherical trigonometry.

Consider a point-like particle in one-dimensional space. Initially, the particle is in rest at the origin of coordinates. It can be moved with acceleration from interval $\left (- a , a\right )$. Let $M\left (t\right )$ be a set of all possible physical states $\left (x, v\right )$ of positions $x$ and velocities $v$, which particle can achieve after time $t$ is elapsed. If we plot set $M\left (t\right )$ into $v(x)$ coordinate system we get surface $S\left (t\right )$. Find analytic expression for boundaries of $S\left (t\right )$.

Bonus: Find area of $S\left (t\right )$ as a funcion of time.}

1. Suppose we have a horizontal plane with a small hole. Through this hole goes a rope with length $l$ on which a weight of mass $M$ is hung. You may consider the weight to be a mass point. One the other end of the rope there is a second mass point with mass $m$. The rope between them is stretched thanks to the weight of mass $M$. Initially, the whole setup is in rest while the part of the rope below the plane is vertical. Then we grant the mass point on the plane velocity $v$ in a horizontal direction perpendicular to the rope as we let the system go free. Neglect all friction in this problem. Choose appropriate coordinates and find the Lagrangian for this situation.
   
2. Suppose we have an iron rod bent to a shape of a parabola given by the equation $y = x^2$. The gravity of Earth points in the negative direction of the $y$ axis. A mass point of mass $M$ can move freely along the parabola. A second mass point with mass $m$ is connected to the first by a rigid rod of length $l$. This way we have created a pendulum with a hinge sliding along the rod. The system can move only in the plane of the parabola. Find appropriate generalized coordinates and the Lagrangian for this situation.
   
3. Suppose we have line along which slides a mass point with mass $m$ (without friction). The angle between the line and the horizontal plane is $\alpha $. Find appropriate generalized coordinates and the Lagrangian for this situation. Then set up Lagrangian equations of the second kind, double-integrate them and find the solution. Do not forget about the constants of integration and explain their physical meaning. What will be their values if the mass point starts at rest at the height $h$?

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  $T\_{max}$.

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.

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?

1. Suppose we have a dumbbell consisting of two mass points with masses $m$ and $M$ connected via a massless rod. This dumbbell is in a free fall. Write a constraint function and Lagrangian equations of the first kind for this object.
   
2. Suppose we have a triangular prism with mass $M$ on a horizontal platform as in the picture. A mass point with the mass $m$ is sliding down a side of the prism. The angle between said side and the platform is $\alpha $. You may neglect friction.
   

• Set up Lagrangian equations of the first kind for this situation.
• Show that, for zero initial speed of the mass point, the total momentum of this system in the direction of $x$ axis is zero.
• Solve the system of (Lagrangian) equations and find the time-dependent equations for the speeds of the prism and the mass point.
• Find the ratio between these two speeds.

1. Set up Lagrangian equations of the first kind for a simple pendulum. Show that the law of conservation of energy holds for this situation.

Jachym was launching fireworks. We can imagine it as a beacon, that is at some point shot straight up with velocity $v$, and explodes after a certain delay. Jachym was standing a distance $x$ from the launch site when he heard the launch of the fireworks. After a delay $t_1$ he saw the explosion and after another delay $t_2$ he heard the explosion. Calculate the velocity $v$.

When James Bond let go of agent 006 Alec Treveljan from the top of the Arecibo radiotelescope in the final scene of the film Golden Eye, the falling agent started screaming with a frequency $f$. How does the frequency agent 007 hears at the top of the telescope change as a function of time. Neglect air resistance.

Hint: Ask Mr. Doppler