A light spring with stiffness coefficient $k$ is attached at one end to a vertical axis around which it can rotate freely, and at the other end to a small weight of mass $m$. The whole system is on a horizontal smooth table, the spring is not stretched. A shock imparts to the weight a velocity $v_{0}$ directed perpendicular to the spring. Find the minimum and maximum distance from the weight to the axis if the velocity of the weight at the maximum distance from the axis is $v_{1}$.

What energy must be given to a spacecraft of mass 1 t in order for it to enter a circular orbit with radius $R = 16000 km$ after starting from the Earth's surface? Consider that the potential energy of a body in the Earth's gravity field at a distance from the Earth's center $R \geq R_{earth}$ is $E_{n} = - \frac{GmM_{earth}}{R}$, where $M_{z}$ is the Earth's mass.

A particle of mass $m$ moves under the action of some force along a circular orbit of radius $R$ with constant velocity. The potential energy $U$ of the particle in the field of this force depends only on the distance to the center of the orbit as follows:
a) $U(R) = kR (k > 0)$;
b) $U(R)=kR^{2} \: (k>0)$.
Find the velocity of the particle in cases a) and b).

Compare the potential energies of a body on the surface of the Earth and on the surface of the Moon, Accept that the acceleration of free fall on the Moon is six times less than on the Earth, and the radius of the Moon is three times less than the radius of the Earth.

A satellite flies around the earth in a circular orbit with radius $R = 25600 km$. How many times is the satellite's speed different from the first space velocity?

A small ball of mass m, attached to a spring with stiffness coefficient $k$, makes harmonic oscillations in the direction of the spring axis with amplitude $A$. When the spring was compressed, a second spring pendulum of the same mass is placed in the path of the ball, both springs having a common axis and the equilibrium positions of the balls coinciding. Describe the further motion of the system in the following cases:
(a) The shock is central and perfectly elastic;
b) the impact is central and absolutely inelastic (i.e. the balls "stick together").
At the initial moment the second ball rests in the equilibrium position.

In a deep vertical shaft underground there are two weights of equal mass connected by a weightless non-stretchable thread, which is thrown over a fixed block. Find the period of vertical oscillation of the weights if the radius of the Earth is 6400 km, the acceleration of free fall at the surface is $10 m/s^{2}$.

A projectile fired from a cannon mounted at an angle of $45^{\circ}$ to the horizon on a flat horizontal plain breaks at the upper point of its trajectory into two fragments of equal mass. The first fragment falls to the ground directly below the point of bursting 15 s later. At what distance from the cannon will the second fragment fall if 15 s elapsed from the moment of firing to the moment of rupture? Neglect air resistance.