To determine two characteristic “cosmic” velocities associated with the size and gravitational field of a certain planet. We will consider the planet to be one ball.
Rice. 5.8. Different trajectories of satellites around the Earth
First cosmic speed they call such a horizontally directed minimum speed at which a body could move around the Earth in a circular orbit, that is, turn into an artificial satellite of the Earth.
This, of course, is an idealization, firstly, the planet is not a ball, and secondly, if the planet has a sufficiently dense atmosphere, then such a satellite - even if it can be launched - will burn up very quickly. Another thing is that, say, an Earth satellite flying in the ionosphere at an average altitude above the surface of 200 km has an orbital radius that differs from the average radius of the Earth by only about 3%.
A satellite moving in a circular orbit with a radius (Fig. 5.9) is acted upon by the gravitational force of the Earth, giving it normal acceleration
Rice. 5.9. Movement of an artificial Earth satellite in a circular orbit
According to Newton's second law we have
If the satellite moves close to the Earth's surface, then
Therefore, for on Earth we get
It can be seen that it is really determined by the parameters of the planet: its radius and mass.
The period of revolution of a satellite around the Earth is
where is the radius of the satellite’s orbit, and is its orbital speed.
The minimum value of the orbital period is achieved when moving in an orbit whose radius is equal to the radius of the planet:
so the first escape velocity can be defined this way: the speed of a satellite in a circular orbit with a minimum period of revolution around the planet.
The orbital period increases with increasing orbital radius.
If the period of revolution of a satellite is equal to the period of revolution of the Earth around its axis and their directions of rotation coincide, and the orbit is located in the equatorial plane, then such a satellite is called geostationary.
A geostationary satellite constantly hangs over the same point on the Earth's surface (Fig. 5.10).
Rice. 5.10. Movement of a geostationary satellite
In order for a body to leave the sphere of gravity, that is, to move to such a distance where attraction to the Earth ceases to play a significant role, it is necessary second escape velocity(Fig. 5.11).
Second escape velocity they call the lowest speed that must be imparted to a body so that its orbit in the Earth’s gravitational field becomes parabolic, that is, so that the body can turn into a satellite of the Sun.
Rice. 5.11. Second escape velocity
In order for a body (in the absence of environmental resistance) to overcome gravity and go into outer space, it is necessary that the kinetic energy of the body on the surface of the planet be equal to (or exceed) the work done against the forces of gravity. Let's write the law of conservation of mechanical energy E such a body. On the surface of the planet, specifically the Earth
The speed will be minimal if the body is at rest at an infinite distance from the planet
Equating these two expressions, we get
whence for the second escape velocity we have
To impart the required speed (first or second cosmic speed) to the launched object, it is advantageous to use the linear speed of the Earth’s rotation, that is, launch it as close as possible to the equator, where this speed, as we have seen, is 463 m/s (more precisely 465.10 m/s ). In this case, the direction of launch must coincide with the direction of rotation of the Earth - from west to east. It is easy to calculate that in this way you can gain several percent in energy costs.
Depending on the initial speed imparted to the body at the throwing point A on the surface of the Earth, the following types of movement are possible (Fig. 5.8 and 5.12):
Rice. 5.12. Shapes of particle trajectory depending on throwing speed
The movement in the gravitational field of any other cosmic body, for example, the Sun, is calculated in exactly the same way. In order to overcome the gravitational force of the luminary and leave the solar system, an object at rest relative to the Sun and located from it at a distance equal to the radius of the earth's orbit (see above), must be given a minimum speed, determined from the equality
where, recall, is the radius of the Earth's orbit, and is the mass of the Sun.
This leads to a formula similar to the expression for the second escape velocity, where it is necessary to replace the mass of the Earth with the mass of the Sun and the radius of the Earth with the radius of the Earth’s orbit:
Let us emphasize that this is the minimum speed that must be given to a stationary body located in the Earth's orbit in order for it to overcome the gravity of the Sun.
Note also the connection
with the Earth's orbital speed. This connection, as it should be - the Earth is a satellite of the Sun, is the same as between the first and second cosmic velocities and .
In practice, we launch a rocket from the Earth, so it obviously participates in orbital motion around the Sun. As shown above, the Earth moves around the Sun at linear speed
It is advisable to launch the rocket in the direction of the Earth's movement around the Sun.
The speed that must be imparted to a body on Earth in order for it to leave the solar system forever is called third escape velocity .
The speed depends on the direction in which the spacecraft leaves the zone of gravity. At an optimal start, this speed is approximately = 6.6 km/s.
The origin of this number can also be understood from energy considerations. It would seem that it is enough to tell the rocket its speed relative to the Earth
in the direction of the Earth's movement around the Sun, and it will leave the solar system. But this would be correct if the Earth did not have its own gravitational field. The body must have such a speed having already moved away from the sphere of gravity. Therefore, calculating the third escape velocity is very similar to calculating the second escape velocity, but with an additional condition - a body at a great distance from the Earth must still have a speed:
In this equation, we can express the potential energy of a body on the surface of the Earth (the second term on the left side of the equation) in terms of the second escape velocity in accordance with the previously obtained formula for the second escape velocity
From here we find
Additional Information
http://www.plib.ru/library/book/14978.html - Sivukhin D.V. General course of physics, volume 1, Mechanics Ed. Science 1979 - pp. 325–332 (§61, 62): formulas for all cosmic velocities (including the third) were derived, problems about the motion of spacecraft were solved, Kepler's laws were derived from the law of universal gravitation.
http://kvant.mirror1.mccme.ru/1986/04/polet_k_solncu.html - Magazine “Kvant” - flight of a spacecraft to the Sun (A. Byalko).
http://kvant.mirror1.mccme.ru/1981/12/zvezdnaya_dinamika.html - Kvant magazine - stellar dynamics (A. Chernin).
http://www.plib.ru/library/book/17005.html - Strelkov S.P. Mechanics Ed. Science 1971 - pp. 138–143 (§§ 40, 41): viscous friction, Newton's law.
http://kvant.mirror1.mccme.ru/pdf/1997/06/kv0697sambelashvili.pdf - “Kvant” magazine - gravitational machine (A. Sambelashvili).
http://publ.lib.ru/ARCHIVES/B/""Bibliotechka_""Kvant""/_""Bibliotechka_""Kvant"".html#029 - A.V. Bialko "Our planet - Earth". Science 1983, ch. 1, paragraph 3, pp. 23–26 - provides a diagram of the position of the solar system in our galaxy, the direction and speed of movement of the Sun and the Galaxy relative to the cosmic microwave background radiation.
Ministry of Education and Science of the Russian Federation
State educational institution of higher professional education "St. Petersburg State University of Economics and Finance"
Department of Technology Systems and Commodity Science
Report on the course of the concept of modern natural science on the topic “Cosmic velocities”
Performed:
Checked:
Saint Petersburg
Cosmic speeds.
Space velocity (first v1, second v2, third v3 and fourth v4) is the minimum speed at which any body in free motion can:
v1 - become a satellite of a celestial body (that is, the ability to orbit around the NT and not fall on the surface of the NT).
v2 - overcome the gravitational attraction of a celestial body.
v3 - leave the solar system, overcoming the gravity of the Sun.
v4 - leave the Milky Way galaxy.
First escape velocity or Circular velocity V1- the speed that must be given to an object without an engine, neglecting the resistance of the atmosphere and the rotation of the planet, in order to put it into a circular orbit with a radius equal to the radius of the planet. In other words, the first escape velocity is the minimum speed at which a body moving horizontally above the surface of the planet will not fall on it, but will move in a circular orbit.
To calculate the first escape velocity, it is necessary to consider the equality of the centrifugal force and the gravitational force acting on an object in a circular orbit.
where m is the mass of the object, M is the mass of the planet, G is the gravitational constant (6.67259·10−11 m³·kg−1·s−2), is the first escape velocity, R is the radius of the planet. Substituting numerical values (for the Earth M = 5.97 1024 kg, R = 6,378 km), we find
7.9 km/sThe first escape velocity can be determined through the acceleration of gravity - since g = GM/R², then
Second escape velocity (parabolic velocity, escape velocity)- the lowest speed that must be given to an object (for example, a spacecraft), the mass of which is negligible relative to the mass of a celestial body (for example, a planet), in order to overcome the gravitational attraction of this celestial body. It is assumed that after a body acquires this speed, it does not receive non-gravitational acceleration (the engine is turned off, there is no atmosphere).
The second cosmic velocity is determined by the radius and mass of the celestial body, therefore it is different for each celestial body (for each planet) and is its characteristic. For the Earth, the second escape velocity is 11.2 km/s. A body that has such a speed near the Earth leaves the vicinity of the Earth and becomes a satellite of the Sun. For the Sun, the second escape velocity is 617.7 km/s.
The second escape velocity is called parabolic because bodies with a second escape velocity move along a parabola.
Derivation of the formula:
To obtain the formula for the second cosmic velocity, it is convenient to reverse the problem - ask what speed a body will receive on the surface of the planet if it falls onto it from infinity. Obviously, this is exactly the speed that must be given to a body on the surface of the planet in order to take it beyond the limits of its gravitational influence.
Let's write down the law of conservation of energy
where on the left are the kinetic and potential energies on the surface of the planet (potential energy is negative, since the reference point is taken at infinity), on the right is the same, but at infinity (a body at rest on the border of gravitational influence - the energy is zero). Here m is the mass of the test body, M is the mass of the planet, R is the radius of the planet, G is the gravitational constant, v2 is the second escape velocity.
Resolving with respect to v2, we get
There is a simple relationship between the first and second cosmic velocities:
Third escape velocity- the minimum required speed of a body without an engine, allowing it to overcome the gravity of the Sun and, as a result, go beyond the boundaries of the Solar system into interstellar space.
Taking off from the surface of the Earth and making the best use of the orbital motion of the planet, a spacecraft can reach a third of escape velocity already at 16.6 km/s relative to the Earth, and when launching from the Earth in the most unfavorable direction, it must be accelerated to 72.8 km/s. Here, for the calculation, it is assumed that the spacecraft acquires this speed immediately on the surface of the Earth and after that does not receive non-gravitational acceleration (the engines are turned off and there is no atmospheric resistance). With the most energetically favorable launch, the object’s speed should be co-directional with the speed of the Earth’s orbital motion around the Sun. The orbit of such a device in the Solar System is a parabola (the speed decreases to zero asymptotically).
Fourth cosmic speed- the minimum required speed of a body without an engine, allowing it to overcome the gravity of the Milky Way galaxy. The fourth escape velocity is not constant for all points of the Galaxy, but depends on the distance to the central mass (for our galaxy this is the object Sagittarius A*, a supermassive black hole). According to rough preliminary calculations, in the region of our Sun, the fourth cosmic speed is about 550 km/s. The value strongly depends not only (and not so much) on the distance to the center of the galaxy, but on the distribution of masses of matter throughout the Galaxy, about which there is no accurate data yet, due to the fact that visible matter makes up only a small part of the total gravitating mass, and the rest is hidden mass .
First cosmic velocity (circular velocity)- the minimum speed that must be given to an object in order to launch it into a geocentric orbit. In other words, the first escape velocity is the minimum speed at which a body moving horizontally above the surface of the planet will not fall on it, but will move in a circular orbit.
Computation and Comprehension
In an inertial reference frame, an object moving in a circular orbit around the Earth will be subject to only one force - the Earth's gravitational force. In this case, the movement of the object will be neither uniform nor uniformly accelerated. This happens because speed and acceleration (not scalar, but vector quantities) in this case do not satisfy the conditions of uniformity/uniform acceleration of movement - that is, movement with a constant (in magnitude and direction) speed/acceleration. Indeed, the velocity vector will be constantly directed tangentially to the surface of the Earth, and the acceleration vector will be perpendicular to it to the center of the Earth, while as they move along the orbit, these vectors will constantly change their direction. Therefore, in an inertial reference frame, such motion is often called “motion in a circular orbit with a constant modulo speed."
Often, for convenience, calculations of the first cosmic velocity proceed to considering this movement in a non-inertial reference frame - relative to the Earth. In this case, the object in orbit will be at rest, since two forces will act on it: centrifugal force and gravitational force. Accordingly, to calculate the first escape velocity, it is necessary to consider the equality of these forces.
More precisely, one force acts on the body - the force of gravity. Centrifugal force acts on the Earth. The centripetal force, calculated from the condition of rotational motion, is equal to the gravitational force. The speed is calculated based on the equality of these forces.
, ,
Where m- mass of the object, M- mass of the planet, G- gravitational constant, - first escape velocity, R- radius of the planet. Substituting numerical values (for Earth M= 5.97 10 24 kg, R= 6,371 km), we find
7.9 km/sThe first escape velocity can be determined through the acceleration of gravity. Because the , That
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An excerpt characterizing the first cosmic velocity
And he again turned to Pierre.“Sergei Kuzmich, from all sides,” he said, unbuttoning the top button of his vest.
Pierre smiled, but it was clear from his smile that he understood that it was not Sergei Kuzmich’s anecdote that interested Prince Vasily at that time; and Prince Vasily realized that Pierre understood this. Prince Vasily suddenly muttered something and left. It seemed to Pierre that even Prince Vasily was embarrassed. The sight of this old man of the world's embarrassment touched Pierre; he looked back at Helen - and she seemed embarrassed and said with her eyes: “Well, it’s your own fault.”
“I must inevitably step over it, but I can’t, I can’t,” thought Pierre, and he started talking again about an outsider, about Sergei Kuzmich, asking what the joke was, since he didn’t hear it. Helen answered with a smile that she didn’t know either.
When Prince Vasily entered the living room, the princess was quietly talking to the elderly lady about Pierre.
- Of course, c "est un parti tres brillant, mais le bonheur, ma chere... - Les Marieiages se font dans les cieux, [Of course, this is a very brilliant party, but happiness, my dear..." - Marriages are made in heaven,] - answered elderly lady.
Prince Vasily, as if not listening to the ladies, walked to the far corner and sat down on the sofa. He closed his eyes and seemed to be dozing. His head fell and he woke up.
“Aline,” he said to his wife, “allez voir ce qu"ils font. [Alina, look what they are doing.]
The princess went to the door, walked past it with a significant, indifferent look and looked into the living room. Pierre and Helene also sat and talked.
“Everything is the same,” she answered her husband.
Prince Vasily frowned, wrinkled his mouth to the side, his cheeks jumped with his characteristic unpleasant, rude expression; He shook himself, stood up, threw his head back and with decisive steps, past the ladies, walked into the small living room. With quick steps, he joyfully approached Pierre. The prince's face was so unusually solemn that Pierre stood up in fear when he saw him.
- God bless! - he said. - My wife told me everything! “He hugged Pierre with one hand and his daughter with the other. - My friend Lelya! I'm very, very happy. – His voice trembled. – I loved your father... and she will be a good wife for you... God bless you!...
He hugged his daughter, then Pierre again and kissed him with a foul-smelling mouth. Tears actually wet his cheeks.
“Princess, come here,” he shouted.
The princess came out and cried too. The elderly lady was also wiping herself with a handkerchief. Pierre was kissed, and he kissed the hand of the beautiful Helene several times. After a while they were left alone again.
“All this had to be this way and could not have been otherwise,” thought Pierre, “so there is no point in asking whether it is good or bad? Good, because definitely, and there is no previous painful doubt.” Pierre silently held his bride's hand and looked at her beautiful breasts rising and falling.
“Uniform and uneven movement” - t 2. Uneven movement. Yablonevka. L 1. Uniform and. L2. t 1. L3. Chistoozernoe. t 3. Uniform movement. =.
“Curvilinear motion” - Centripetal acceleration. UNIFORM MOTION OF A BODY AROUND A CIRCLE There are: - curvilinear motion with a constant velocity; - movement with acceleration, because speed changes direction. Direction of centripetal acceleration and velocity. Motion of a point in a circle. Movement of a body in a circle with a constant absolute speed.
“Motion of bodies on a plane” - Evaluate the obtained values of unknown quantities. Substitute numerical data into a general solution and perform calculations. Make a drawing, depicting interacting bodies on it. Perform an analysis of the interaction of bodies. Ftr. Movement of a body along an inclined plane without friction. Study of the movement of a body on an inclined plane.
“Support and movement” - An ambulance brought a patient to us. Slender, stooped, strong, strong, fat, clumsy, dexterous, pale. Game situation “Concilium of doctors”. Sleep on a hard bed with a low pillow. “Body support and movement. Rules for maintaining correct posture. Correct posture when standing. Children's bones are soft and elastic.
"Space Speed" - V1. THE USSR. That's why. April 12, 1961 Message to extraterrestrial civilizations. Third escape velocity. On board Voyager 2 is a disk with scientific information. Calculation of the first escape velocity at the Earth's surface. The first manned flight into space. Voyager 1 trajectory. The trajectory of bodies moving at low speed.
“Body dynamics” - What underlies dynamics? Dynamics is a branch of mechanics that examines the causes of the movement of bodies (material points). Newton's laws apply only to inertial frames of reference. Frames of reference in which Newton's first law is satisfied are called inertial. Dynamics. In what frames of reference do Newton's laws apply?
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Of our planet. The object will move unevenly and unevenly accelerated. This happens because the acceleration and speed in this case will not satisfy the conditions with a constant speed/acceleration in direction and magnitude. These two vectors (velocity and acceleration) will constantly change their direction as they move along the orbit. Therefore, such movement is sometimes called movement at a constant speed in a circular orbit.
The first cosmic speed is the speed that must be given to a body in order to put it into a circular orbit. At the same time, it will become similar. In other words, the first cosmic speed is the speed at which a body moving above the Earth’s surface will not fall on it, but will continue to move in orbit.
For ease of calculation, we can consider this motion as occurring in a non-inertial reference frame. Then the body in orbit can be considered to be at rest, since two gravity will act on it. Consequently, the first will be calculated based on considering the equality of these two forces.
It is calculated according to a certain formula, which takes into account the mass of the planet, the mass of the body, and the gravitational constant. Substituting the known values into a certain formula, we get: the first cosmic speed is 7.9 kilometers per second.
In addition to the first cosmic speed, there are second and third speeds. Each of the cosmic velocities is calculated using certain formulas and is interpreted physically as the speed at which any body launched from the surface of planet Earth becomes either an artificial satellite (this will happen when the first cosmic velocity is reached) or leaves the Earth’s gravitational field (this happens when it reaches the second cosmic velocity), or will leave the Solar system, overcoming the gravity of the Sun (this happens at the third cosmic velocity).
Having gained a speed of 11.18 kilometers per second (the second cosmic speed), it can fly towards the planets in the solar system: Venus, Mars, Mercury, Saturn, Jupiter, Neptune, Uranus. But to achieve any of them, their movement must be taken into account.
Previously, scientists believed that the motion of the planets was uniform and occurred in a circle. And only I. Kepler established the real shape of their orbits and the pattern according to which the speeds of movement of celestial bodies change as they rotate around the Sun.
The concept of cosmic velocity (first, second or third) is used when calculating the movement of an artificial body in any planet or its natural satellite, as well as the Sun. This way you can determine the escape velocity, for example, for the Moon, Venus, Mercury and other celestial bodies. These speeds must be calculated using formulas that take into account the mass of the celestial body, the gravitational force of which must be overcome
The third cosmic one can be determined based on the condition that the spacecraft must have a parabolic trajectory of motion in relation to the Sun. To do this, during launch at the surface of the Earth and at an altitude of about two hundred kilometers, its speed should be approximately 16.6 kilometers per second.
Accordingly, cosmic velocities can also be calculated for the surfaces of other planets and their satellites. So, for example, for the Moon, the first cosmic one will be 1.68 kilometers per second, the second - 2.38 kilometers per second. The second escape velocity for Mars and Venus, respectively, is 5.0 kilometers per second and 10.4 kilometers per second.