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.
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.
Since ancient times, people have been interested in the problem of the structure of the world. Back in the 3rd century BC, the Greek philosopher Aristarchus of Samos expressed the idea that the Earth revolves around the Sun, and tried to calculate the distances and sizes of the Sun and Earth from the position of the Moon. Since the evidential apparatus of Aristarchus of Samos was imperfect, the majority remained supporters of the Pythagorean geocentric system of the world.
Almost two millennia passed, and the Polish astronomer Nicolaus Copernicus became interested in the idea of a heliocentric structure of the world. He died in 1543, and soon his life's work was published by his students. Copernicus' model and tables of the positions of celestial bodies, based on the heliocentric system, reflected the state of affairs much more accurately.
Half a century later, the German mathematician Johannes Kepler, using the meticulous notes of the Danish astronomer Tycho Brahe on observations of celestial bodies, derived the laws of planetary motion that eliminated the inaccuracies of the Copernican model.
The end of the 17th century was marked by the works of the great English scientist Isaac Newton. Newton's laws of mechanics and universal gravitation expanded and gave theoretical justification to the formulas derived from Kepler's observations.
Finally, in 1921, Albert Einstein proposed the general theory of relativity, which most accurately describes the mechanics of celestial bodies at the present time. Newton's formulas of classical mechanics and the theory of gravity can still be used for some calculations that do not require great accuracy, and where relativistic effects can be neglected.
Thanks to Newton and his predecessors, we can calculate:
- what speed must the body have to maintain a given orbit ( first escape velocity)
- at what speed must a body move in order for it to overcome the gravity of the planet and become a satellite of the star ( second escape velocity)
- the minimum required speed for leaving the planetary system ( third escape velocity)
Any object, being thrown up, sooner or later ends up on the earth's surface, be it a stone, a sheet of paper or a simple feather. At the same time, a satellite launched into space half a century ago, a space station or the Moon continue to rotate in their orbits, as if they were not affected by our planet at all. Why is this happening? Why is the Moon not in danger of falling to the Earth, and why is the Earth not moving towards the Sun? Are they really not affected by universal gravity?
From the school physics course we know that universal gravity affects any material body. Then it would be logical to assume that there is some force that neutralizes the effect of gravity. This force is usually called centrifugal. Its effect can be easily felt by tying a small weight to one end of the thread and untwisting it in a circle. Moreover, the higher the rotation speed, the stronger the tension of the thread, and the slower we rotate the load, the greater the likelihood that it will fall down.
Thus, we are very close to the concept of “cosmic velocity”. In a nutshell, it can be described as the speed that allows any object to overcome the gravity of a celestial body. The role can be a planet, its or another system. Every object that moves in orbit has escape velocity. By the way, the size and shape of the orbit depend on the magnitude and direction of the speed that the given object received at the time the engines were turned off, and the altitude at which this event occurred.
There are four types of escape velocity. The smallest of them is the first. This is the lowest speed it must have for it to enter a circular orbit. Its value can be determined by the following formula:
V1=õ/r, where
µ - geocentric gravitational constant (µ = 398603 * 10(9) m3/s2);
r is the distance from the launch point to the center of the Earth.
Due to the fact that the shape of our planet is not a perfect sphere (at the poles it seems to be slightly flattened), the distance from the center to the surface is greatest at the equator - 6378.1. 10(3) m, and the least at the poles - 6356.8. 10(3) m. If we take the average value - 6371. 10(3) m, then we get V1 equal to 7.91 km/s.
The more the cosmic velocity exceeds this value, the more elongated the orbit will acquire, moving away from the Earth to an ever greater distance. At some point, this orbit will break, take the shape of a parabola, and the spacecraft will set off to plow the expanses of space. In order to leave the planet, the ship must have a second escape velocity. It can be calculated using the formula V2=√2µ/r. For our planet, this value is 11.2 km/s.
Astronomers have long determined what the escape velocity is, both the first and the second, for each planet of our home system. They can be easily calculated using the above formulas if you replace the constant µ with the product fM, in which M is the mass of the celestial body of interest, and f is the gravitational constant (f = 6.673 x 10(-11) m3/(kg x s2).
The third cosmic speed will allow anyone to overcome the gravity of the Sun and leave their native solar system. If you calculate it relative to the Sun, you get a value of 42.1 km/s. And in order to enter solar orbit from Earth, you will need to accelerate to 16.6 km/s.
And finally, the fourth escape velocity. With its help, you can overcome the gravity of the galaxy itself. Its magnitude varies depending on the coordinates of the galaxy. For ours, this value is approximately 550 km/s (if calculated relative to the Sun).
What are artificial earth satellites?
What purpose do they have?
Let's calculate the speed that must be imparted to an artificial Earth satellite so that it moves in a circular orbit at a height h above the Earth.
At high altitudes, the air is very rarefied and offers little resistance to bodies moving in it. Therefore, we can assume that a satellite of mass m is affected only by gravitational force directed towards the center of the Earth (Fig. 3.8).
According to Newton's second law, m cs = .
The centripetal acceleration of the satellite is determined by the formula 
where h is the height of the satellite above the Earth's surface. The force acting on the satellite, according to the law of universal gravitation, is determined by the formula 
where M is the mass of the Earth.
Substituting the found expressions for F and a into the equation for Newton’s second law, we obtain 
From the resulting formula it follows that the speed of the satellite depends on its distance from the Earth’s surface: the greater this distance, the lower the speed it will move in a circular orbit. It is noteworthy that this speed does not depend on the mass of the satellite. This means that any body can become a satellite of the Earth if it is given a certain speed. In particular, at h = 2000 km = 2 10 6 m, the speed is υ ≈ 6900 m/s.
By substituting the value of G and the values of M and R for the Earth into formula (3.7), we can calculate the first escape velocity for the Earth’s satellite:
υ 1 ≈ 8 km/s.
If such a speed is imparted to a body in the horizontal direction at the surface of the Earth, then in the absence of an atmosphere it will become an artificial satellite of the Earth, revolving around it in a circular orbit.
Only sufficiently powerful space rockets can convey such speed to satellites. Currently, thousands of artificial satellites orbit the Earth.
Any body can become an artificial satellite of another body (planet) if it is given the necessary speed.
Questions for the paragraph
1. What determines the first escape velocity?
2. What forces act on the satellite of any planet?
3. Can we say that the Earth is a satellite of the Sun?
4. Derive an expression for the orbital period of the planet’s satellite.
5 How does the speed of a spacecraft change when entering the dense layers of the atmosphere? Are there any contradictions with formula (3.6)?