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 Earth satellite.
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 should 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.
The minimum speed that must be imparted to a physical body (for example, a spacecraft) so that it can overcome the gravitational attraction of a celestial object (for example, a planet or star) and forever leave the sphere of its gravitational action is called parabolic speed (a body having such a speed moves along parabolic trajectory). Parabolic speed decreases with increasing distance from a celestial object. The parabolic speed at the surface of a celestial object is called the second cosmic speed. For the Earth, the second escape velocity is 11.18 kilometers per second. The parabolic speed at an altitude of 300 kilometers above the Earth's surface (sea level) is 10.93 kilometers per second, at an altitude of 1000 kilometers - 6.98 kilometers per second. For the Sun, the second cosmic speed is 617.7 kilometers per second, and the parabolic speed at a distance of 1 astronomical unit from our star (the average radius of the Earth's orbit) is 42.1 kilometers per second. For the largest planet in the solar system (Jupiter), the second escape velocity is 59.5 kilometers per second, for the smallest (Mercury) - 4.2 kilometers per second.
What is the third escape velocity?
The third cosmic speed is the minimum speed that must be imparted to a body (for example, a spacecraft) near the surface of the Earth so that it can, having overcome the gravitational attraction of the Earth and the Sun, leave the solar system forever. The third cosmic speed is approximately 16.6 kilometers per second (when launched at an altitude of 200 kilometers above the earth's surface), and the direction of the body's speed relative to the Earth must coincide with the direction of the speed of the Earth's orbital motion.
What does classical mechanics study?
Classical mechanics studies the motion of macroscopic bodies at speeds small compared to the speed of light. Classical mechanics is based on Newton's laws. The movement of microparticles (method of description and laws of motion) in given external fields is studied by quantum mechanics, and the laws of mechanical motion of bodies (particles) at speeds comparable to the speed of light are studied by relativistic mechanics, based on the special theory of relativity.
What keeps the Moon in Earth orbit?
Our natural satellite is prevented from falling to Earth by its orbital speed, which exceeds the first cosmic speed. But escaping from the gravitational embrace of the Earth and leaving its environs forever is prevented by the Earth’s gravity, to overcome which the Moon’s orbital speed is not high enough (less than the second cosmic speed).
Second escape velocity (parabolic velocity, release velocity, escape velocity)- the lowest speed that must be given to an object (for example, a spacecraft), the mass of which is negligible compared to the mass of a celestial body (for example, a planet), in order to overcome the gravitational attraction of this celestial body and leave a closed orbit around it. It is assumed that after a body acquires this speed, it no longer receives 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 cosmic velocity is called parabolic because bodies that have a speed at launch exactly equal to the second cosmic velocity move in a parabola relative to the celestial body. However, if a little more energy is given to the body, its trajectory ceases to be a parabola and becomes a hyperbola. If it is a little less, then it turns into an ellipse. In general, they are all conic sections.
If a body is launched vertically upward at a second cosmic speed or higher, it will never stop and begin to fall back.
The same speed is acquired at the surface of a celestial body by any cosmic body that was at rest at an infinitely large distance and then began to fall.
The second cosmic speed was first achieved by a USSR spacecraft on January 2, 1959 (Luna-1).
Calculation
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 its gravitational influence.
m v 2 2 2 − G m M R = 0 , (\displaystyle (\frac (mv_(2)^(2))(2))-G(\frac (mM)(R))=0,) R = h + r (\displaystyle R=h+r)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- mass of the test body, M- mass of the planet, r- radius of the planet, h - length from the base of the body to its center of mass (height above the surface of the planet), G- gravitational constant, v 2 - second escape velocity.
Solving this equation for v 2, we get
v 2 = 2 G M R . (\displaystyle v_(2)=(\sqrt (2G(\frac (M)(R)))).)There is a simple relationship between the first and second cosmic velocities:
v 2 = 2 v 1 . (\displaystyle v_(2)=(\sqrt (2))v_(1).)The square of the escape velocity is equal to twice the Newtonian potential at a given point (for example, on the surface of a celestial body):
v 2 2 = − 2 Φ = 2 G M R . (\displaystyle v_(2)^(2)=-2\Phi =2(\frac (GM)(R)).)The first escape velocity is the minimum speed at which a body moving horizontally above the surface of the planet will not fall onto it, but will move in a circular orbit.
Let's consider the motion of a body in a non-inertial frame of reference - 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.
where m is the mass of the object, M is the mass of the planet, G is the gravitational constant (6.67259 10 −11 m? kg −1 s −2),
The first escape velocity, R is the radius of the planet. Substituting numerical values (for Earth 7.9 km/s
The first escape velocity can be determined through the acceleration of gravity - since g = GM/R?, then
The second cosmic velocity is the lowest speed that must be given to an object whose mass is negligible compared to the mass of a celestial body in order to overcome the gravitational attraction of this celestial body and leave a circular orbit around it.
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. 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, v 2 is the second escape velocity.
There is a simple relationship between the first and second cosmic velocities:
The square of the escape velocity is equal to twice the Newtonian potential at a given point:
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We, earthlings, are accustomed to standing firmly on the ground and not flying away anywhere, and if we throw some object into the air, it will definitely fall to the surface. It’s all to blame for the gravitational field created by our planet, which bends space-time and forces an apple thrown to the side, for example, to fly along a curved trajectory and intersect with the Earth.
Any object creates a gravitational field around itself, and for the Earth, which has an impressive mass, this field is quite strong. That is why powerful multi-stage space rockets are being built, capable of accelerating spaceships to the high speeds needed to overcome the planet’s gravity. The meaning of these velocities is called the first and second cosmic velocities.
The concept of the first cosmic velocity is very simple - this is the speed that must be given to a physical object so that, moving parallel to the cosmic body, it cannot fall on it, but at the same time remains in a constant orbit.
The formula for finding the first escape velocity is not complicated: WhereV G M– mass of the object;R– radius of the object;
Try to substitute the necessary values into the formula (G - the gravitational constant is always equal to 6.67; the mass of the Earth is 5.97·10 24 kg, and its radius is 6371 km) and find the first escape velocity of our planet.
As a result, we get a speed of 7.9 km/s. But why, moving at exactly this speed, will the spacecraft not fall to Earth or fly into outer space? It will not fly into space due to the fact that this speed is still too low to overcome the gravitational field, but it will fall to Earth. But only because of its high speed, it will always “avoid” a collision with the Earth, while at the same time continuing its “fall” in a circular orbit caused by the curvature of space.
This is interesting: The International Space Station works on the same principle. The astronauts on it spend all their time in a constant and incessant fall, which does not end tragically due to the high speed of the station itself, which is why it consistently “misses” the Earth. The speed value is calculated based on .
But what if we want the spacecraft to leave the boundaries of our planet and not be dependent on its gravitational field? Accelerate it to the second cosmic speed! So, the second escape velocity is the minimum speed that must be given to a physical object in order for it to overcome the gravitational attraction of a celestial body and leave its closed orbit.
The value of the second escape velocity also depends on the mass and radius of the celestial body, so it will be different for each object. For example, to overcome the gravitational attraction of the Earth, the spacecraft needs to reach a minimum speed of 11.2 km/s, Jupiter - 61 km/s, the Sun - 617.7 km/s.
The escape velocity (V2) can be calculated using the following formula:
Where V– first escape velocity;G– gravitational constant;M– mass of the object;R– radius of the object;
But if the first escape velocity of the object under study (V1) is known, then the task becomes much easier, and the second escape velocity (V2) is quickly found using the formula:
This is interesting: second cosmic black hole formula more299,792 km/c, that is, greater than the speed of light. That is why nothing, not even light, can escape beyond its limits.
In addition to the first and second comic speeds, there are the third and fourth, which must be achieved in order to go beyond the boundaries of our Solar system and galaxy, respectively.
Illustration: bigstockphoto | 3DSculptor
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