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)
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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First cosmic speed is the minimum speed that must be imparted to a space projectile in order for it to enter low-Earth orbit.
Any object that we throw horizontally, after flying a certain distance, will fall to the ground. If you throw this object harder, it will fly longer, fall farther, and the trajectory of its flight will be flatter. If you successively give an object greater and greater speed, at a certain speed the curvature of its trajectory will become equal to the curvature of the Earth's surface. The earth is a sphere, as the ancient Greeks knew. What will this mean? This will mean that the surface of the Earth will seem to run away from a thrown object at the same speed with which it will fall on the surface of our planet. That is, an object thrown at a certain speed will begin to circle around the Earth at a certain constant height. If you neglect air resistance, the rotation will never stop. The launched object will become an artificial Earth satellite. The speed at which this happens is called the first cosmic speed.
The first escape velocity for our planet is easy to calculate by considering the forces that act on a body launched above the Earth’s surface at a certain speed.
The first force is the force of gravity, directly proportional to the mass of the body and the mass of our planet and inversely proportional to the square of the distance between the center of the Earth and the center of gravity of the launched body. This distance is equal to the sum of the earth's radius and the height of the object above the earth's surface.
The second force is centripetal. It is directly proportional to the square of the flight speed and body mass and inversely proportional to the distance from the center of gravity of the rotating body to the center of the Earth.
If we equate these forces and make simple transformations that are accessible to a 6th grade schoolchild (or when they start studying algebra in Russian schools these days?), it turns out that the first cosmic velocity is proportional to the square root of the partial division of the Earth's mass by the distance from the flying body to the center Earth. Substituting the appropriate data, we find that the first escape velocity at the Earth’s surface is 7.91 kilometers per second. As the flight altitude increases, the first escape velocity decreases, but not too much. So, at an altitude of 500 kilometers above the Earth’s surface it will be 7.62 kilometers per second.
The same reasoning can be repeated for any round (or almost round) celestial body: the Moon, planets, asteroids. The smaller the celestial body, the lower its first escape velocity. Thus, in order to become an artificial satellite of the Moon, a speed of only 1.68 kilometers per second will be required, almost five times less than on Earth.
The launch of a satellite into orbit around the Earth is carried out in two stages. The first stage lifts the satellite to a high altitude and partially accelerates it. The second stage brings the satellite's speed to the first cosmic speed and puts it into orbit. Why the rocket takes off was written in.
Once placed into orbit around the Earth, the satellite can orbit around it without the help of engines. It seems to be falling all the time, but cannot reach the surface of the Earth. It is precisely because the Earth’s satellite constantly seems to be falling that a state of weightlessness arises in it.
In addition to the first escape velocity, there are also second, third and fourth escape velocity. If the spacecraft reaches second space speed (about 11 km/sec), it can leave near-Earth space and fly to other planets.
Having developed third space speed (16.65 km/sec) the spacecraft will leave the solar system, and fourth space speed (500 - 600 km/sec) is the limit over which a spaceship can make an intergalactic flight.
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“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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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 what 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 boundaries.
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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