What is 1 escape velocity equal to? Cosmic speeds

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.

Humanity has long been striving for space. But how to break away from the Earth? What prevented man from flying to the stars?

As we already know, this was prevented by gravity, or the gravitational force of the Earth - the main obstacle to space flights.

Earth gravity

All physical bodies located on Earth are subject to the action law of universal gravitation . According to this law, they all attract each other, that is, they act on each other with a force called gravitational force, or gravity .

The magnitude of this force is directly proportional to the product of the masses of the bodies and inversely proportional to the square of the distance between them.

Since the mass of the Earth is very large and significantly exceeds the mass of any material body located on its surface, the gravitational force of the Earth is significantly greater than the gravitational force of all other bodies. We can say that compared to the gravitational force of the Earth they are generally invisible.

The earth attracts absolutely everything to itself. Whatever object we throw upward, under the influence of gravity it will definitely return to Earth. Drops of rain fall down, water flows from the mountains, leaves fall from the trees. Any item we drop also falls to the floor, not the ceiling.

The main obstacle to space flights

Earth's gravity prevents aircraft from leaving the Earth. And it is not easy to overcome it. But man learned to do it.

Let's observe the ball lying on the table. If he rolls off the table, the gravity of the Earth will cause him to fall to the floor. But if we take the ball and forcefully throw it into the distance, it will not fall immediately, but after some time, describing a trajectory in the air. Why was he able to overcome gravity at least for a short time?

And this is what happened. We applied a force to it, thereby imparting acceleration, and the ball began to move. And the more acceleration the ball receives, the higher its speed will be and the further and higher it can fly.

Let us imagine a cannon mounted on the top of a mountain, from which projectile A is fired at high speed. Such a projectile is capable of flying several kilometers. But in the end, the projectile will still fall to the ground. Its trajectory under the influence of gravity has a curved appearance. Projectile B leaves the cannon at higher speed. Its flight path is more elongated, and it will land much further. The more speed a projectile receives, the straighter its trajectory becomes and the greater the distance it travels. And finally, at a certain speed, the trajectory of projectile C takes the shape of a closed circle. The projectile makes one circle around the Earth, another, a third and no longer falls on the Earth. It becomes an artificial satellite of the Earth.

Of course, no one sends cannon shells into space. But spacecraft that have reached a certain speed become Earth satellites.

First escape velocity

What speed must a spacecraft achieve to overcome gravity?

The minimum speed that must be imparted to an object in order to put it into a near-Earth circular (geocentric) orbit is called first escape velocity .

Let's calculate the value of this speed relative to the Earth.

A body in orbit is acted upon by a gravitational force directed toward the center of the Earth. It is also a centripetal force trying to attract this body to the Earth. But the body does not fall to the Earth, since the action of this force is balanced by another force - centrifugal, which tries to push it out. Equating the formulas of these forces, we calculate the first escape velocity.

Where m – mass of the object in orbit;

M – mass of the Earth;

v 1 – first escape velocity;

R – radius of the Earth

G – gravitational constant.

M = 5.97 10 24 kg, R = 6,371 km. Hence, v 1 ≈ 7.9 km/s

The value of the first earth's cosmic velocity depends on the radius and mass of the Earth and does not depend on the mass of the body being launched into orbit.

Using this formula, you can calculate the first cosmic velocities for any other planet. Of course, they differ from the first escape velocity of the Earth, since celestial bodies have different radii and masses. For example, the first escape velocity for the Moon is 1680 km/s.

An artificial Earth satellite is launched into orbit by a space rocket that accelerates to the first cosmic velocity and higher and overcomes gravity.

Beginning of the space age

The first cosmic speed was achieved in the USSR on October 4, 1957. On this day, earthlings heard the call sign of the first artificial Earth satellite. It was launched into orbit using a space rocket created in the USSR. It was a metal ball with antennae, weighing only 83.6 kg. And the rocket itself had enormous power for that time. After all, in order to launch just 1 additional kilogram of weight into orbit, the weight of the rocket itself had to increase by 250-300 kg. But improvements in rocket designs, engines and control systems soon made it possible to send much heavier spacecraft into Earth orbit.

The second space satellite, launched in the USSR on November 3, 1957, already weighed 500 kg. On board there was complex scientific equipment and the first living creature - the dog Laika.

The space age began in human history.

Second escape velocity

Under the influence of gravity, the satellite will move horizontally above the planet in a circular orbit. It will not fall to the surface of the Earth, but it will not move to another, higher orbit. And in order for him to do this, he needs to be given a different speed, which is called second escape velocity . This speed is called parabolic, escape speed , release speed . Having received such a speed, the body will cease to be a satellite of the Earth, will leave its surroundings and become a satellite of the Sun.

If the speed of a body when starting from the Earth's surface is higher than the first escape velocity, but lower than the second, its near-Earth orbit will have the shape of an ellipse. And the body itself will remain in low-Earth orbit.

A body that has received a speed equal to the second escape velocity when starting from the Earth will move along a trajectory shaped like a parabola. But if this speed even slightly exceeds the value of the second escape velocity, its trajectory will become a hyperbola.

The second escape velocity, like the first, has different meanings for different celestial bodies, since it depends on the mass and radius of this body.

It is calculated by the formula:

The relationship between the first and second escape velocity remains

For the Earth, the second escape velocity is 11.2 km/s.

The first rocket to overcome gravity was launched on January 2, 1959 in the USSR. After 34 hours of flight, she crossed the orbit of the Moon and entered interplanetary space.

The second space rocket towards the Moon was launched on September 12, 1959. Then there were rockets that reached the surface of the Moon and even made a soft landing.

Subsequently, spacecraft went to other planets.

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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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:

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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

The 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 .