escape velocity

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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:

Find

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18. Ethical aspect of speech culture. Speech etiquette and communication culture. Formulas of speech etiquette. Etiquette formulas for acquaintance, introduction, greeting and farewell. “You” and “You” as forms of address in Russian speech etiquette. National features of speech etiquette.- 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. Centripetal force calculated from the condition rotational movement, is equal to the force of gravity. The speed is calculated based on the equality of these forces.

$m\backslash frac(v\_1^2)(R)=G\backslash frac(Mm)(R^2)$, $v\_1=\backslash sqrt(G\backslash frac(M)(R))$,

Where m- mass of the object, M- mass of the planet, G- gravitational constant, $v\_1$- 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

$v\_1\backslash approx$ 7.9 km/s

The first escape velocity can be determined through the acceleration of gravity. Because the $g\; =\; \backslash frac(GM)(R^2)$, That

$v\_1=\backslash sqrt(gR)$.

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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 for you good wife…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.

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1 first escape velocity = 7899.9999999999 meters per second [m/s]

Initial value

Converted value

meter per second meter per hour meter per minute kilometer per hour kilometer per minute kilometer per second centimeter per hour centimeter per minute centimeter per second millimeter per hour millimeter per minute millimeter per second foot per hour foot per minute foot per second yard per hour yard per minute yard per second mile per hour mile per minute miles per second knot knot (UK) speed of light in vacuum first escape velocity second escape velocity third escape velocity speed of rotation of the Earth speed of sound in fresh water speed of sound in sea ​​water(20°C, depth 10 meters) Mach number (20°C, 1 atm) Mach number (SI standard)

Thermal efficiency and fuel efficiency

More about speed

General information

Speed ​​is a measure of the distance traveled in a certain time. Speed ​​can be a scalar quantity or a vector quantity - the direction of movement is taken into account. The speed of movement in a straight line is called linear, and in a circle - angular.

Speed ​​measurement

Average speed v found by dividing the total distance traveled ∆ x on total time ∆t: v = ∆x/∆t.

In the SI system, speed is measured in meters per second. Kilometers per hour are also widely used in metric system and miles per hour in the US and UK. When, in addition to the magnitude, the direction is also indicated, for example, 10 meters per second to the north, then we're talking about about vector speed.

The speed of bodies moving with acceleration can be found using the formulas:

• a, With initial speed u during the period ∆ t, has a finite speed v = u + a×∆ t.
• A body moving with constant acceleration a, with initial speed u and final speed v, It has average speed ∆v = (u + v)/2.

Average speeds

Speed ​​of light and sound

According to the theory of relativity, the speed of light in a vacuum is the highest speed at which energy and information can travel. It is denoted by the constant c and is equal to c= 299,792,458 meters per second. Matter cannot move at the speed of light because it would require an infinite amount of energy, which is impossible.

The speed of sound is usually measured in an elastic medium, and is equal to 343.2 meters per second in dry air at a temperature of 20 °C. The speed of sound is lowest in gases and highest in solids X. It depends on the density, elasticity, and shear modulus of the substance (which shows the degree of deformation of the substance under shear load). Mach number M is the ratio of the speed of a body in a liquid or gas medium to the speed of sound in this medium. It can be calculated using the formula:

M = v/a,

Where a is the speed of sound in the medium, and v- body speed. Mach number is commonly used in determining speeds close to the speed of sound, such as airplane speeds. This value is not constant; it depends on the state of the medium, which, in turn, depends on pressure and temperature. Supersonic speed is a speed exceeding Mach 1.

Vehicle speed

Below are some vehicle speeds.

• Passenger aircraft with turbofan engines: The cruising speed of passenger aircraft is from 244 to 257 meters per second, which corresponds to 878–926 kilometers per hour or M = 0.83–0.87.
• High-speed trains (like the Shinkansen in Japan): these trains reach maximum speeds from 36 to 122 meters per second, that is, from 130 to 440 kilometers per hour.

Animal speed

The maximum speeds of some animals are approximately equal to:

Human speed

• Humans walk at speeds of about 1.4 meters per second, or 5 kilometers per hour, and run at speeds of up to about 8.3 meters per second, or 30 kilometers per hour.

Examples of different speeds

Four-dimensional speed

In classical mechanics, vector velocity is measured in three-dimensional space. According to the special theory of relativity, space is four-dimensional, and the measurement of speed also takes into account the fourth dimension - space-time. This speed is called four-dimensional speed. Its direction may change, but its magnitude is constant and equal to c, that is, the speed of light. Four-dimensional speed is defined as

U = ∂x/∂τ,

Where x represents a world line - a curve in space-time along which a body moves, and τ is the "proper time" equal to the interval along the world line.

Group speed

Group velocity is the speed of wave propagation, describing the speed of propagation of a group of waves and determining the speed of wave energy transfer. It can be calculated as ∂ ω /∂k, Where k is the wave number, and ω - angular frequency. K measured in radians/meter, and the scalar frequency of wave oscillation ω - in radians per second.

Hypersonic speed

Hypersonic speed is a speed exceeding 3000 meters per second, that is, many times faster than the speed of sound. Solid bodies moving at such speeds acquire the properties of liquids, since, thanks to inertia, the loads in this state are stronger than the forces that hold the molecules of a substance together during collisions with other bodies. At ultrahigh hypersonic speeds, two colliding solids turn into gas. In space, bodies move at exactly this speed, and engineers who design spacecraft orbital stations and spacesuits must take into account the possibility of a station or astronaut colliding with space debris and other objects when working in outer space. In such a collision, the skin of the spacecraft and the spacesuit suffer. Hardware developers are conducting hypersonic collision experiments in special laboratories to determine how severe impacts the suits, as well as the skin and other parts of the spacecraft, such as fuel tanks and solar panels, testing their strength. To do this, spacesuits and skin are exposed to impacts different objects from special installation with supersonic speeds exceeding 7500 meters per second.

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 plants are built space rockets, capable of accelerating spaceships to high speeds, which are needed to overcome the gravity of the planet. The meaning of these velocities is what is called the first and second cosmic velocities.

The concept of first cosmic velocity is very simple - this is the speed that must be given physical object, so that it, moving parallel to the cosmic body, could not fall on it, but at the same time would remain 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? He will not fly into space due to the fact that given speed is still too small to overcome the gravitational field, but it will fall to Earth. But only because 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. 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 cosmic 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 pull of the Earth, spacecraft it is necessary to reach a minimum speed of 11.2 km/s, Jupiter - 61 km/s, 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 limits of our solar system and galaxies respectively.

Illustration: bigstockphoto | 3DSculptor

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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 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 accessible to a 6th grade student (or when Russian school are they starting to study algebra these days?), it turns out that the first escape velocity is proportional to square root from the partial division of the mass of the Earth by the distance from the flying body to the center of the 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 less heavenly body, the lower the first escape velocity for him. 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 greater height 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 spaceship 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.