Formula for finding the first escape velocity. Life of wonderful names

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 makes it impossible aircraft leave 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 with 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 meaning of the first earthly escape velocity depends on the radius and mass of the Earth and does not depend on the mass of the body 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 signs of the first artificial satellite Earth. 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.

Second space satellite, launched in the USSR on November 3, 1957, already weighed 500 kg. On board it was complex scientific equipment and the first Living being- 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, for different celestial bodies has different meaning, 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.

This 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.

Useful information about escape velocity:

If at the moment of entering orbit the spacecraft has a speed equal to First cosmic speed, perpendicular to the direction of the center of the Earth, then its orbit (in the absence of any other forces) will be circular. If the speed of the vehicle is equal to less than , then its orbit has the shape of an ellipse, and the point of entry into orbit is located at the apogee. If this point is at an altitude of about 160 km, then immediately after entering orbit the satellite enters the underlying dense layers of the atmosphere and burns up. That is, for the specified height first Cosmic speeds is the minimum for a spacecraft to become a satellite of the Earth. On high altitudes a spacecraft can become a satellite even at a speed slightly lower First Space Speed, calculated for this height. So, at an altitude of 300 km, it is enough for a spacecraft to have a speed 45 m/sec less than First Space Speed

There is also:

Second escape velocity:

In the formula we used:

Gravitational constant

Ministry of Education and Science of the Russian Federation

State educational institution higher vocational education"St. Petersburg State University 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:

Checked:

Saint Petersburg

Cosmic speeds.

Cosmic velocity (first v1, second v2, third v3 and fourth v4) is the minimum speed at which any body in free movement will be able:

v1 - become a companion celestial body(that is, the ability to orbit around the NT and not fall onto 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 galaxy Milky Way.

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 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*, the 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 constitutes only a small part the total gravitating mass, and everything else is hidden mass.

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 that 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 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 away into space? 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 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 formula there is more black hole299,792 km/c, that is more speed Sveta. 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

If you find an error, please highlight a piece of text and click Ctrl+Enter.

“Uniform and uneven movement” - t 2. Uneven movement. Yablonevka. L 1. Uniform and. L2. t 1. L3. Chistoozernoe. t 3. Uniform movement. =.

“Curvilinear motion” - Centripetal acceleration. UNIFORM MOTION OF A BODY AROUND A CIRCLE There are: - curvilinear motion with a constant velocity; - movement with acceleration, because speed changes direction. Direction of centripetal acceleration and velocity. Motion of a point in a circle. Movement of a body in a circle with a constant absolute speed.

“Motion of bodies on a plane” - Evaluate the obtained values ​​of unknown quantities. Substitute numerical data into the solution general view, make calculations. Make a drawing, depicting interacting bodies on it. Perform an analysis of the interaction of bodies. Ftr. Body movement inclined plane without friction. Study of the movement of a body on an inclined plane.

“Support and movement” - Contact us ambulance brought the patient. Slender, stooped, strong, strong, fat, clumsy, dexterous, pale. Game situation “Concilium of doctors”. Sleep on a hard bed with a low pillow. “Body support and movement. Rules for maintaining correct posture. Correct posture while standing. Children's bones are soft and elastic.

"Space Speed" - V1. THE USSR. That's why. April 12, 1961 Message to extraterrestrial civilizations. Third escape velocity. On board Voyager 2 is a disk with scientific information. Calculation of the first escape velocity at the Earth's surface. The first manned flight into space. Voyager 1 trajectory. The trajectory of bodies moving at low speed.

“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?

There are 20 presentations in total