Escape velocity

The escape velocity (also called speed of escape , parabolic speed , speed of escape , or speed of exhaust , in English escape velocity ) of a Planet is the Speed which, if it is assigned with an object on the surface of this planet, will lead so that he escapes gravitational attraction definitively from this planet (this by supposing negligible the resistance of the atmosphere). Formulated differently, it is the minimal Speed which a body must reach theoretically to move away indefinitely from a Astre in spite of the gravitational attraction of the latter. The escape velocity of a planet is as speed as a body, initially at rest and remotely infinite, acquires while falling to surface from planet.

The escape velocity is calculated according to the following formula:

$v = \ sqrt {\ frac {2GM} {R}}$

Where $G$ is the gravitational Constante universal (6,6742×10^-11 m³·kg^-1·s^-2), $M$ is the mass of planet, and $R$ its ray. The escape velocity increases thus when the mass of planet increases and also when its ray decreases.

Demonstration of the relation

One leaves the principle according to which the mechanical energy of a body is constant during time. At the distance

R

, the speed of the body is the escape velocity. At an infinite distance, its speed and its potential energy of gravitation are null. Its mechanical energy is thus null.

$E_m = E_c + E_p = \ frac {1} {2} mv^2 - \ frac {GMm} {R} = 0$

The masses are simplified and the formula indicated is obtained.

As by definition, the escape velocity is speed necessary to withdraw itself completely from the gravity of a planet or more generally of an unspecified body starting from its surface, this one is higher than the speed of placing in orbit since a body in orbit still undergoes the gravity of the body in question. The speed of placing in orbit is:

v = \ sqrt {\ frac {GM} {R}}

To show it, apply the basic principle of dynamics to the satellite to put into orbit:

$m~a = \ frac {GMm}$

In the Reference mark of Frenet related to the satellite in orbit, the normal acceleration is written:

$has = \ frac {R}$

The masses are simplified again and the announced formula well is obtained.

It should be noted that a body in altitude requires a speed lower than that of release to withdraw itself from gravity. Necessary speed in this case is obtained by the formula:

v = \ sqrt {\ frac {2GM} {D}}

where D is the distance to the center of planet or the body from which one wishes to release oneself.

Remarkable values escape velocity

The escape velocity of a body leaving the surface of the Ground, known as also cosmic second speed ( second space velocity ), is about 11,186 kilometers a second (that is to say approximately 40269 km/h) compared to a geocentric inertial reference mark. By comparison, that of Jupiter is of 59,5 km/s. the probe Luna 1 was, in 1959, the first object built by the man to reach the escape velocity terrestrial at the time of its way in direction of the the Moon.

The escape velocity of a body leaving the Solar system, known as also the third speed cosmic ( third cosmic velocity ), is about 16,6 kilometers a second compared to a geocentric inertial reference mark.

Remarks

Contrary to a spread belief, there is not no need that this speed is vertical: the escape velocity is a quantity vectorial Scalaire and not . It acts in fact of a kinetic energy of release, but as this one is proportional to the mass of the object, it is convenient to characterize it by the speed which is associated for him. It does not matter the direction towards which the body moves, provided all the same that it is not directly towards planet!

One can also speak parabolic speed: it is the value, expressed according to a planet, speed which it is necessary to give to an object so that the trajectory of this object subjected exclusively to the attraction of this planet is a parabola (which could be degenerated).

Above this speed, the object will leave the orbit of planet and will move away. In lower part, the object remains related to planet; it will thus be put into elliptic orbit, and will or not return to be crushed on planet according to the characteristics of this orbit: in this case, the direction plays a part as well as the starting point and speed.

Notes and references of the article

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