escape

In celestial mechanics, escape velocity or escape speed is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a primary body, thus reaching an infinite distance from it. It is typically stated as an ideal speed, ignoring atmospheric friction. Although the term "escape velocity" is common, it is more accurately described as a speed than a velocity because it is independent of direction; the escape speed increases with the mass of the primary body and decreases with the distance from the primary body. The escape speed thus depends on how far the object has already traveled, and its calculation at a given distance takes into account that without new acceleration it will slow down as it travels—due to the massive body's gravity—but it will never quite slow to a stop.
A rocket, continuously accelerated by its exhaust, can escape without ever reaching escape speed, since it continues to add kinetic energy from its engines. It can achieve escape at any speed, given sufficient propellant to provide new acceleration to the rocket to counter gravity's deceleration and thus maintain its speed.
More generally, escape velocity is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero; an object which has achieved escape velocity is neither on the surface, nor in a closed orbit (of any radius). With escape velocity in a direction pointing away from the ground of a massive body, the object will move away from the body, slowing forever and approaching, but never reaching, zero speed. Once escape velocity is achieved, no further impulse need be applied for it to continue in its escape. In other words, if given escape velocity, the object will move away from the other body, continually slowing, and will asymptotically approach zero speed as the object's distance approaches infinity, never to come back. Speeds higher than escape velocity retain a positive speed at infinite distance. Note that the minimum escape velocity assumes that there is no friction (e.g., atmospheric drag), which would increase the required instantaneous velocity to escape the gravitational influence, and that there will be no future acceleration or extraneous deceleration (for example from thrust or from gravity of other bodies), which would change the required instantaneous velocity.
Escape speed at a distance d from the center of a spherically symmetric primary body (such as a star or a planet) with mass M is given by the formula





v

e


=




2
G
M

d



=


2
g
d




{\displaystyle v_{e}={\sqrt {\frac {2GM}{d}}}={\sqrt {2gd}}}
where G is the universal gravitational constant (G ≈ 6.67×10−11 m3·kg−1·s−2) and g is the local gravitational acceleration (or the surface gravity, when d = r). The escape speed is independent of the mass of the escaping object. For example, the escape speed from Earth's surface is about 11.186 km/s (40,270 km/h; 25,020 mph; 36,700 ft/s) and the surface gravity is 9.8 m/s (35 km/h; 22 mph; 32 ft/s).
When given an initial speed



V


{\displaystyle V}
greater than the escape speed




v

e


,


{\displaystyle v_{e},}
the object will asymptotically approach the hyperbolic excess speed




v




,


{\displaystyle v_{\infty },}
satisfying the equation:







v






2


=

V

2






v

e




2


.


{\displaystyle {v_{\infty }}^{2}=V^{2}-{v_{e}}^{2}.}
In these equations atmospheric friction (air drag) is not taken into account.

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