Escape velocity

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In physics, breakout speed is the minimum velocity required for an object to escape the gravitational influence of a very large body.

The escape velocity from Earth is about 11,186 km/sec (6,951 mi/sec, 40,270 km/h, 25,020 mph) on the surface. More generally, the escape velocity is the rate at which the kinetic energy of an object and its gravitational potential energy equal to zero; an object that has reached a loose velocity is not on the surface, nor in a closed orbit (of any radius). With the speed of escaping in the direction that leads away from the land of a large body, the object will move away from the body, slow down forever and approach, but never reach, zero speed. Once the escape speed is reached, no further impetus should be applied in order to continue its escape. In other words, given the velocity, the object will move away from the other body, continue to slow down, and will approach the asymptotic zero velocity as the object's distance approaches infinity, never to return. The higher velocity of the free velocity has a positive speed at infinity. Note that the minimum escape velocity assumes that there is no friction (for example, atmospheric resistance), which will increase the instantaneous velocity required to avoid the effects of gravity, and that there will be no additional source of future speed (eg, impulse), which will reduce the momentary speed required.

Untuk tubuh bulat simetris dan masif seperti bintang, atau planet, kecepatan lepas untuk tubuh itu, pada jarak tertentu, dihitung dengan rumus

${\ displaystyle v_ {e} = {\ sqrt {\ frac {2GM} {r}}},} Â Â$

where G is the universal gravitational constant ( G ? 6.67ÃÆ' â € "10 ^-11 m ³ Â · Kg ^-1 Ã, Â · s ^-2 ), M body mass to escape, and r the distance from the center of the body mass to the object. The relationship does not depend on the mass of the object coming out of the big body. Conversely, the body falling under the mass gravitational pull of the M , from infinity, begins with a zero velocity, will attack a large object at a speed equal to the escape velocity given by the same formula.

Ketika diberi thatcepted ${\ displaystyle V} Â$ lebih besar dari kecepatan lepas ${\ displaystyle v_ {e},} Â$ objek akan secara asimtotik mendekati kelebihan quecepat hiperbolik ${\ displaystyle v _ {\ infty},} Â$ memuaskan persamaannya:

${\ displaystyle {v _ {\ infty}} ^ 2 = V2 - {v_ {e}} 2.} Â Â$

In this equation atmospheric friction (air resistance) is not taken into account. A rocket that moves out of the gravitational well actually does not need to reach escape velocity to escape but can achieve the same result (escape) at any speed with the appropriate propulsion mode and propellant sufficient to provide an acceleration force on the object to escape self.. The escape velocity is only necessary to send a ballistic object on a path that will allow the object to escape from the gravitational well of the mass M .

Video Escape velocity

Overview

The existence of escape velocity is a consequence of limited energy conservation and energy depth fields. For objects with a given total energy, which moves subject to conservative forces (such as static gravity fields) is only possible for objects to achieve a combination of locations and speeds that have total energy; and places that have a higher potential energy than this can not be reached at all. By adding velocity (kinetic energy) to the object it extends the possibilities of an accessible location, until, with enough energy, they become infinite.

For a given gravitational potential energy at a given position, the escape velocity is the minimum velocity of an object without propulsion must be able to "escape" from gravity (ie gravity which will never succeed in retracting). The actual escaping speed is the speed (not the speed) because it does not specify the direction: no matter what direction the journey takes, it can escape the gravitational field (provided the path does not cut the planet).

The simplest way to lower the runaway speed formula is to use energy conservation. For the sake of simplicity, unless stated otherwise, we assume that an object is trying to escape from a uniform spherical planet by moving away from it and that the only significant force acting on a moving object is planetary gravity. In its original state, i , imagine that a spacecraft m is at a distance r from the center of the planet's mass, whose Mass is M . The initial speed is the same as the runoff speed, $Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{    Â     v                       e                                {\ displaystyle v_ {e}}  Â} . In the latter state, f , it will be an infinite distance from the planet, and its speed will be very small and assumed 0. Kinetic energy K and gravity potential energy U _g is the only type of energy we will face, so with energy conservation,

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â(Â\; Â\; Â\; Â\; Â\; Â\; Â\; Â\; ÂKÂ\; Â\; Â\; Â\; Â\; Â\; Â\; ÂÂ\; Â\; Â\; Â\; Â\; Â\; Â}$ _{          U                        g                                  )                         me                          =        (          K                          U                        g                                  )                 Â                                         {\ displaystyle (K U_ {g}) _ {i} = (K U_ {g}) _ {f} \,}  Â}

K _? = 0 karena kecepatan akhir adalah nol, dan U _g? = 0 karena jarak akhirnya adalah tak terhingga , jadi

${\ displaystyle {\ begin {aligned} \ Rightarrow {} & amp; {\ frac {1} {2}} mv_ {e} ^ {2} {\ frac { -GMm} {r}} = 0 0 \\ [3pt] \ Rightarrow {} & amp; v_ {e} = {\ sqrt {\ frac {2GM} {r}}} = {\ sqrt {\ frac {2 \ mu} {r}}} \ end {aligned}}}  ÂÂ$

Where? is the standard gravity parameter.

The same result is obtained by relativistic calculations, in which case the variable r represents radial coordinates or reduces the circumference of the Schwarzschild metric.

Slightly more formalized, "escape velocity" is the initial velocity required to go from a starting point in the gravitational potential field to infinity and ends at infinity with zero remaining velocity, without additional acceleration. All speed and speed are measured with respect to the field. In addition, the escape velocity at a point in space is equal to the speed that the object will have if it starts at rest from an infinite distance and is pulled by gravity to that point.

In general usage, the starting point is on the surface of the planet or the moon. On Earth's surface, its velocity is about 11.2 km/sec, which is approximately 33 times the speed of sound (Mach 33) and several times the speed of the rifle bullet (up to 1.7 km/sec). However, at an altitude of 9,000 km in "space", it's a little less than 7.1 km/sec.

Free velocity from the masses of escaped objects. It does not matter if the mass is 1 kg or 1,000 kg; what is different is the amount of energy needed. For bulk objects ${\ displaystyle m}$ GMm/r , the function of the object mass (where r < is the radius of the Earth, G is the gravitational constant, and M is the mass of the Earth, M = 5.9736 ÃÆ'â € "10 ²⁴ kg ). The associated quantity is the energy of a specific orbital which is essentially the amount of kinetic energy and potential shared by the mass. An object has reached a loose velocity when the specific orbital energy is greater or equal to zero.

Maps Escape velocity

Running speed in various situations

From the body surface

Explain alternate untuk kecepate lepas ${\ displaystyle v_ {e}} Â Â$ sangat berguna di permukaan pada tubuh adalah:

${\ displaystyle v_ {e} = {\ sqrt {2gr \,}}} Â Â$

where r is the distance between the center of the body and the point at which the escape velocity is being calculated and g is the gravitational acceleration at that distance (ie, surface gravity).

Untuk tubuh dengan distribusi massa sferis simetris, kecepatan lepas ${\ displaystyle v_ {e}} Â Â$ let's allow sebanding dengan radius dengan asumsi kepadatan konstan, dan sebanding dengan akar kuadrat dari kerapatan rata-rata?

${\ displaystyle v_ {e} = Kr {\ sqrt {\ rho}}} Â Â$

di mana ${\ displaystyle K = {\ sqrt {{\ frac {8} {3}} \ pi G}}} Â Â$ ? 2,364 ÃÆ'â € "10 ^-5 m ^1,5 Ã, Â · kg ^-0,5 Ã, Â · s ^{- 1}

Dari badan yang berputar

The breakout velocity relative to the surface of the rotating body depends on the direction in which the escaping body is running. For example, since the Earth's rotational speed is 465 m/s at the equator, a rocket launched tangentially from the equator of the Earth to the east requires an initial velocity of about 10,735 km/sec relative to Earth to escape while the rocket launched tangentially from the equator of Earth to the west requires an initial velocity of about 11,665 km/s relative to Earth . The surface velocity decreases with the geographic latitude cosine, so the launch facility is often located as close to the equator as possible, eg. American Cape Canaveral (latitude 28Ã, Â ° 28'Ã, N) and French Guyana Space Center (latitude 5 Â ° 14'Ã, N).

Practical considerations

In most situations it is impractical to achieve almost instantaneous escaping speed, due to implied acceleration, and also because if any atmospheric hypersonic velocity is involved (at Earth speeds of 11.2 km/s, or 40.320 km/h) it will cause most objects burning due to aerodynamic heating or tearing up by atmospheric uptake. For a true escaping orbit, the spacecraft will accelerate steadily out of the atmosphere until it reaches a corresponding loose speed for its altitude (which will be less than the surface). In many cases, the spacecraft may be first placed in a parking orbit (eg low Earth orbit, LEO, at 160-2,000 km) and then accelerated to the escape velocity at that altitude, which will be slightly lower (about 11.0 km/s at LEO 200 km). The additional change required in speed, however, is much less because the spacecraft already has a significant orbital velocity (in Earth's low orbit speed of about 7.8 km/sec, or 28,080 km/h).

From an orbiting body

The escape speed at a certain height is ${\ displaystyle {\ sqrt {2}}}$ the first cosmic velocity , whereas in this context the escape velocity is referred to as second cosmic speed .

For the body in an elliptical orbit that wants to accelerate to an orbital escape, the required speed will vary, and will be greatest in the periapsis when the body is closest to the central body. However, the body's orbital velocity will also be at its highest point at this point, and a change in the required speed will be at its lowest, as explained by the Oberth effect.

Barycentric exit speed

The escape velocity technically can be measured as relative to the other, the central body or relative to the center of mass or the barycenter of the body system. So for a two-body system, the term freelance can be ambiguous, but it is usually meant to mean the speed of bleachric release from the smaller body. In the gravitational field, breakout speed refers to the escape velocity of the zero mass test particle relative to the mass-producing barycenter of the field. In most situations involving the spacecraft the difference is negligible. For the same mass as the Saturn V rocket, the relative release speed of the launch pad is 253.5 am/s (8 nanometers per year) faster than the relative speed relative to the common mass center.

High speed low speed

Mengabaikan semua faktor selain gaya gravitasi antara tubuh dan objek, sebuah objek yang diproyeksikan secara vertikal dengan kecepatan ${\ displaystyle v}  ÂÂ$ dari permukaan tubuh bola dengan kecepatan lepas ${\ displaystyle v_ {e}}  ÂÂ$ dan radius ${\ displaystyle R}  ÂÂ$ akan mencapai tinggi maksimum $            {\displaystyle         h      }        \{\backslash \; displaystyle\; h\}\;  ÂÂ$ memuaskan persamaannya

${\ displaystyle v = v_ {e} {\ sqrt {\ frac {h} {R h}}} \,}  ÂÂ$

yang mana, pemecahan untuk h menghasilkan

${\ displaystyle h = {\ frac {x2}} {1-x2}} \ R \,} Â Â$

di mana ${\ displaystyle x = {\ frac {v} {v_ {e}}}} Â$ adalah rasio dari kecepatan asli ${\ displaystyle v} Â$ untuk kecepata lepas ${\ displaystyle v_ {e}.} Â Â$

Unlike loose velocity, the direction (upward vertical) is important to reach the maximum height.

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Path

If an object reaches escape velocity, but is not directed directly from the planet, it follows a curved track or path. Although this trajectory does not form a closed form, it can be called an orbit. Assuming that gravity is the only significant force in the system, the velocity of this object at any point in the path will equal the velocity of at that point due to conservation of energy, the total energy must always be 0, which implying that it always has a freelance rate; see the above derivation. The shape of the trajectory will be a parabola whose focus lies at the center of the planet's mass. The actual escape requires a path with a path that does not intersect with the planet, or its atmosphere, because this will cause the object to fall. As it moves away from the source, this path is called the run orbit. The runaway orbit is known as C3 = 0 orbit. C3 is the characteristic energy, = - GM/2a , where a is the semi-major axis, which is not limited to the parabolic path.

If the body has a velocity greater than the freelance speed it will form a hyperbolic path and will have an excessive hyperbolic speed, equivalent to the extra energy the body has. A relatively small additional delta - v above that is required to accelerate to escape velocity can produce a relatively large speed at infinity. For example, in a place where the escape velocity is 11.2 km/s, the addition of 0.4 km/s produces hyperbolic excess speed of 3.02 km/s:

$Â\; Â{\displaystyle Â\; Â\; Â\; Â\; Â\; Â\; Â}$ _{    Â     v                       ?                          =                       ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,    Â Â  <Â> V                             2        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -    ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                  ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ...                 v                                    e     ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,        Â        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,                                 2        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>                           =                       ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂï <½Â · 11,6                             2        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>      ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÃ, -    ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,             11.2                             2        ÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂÂ,         Â  <Â>                           ?         3.02.               {\ displaystyle v _ {\ infty} = {\ sqrt {V ^ {2} - {v_ {e}} ^ {2}}} = { \ sqrt {11.6 ^ {2} -11.2 ^ {2}}} \ approx 3.02.}  Â}

If the body in a circular orbit (or on an elliptical orbit periapsis) accelerates along its course to release speed, the acceleration point will form the periapsis of the escape route. The direction of the journey will eventually reach 90 degrees towards the acceleration point. If the body accelerates outwardly, the direction of the journey will eventually be at a smaller angle, and is represented by one of the asymptotes of the hyperbolic trajectory it is now taking. This means that time of acceleration is very important if the goal is to escape in a certain direction.

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Multiple bodies

Ketika melarikan diri dari sistem senyawa, seperti bulan yang mengorbit planet atau planet yang mengorbit matahari, sebuah roket yang meninggalkan kecepatura melarikan diri ( ${\ displaystyle v_ {e1}} Â Â$ ) untuk badan pertama (yang mengorbit), (misalnya Bumi) tidak akan melakukan perjalanan that jarak yang tidak terbatas karena membutuhkan kecepatan yang lebih tinggi untuk keluar dari gravitasi tubuh kedua (misalnya Matahari). Dekat Bumi, lintasan roket akan muncul parabola, tetapi masih acan terikat secara gravitasi ke tubuh kedua dan akan memasuki orbit elips di sekitar tubuh itu, denotes kecepted orbital yang sama que tubuh pertama.

di mana ${\ displaystyle k = 1 - {\ frac {1} {\ sqrt {2}}} \ approx,2929} Â$ untuk orbit language.

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Daftar kecepatan pelarian

The last two columns will depend precisely where the orbit's escape velocity is reached, because its orbit is not exactly circular (especially Mercury and Pluto).

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Lower the loose speed using calculus

Biarkan G menjadi konstant gravitasi dan biarkan M menjadi massa bumi (atau tubuh gravitasi lainnya) dan m menjadi massa pelarian tubuh atau proyektil. Di kejauhan r dari pusat gravitasi, tubuh merasakan kekuatan yang menarik

${\ displaystyle F = G {\ frac {Mm} {r2}}}} Â Â$

Pekerjaan yang diperlukan untuk menggerakkan tubuh lebih dari jarak kecil dr terhadap gaya ini karena itu diberikan oleh

${\ displaystyle dW = Fdr = -G {\ frac {Mm} {r2}}}, dr,} Â Â$

di mana tanda minus menunjukkan kekuatan bertindak dalam arti sebaliknya ${\ displaystyle dr} Â Â$ .

Total kerja yang diperlukan untuk menggerakkan tubuh dari permukaan r ₀ dari tubuh gravitasi hingga tak terbatas kemudian

${\ displaystyle W = \ int_ {r_ {0}} ^ {\ infty} -G {\ frac {Mm} {r2} } \, dr = -G {\ frac {Mm} {r_ {0}}} = - mgr_ {0}.} Â Â$

Ini adalah energi kinetik minimal yang diperlukan untuk dapat mencapai tak terhingga, sehingga quecepat pelarian v ₀ memenuhi

${\ displaystyle WK = 0 \ Rightarrow {\ frac {1} {2}} mv_ {0} 2 = G {\ frac {Mm} {r_ {0}}},} Â Â$

yang mengakibatkan

${\ displaystyle v_ {0} = {\ sqrt {\ frac {2GM} {r_ {0}}}} = {\ sqrt {2gr_ {0} }}.} Â Â$

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Lihat juga

• Lubang hitam - sebuah objek dengan kecepata lepas lebih besar dari kecepatan cahaya
• Energi characteristik (C ₃ )
• Anglia Delta-v - kecepatan yang diperlukan untuk melakukan manuver.
• Katapel gravitasi - teknik untuk mengubah lintasan
• Gravitation denotes baik
• Daftar objek buatan dalam orbit heliosentris
• Daftar objek buatan yang meninggalkan Tata Surya
• cannonball Newton
• Efek Oberth - pembakaran propellant jauh di dalam medan gravitasi memberikan perubahan yang lebih tinggi dalam energi kinetik
• Kecepatan orbit
• Masalah dua-tubuh

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Catatan

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Reference

• Roger R. Bate; Donald D. Mueller; Jerry E. White (1971). Dasar-dasar astrodinamika . New York: Publikasi Dover. ISBN, 0-486-60061-0.

src: hg101.kontek.net

Tautan eksternal

• Pelarian kecephal calculator
• Kalkulator penghitungan kecepata numerik berbasis-web

Source of the article : Wikipedia