Name: launching a satellite beyond gravity
Uploaded: 2025-05-22T09:05:41.000Z
Duration: 1 min 26 s
Channel: JoVE

K12 Standards

Pennsylvania State Academic Standards

9-12

Earth and Space Science

Space Systems

3.3.9-12.A

Escape Velocity

Betrachten wir ein Objekt mit der Masse m in einer Entfernung von r1 vom Erdmittelpunkt. Da sich dieses Objekt im Schwerefeld der Erde befindet, besitzt es die potentielle Gravitationsenergie U1.
Angenommen, das Objekt bewegt sich in einer Entfernung von r2 weiter weg, so dass sich seine potentielle Energie zu U2 ändert.
Nach dem Energieerhaltungssatz ist die Summe aus kinetischer Energie und potentieller Energie bei r1 gleich der Summe aus kinetischer Energie und potentieller Energie bei r2.
Betrachtet man die Entfernung r2 als unendlich, wobei das Gravitationsfeld der Erde vernachlässigbar ist, so ist die potentielle Energie U2 Null.
Theoretisch ist K2 gleich Null, da das Objekt im Unendlichen stationär wird. Substituiert man die kinetische Energie und die potentielle Energie bei r1, so ist die Geschwindigkeit des Objekts bei r1 gleich der Quadratwurzel aus dem Zweifachen des Produkts aus der Gravitationskonstante und der Erdmasse, dividiert durch r1.
Nähert man sich r1 dem Radius der Erde an, erhält man einen Ausdruck für die Fluchtgeschwindigkeit des Objekts. Es ist die minimale Anfangsgeschwindigkeit, die ein Objekt benötigt, um dem Schwerefeld der Erde zu entkommen.

fluchtgeschwindigkeit

Die Fluchtgeschwindigkeit eines Objekts wird als die minimale Anfangsgeschwindigkeit definiert, die es benötigt, um der Oberfläche eines anderen Objekts zu entkommen, an das es gravitativ gebunden ist, und nie wieder zurückzukehren. Was wäre zum Beispiel die minimale Geschwindigkeit, mit der ein Satellit von der Erdoberfläche gestartet werden sollte, damit er dem Gravitationsfeld der Erde gerade noch entkommt?

Um die Fluchtgeschwindigkeit zu berechnen, wird angenommen, dass keine Energie durch Reibung verloren geht. In der Praxis muss ein Satellit, der von der Erdoberfläche gestartet wird, nicht nur dem Erdgravitationsfeld entkommen, sondern wird auch durch die Erdatmosphäre abgebremst. Daher ist die rein aus Gravitationsenergieüberlegungen berechnete Fluchtgeschwindigkeit kleiner als die tatsächliche Fluchtgeschwindigkeit.

Es wird auch angenommen, dass der Satellit im Unendlichen, wo die Erdgravitationskraft gleich Null ist, eine Geschwindigkeit von Null hat. Da die Fluchtgeschwindigkeit nicht von der Masse des Satelliten abhängt, ist sie für jedes Objekt gleich, egal ob Satellit oder Ball.

Wenn sich ein Objekt weiter von der Erdoberfläche entfernt befinden würde, z.B. der Mond, bräuchte es eine kleinere Geschwindigkeit, um dem Erdgravitationsfeld zu entkommen. Dies gilt jedoch nicht, wenn die Masse direkt auf die Erde zusteuert und mit ihr kollidiert, was zu Kräften führt, die nicht nur gravitativ sind.

Alternativ kann die Fluchtgeschwindigkeit berechnet werden, indem die Gesamtenergie eines Systems, z.B. der Erde und eines Satelliten, auf Null gesetzt wird. Die Gravitationspotentielle Energie bei großen Entfernungen wird konventionell als Null angenommen. Da die Fluchtgeschwindigkeit als die minimale Geschwindigkeit definiert ist, mit der ein Satellit von der Oberfläche gestartet werden sollte, kann auch seine kinetische Energie bei Unendlichkeit als Null angenommen werden. Wenn sie natürlich positiv ist, würde sie weiter weg gehen und nie zurückkehren. Daher ist die Geschwindigkeit von Interesse, wenn die kinetische Energie im Unendlichen gleich Null ist, d. h. wenn der Körper dem Gravitationsfeld einfach entkommt.

Berechnungen zeigen, dass die Fluchtgeschwindigkeit von der Erdoberfläche aus, unter der Annahme, dass keine Atmosphäre vorhanden ist, etwa 11 km/s beträgt. Im Vergleich dazu beträgt die Fluchtgeschwindigkeit aus dem Gravitationsfeld der Sonne etwa 42 km/s in Erdnähe. Raumfahrzeuge, die von der Erde gestartet werden, um das Sonnensystem zu verlassen, müssten beide Faktoren berücksichtigen.

Dieser Text basiert auf <a href="https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy-and-total-energy">Openstax, University Physics Volume 1, Section 13.3: Gravitational Potential Energy and Total Energy.</a>

Fluchtgeschwindigkeit

Die Fluchtgeschwindigkeit eines Objekts ist definiert als die minimale Anfangsgeschwindigkeit, die es benötigt, um der Oberfläche eines anderen Objekts zu entkommen, auf das es sich ...

Education

Lehre

JoVE Core

Physics

Physik

Unit 1

Einheit 1

Gravitation

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

launching a satellite beyond gravity

Launching a satellite beyond gravity requires enough escape velocity to leave a planet or other body and never return. Escape velocity is the minimum initial speed needed to break free from the gravitational pull of an object that is holding the satellite in orbit or near its surface.

For Earth, the idea is often shown with a satellite launched from the ground. The calculation assumes that no energy is lost to friction. In real life, Earth’s atmosphere also slows the satellite down, so the true launch speed must be higher than the value found from gravity alone.

The escape speed depends on gravity, not on the mass of the object being launched. That means a satellite and a ball would need the same escape velocity if they started from the same place. An object that starts farther from Earth’s surface, such as the Moon, needs a smaller speed to escape Earth’s gravitational field, unless it is moving directly toward Earth and collides with it.

Escape velocity can also be found by using energy. In this method, the total energy of the Earth-satellite system is set to zero. Gravitational potential energy is taken as zero at very large distances, and the satellite is treated as having zero speed at infinity, where Earth’s gravity is zero. The escape condition is the point where the object just makes it out, so its kinetic energy at infinity is also zero.

For Earth, the escape velocity is about 11 km/s if the atmosphere is ignored. For the Sun’s gravitational field at Earth’s distance, the escape velocity is about 42 km/s. Spacecraft leaving Earth to travel out of the solar system must take both Earth’s gravity and the Sun’s gravity into account.

Launching a Satellite Beyond Gravity

gravitation

Newton’s Law Behind Planet Motion

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newton's law of gravitation

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gravitation between spherically symmetric masses

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Gravitational Potential Inside a Shell

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reduced mass coordinates: isolated two-body problem

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Center of Mass in the Two-Body Problem

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acceleration due to gravity on earth

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acceleration due to gravity on other planets

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apparent weight and the earth's rotation

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variation in acceleration due to gravity near the earth's surface

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potential energy due to gravitation

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the principle of superposition and the gravitational field

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escape velocity

Launching a satellite beyond gravity requires enough escape velocity to leave a planet or other body and never return. Escape velocity is the minimum initial speed needed to break free from the gravitational pull of an object that is holding the satellite in orbit or near its surface. For Earth, the idea is often shown with a satellite launched from the ground. The calculation assumes that no energy is lost to friction. In real life, Earth’s atmosphere also slows the satellite down, so the...

circular orbits and critical velocity for satellites

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Orbital Speed and Period in Circular Paths

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kepler's second law of planetary motion

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kepler's third law of planetary motion

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principle of equivalence

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space-time curvature and the general theory of relativity

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Einstein’s Gravity and Mercury’s Orbit

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key terms and concepts

Explore escape velocity, the minimum speed needed for a satellite to leave Earth or the Sun’s gravity. Learn why mass, atmosphere, and starting distance matter.

Launching a Satellite Beyond Gravity | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

3.3.9-12.D

Louisiana Student Standards for Science

High School

Earth Science

Earth’s Place In The Universe

HS-ESS1-1

HS-ESS1-2

HS-ESS1-3

Tennessee Academic Standards

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

PHYS.PS2.9

Virginia  Standards of Learning

HIgh School

ES.2

ES.2.b

ES.2.c

PH.3

PH.3.b

new-jersey-student-learning-standards-for-science

New Jersey Student Learning Standards for Science

Earth's Place in the Universe

HS-ESS1-4

Florida Department of Education

SC.912.E.5

SC.912.E.5.6

SC.912.E.5.10

Physical Science

SC.912.P.10

SC.912.P.10.6

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.H

south-carolina-college-and-career-ready-science-standards

South Carolina College and Career Ready Science Standards

Earth and Space Science (ESS)

Earth’s Place in the Universe

E-ESS1-6.

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Kepler's Second Law of Planetary Motion

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Kepler's Third Law of Planetary Motion

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Nuclear Stability

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Students who demonstrate understanding can develop a model based on evidence to illustrate the life span of the sun and the role of nuclear fusion in the sun's core to release energy in the form of radiation.  Clarification Statement: Emphasis is on the energy transfer mechanisms that allow energy from nuclear fusion in the sun's core to reach Earth.  Examples of evidence for the model include observations of the masses and lifetimes of other stars, as well as the ways that the sun's radiation varies due to sudden solar flares ("space weather"), the 11-year sunspot cycle, and non-cyclic variations over centuries.  Assessment Boundary: Assessment does not include details of the atomic and sub-atomic processes involved with the sun's nuclear fusion.

Emission Spectra

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Atomic Spectroscopy: Absorption, Emission, and Fluorescence

Atomic spectroscopy is a vital tool in elemental analysis, both qualitatively and quantitatively. It can be broadly divided into optical spectroscopy, ...

Detection of Black Holes

Although black holes were theoretically postulated in the 1920s, they remained outside the domain of observational astronomy until the 1970s.
Their ...

Space-Time Curvature and the General Theory of Relativity

In 1905, Albert Einstein published his special theory of relativity. According to this theory, no matter in the universe can attain a speed greater than ...

Schwarzschild Radius and Event Horizon

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The Wave Nature of Light

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Nuclear Binding Energy

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Metal Flame Emission

Metal Flame Emission
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molecular-spectroscopy-absorption-and-emission

Molecular Spectroscopy: Absorption and Emission

Molecules possess discrete energy levels called quantum states. Unlike atoms, which have simpler energy levels, molecules possess additional rotational ...

atomic-emission-spectroscopy-instrumentation

Atomic Emission Spectroscopy: Instrumentation

The instrumentation of atomic emission spectrometry (AES) involves various components, including atomization devices that convert samples into gas-phase ...

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Atomic Absorption Spectroscopy: Radiation and Light Sources

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Students who demonstrate understanding can construct an explanation of the Big Bang theory based on astronomical evidence of light spectra, motion of distant galaxies, and composition of matter in the universe.  Clarification Statement: Emphasis is on the astronomical evidence of the red shift of light from galaxies as an indication that the universe is currently expanding, the cosmic microwave background as the remnant radiation from the Big Bang, and the observed composition of ordinary matter of the universe, primarily found in stars and interstellar gases (from the spectra of electromagnetic radiation from stars), which matches that predicted by the Big Bang theory (3/4 hydrogen and 1/4 helium).  Assessment Boundary:

3.3.9-12.B

Launching a Satellite Beyond Gravity

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