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

Rozważmy obiekt o masie m w odległości r1 od środka Ziemi. Ponieważ obiekt ten znajduje się w polu grawitacyjnym Ziemi, posiada grawitacyjną energię potencjalną U1.
Załóżmy, że obiekt oddala się na odległość r2 tak, że jego energia potencjalna zmienia się na U2.
Zgodnie z prawem zachowania energii, suma energii kinetycznej i energii potencjalnej w r1 jest równa sumie energii kinetycznej i energii potencjalnej w r2.
Przyjmując odległość r2 jako nieskończoność, gdzie pole grawitacyjne Ziemi jest znikome, energia potencjalna U2 wynosi zero.
Teoretycznie K2 jest równe zero, ponieważ obiekt staje się nieruchomy w nieskończoności. Podstawiając energię kinetyczną i energię potencjalną przy r1, prędkość obiektu przy r1 jest równa pierwiastkowi kwadratowemu z dwukrotności iloczynu stałej grawitacyjnej i masy Ziemi podzielonej przez r1.
Przybliżając r1 równą promieniowi Ziemi, otrzymuje się wyrażenie na prędkość ucieczki obiektu. Jest to minimalna prędkość początkowa, jaką obiekt potrzebuje, aby uciec z pola grawitacyjnego Ziemi.

prędkość ucieczki

Prędkość ucieczki obiektu definiuje się jako minimalną prędkość początkową potrzebną obiektowi do ucieczki z powierzchni innego obiektu, z którym jest związany grawitacyjnie, i nigdy nie do powrotu. Na przykład, jaka byłaby minimalna prędkość, z jaką satelita powinien zostać wystrzelony z powierzchni Ziemi, tak aby po prostu uciekł z ziemskiego pola grawitacyjnego?

Aby obliczyć prędkość ucieczki, zakłada się, że siły tarcia nie powodują utraty energii. W praktyce satelita wystrzelony z powierzchni Ziemi nie tylko musi uciec z ziemskiego pola grawitacyjnego, ale również ziemska atmosfera go spowalnia. Zatem prędkość ucieczki obliczona wyłącznie na podstawie energii grawitacyjnej jest mniejsza niż rzeczywista prędkość ucieczki.

Zakłada się również, że satelita ma zerową prędkość w nieskończoności, gdzie siła grawitacji Ziemi wynosi zero. Ponieważ prędkość ucieczki nie zależy od masy satelity, byłaby taka sama dla każdego obiektu, niezależnie od tego, czy jest to satelita czy kula.

Jeśli obiekt znajduje się w większej odległości od powierzchni Ziemi, na przykład Księżyc, potrzebowałby mniejszej prędkości, aby uciec z ziemskiego pola grawitacyjnego. Dzieje się tak, chyba że masa kieruje się bezpośrednio w stronę Ziemi i zderza się z nią, powodując powstanie sił nie tylko grawitacyjnych.

Alternatywnie prędkość ucieczki można obliczyć, przyrównując całkowitą energię układu, np. Ziemi i satelity, do zera. Tradycyjnie przyjmuje się, że energia potencjalna grawitacji na dużych odległościach wynosi zero. Ponieważ prędkość ucieczki definiuje się jako minimalną prędkość, z jaką satelita powinien zostać wystrzelony z powierzchni, można przyjąć, że jego energia kinetyczna w nieskończoności również wynosi zero. Oczywiście, jeśli będzie dodatni, odejdzie dalej i nigdy nie wróci. Zatem interesująca prędkość ma miejsce, gdy energia kinetyczna w nieskończoności wynosi zero; oznacza to, że ciało po prostu ucieka z pola grawitacyjnego.

Obliczenia pokazują, że prędkość ucieczki z powierzchni Ziemi, przy założeniu braku atmosfery, wynosi około 11 km/s. Dla porównania, prędkość ucieczki z pola grawitacyjnego Słońca wynosi około 42 km/s w odległości Ziemi. Statek kosmiczny wystrzeliwany z Ziemi w celu ucieczki z Układu Słonecznego musiałby wziąć pod uwagę oba czynniki.

Ten tekst jest adaptacją <a href="https://openstax.org/books/university-physics-volume-1/pages/13-3-gravitational-potential-energy- and-total-energy">Openstax, University Physics, tom 1, sekcja 13.3: Energia potencjalna grawitacji i energia całkowita.</a>

Prędkość ucieczki

Prędkość ucieczki obiektu definiuje się jako minimalną prędkość początkową, jaką wymaga, aby uciec z powierzchni innego obiektu, do którego się znajduje...

Education

JoVE Core

Physics

Fizyka

Unit 1

Jednostka 1

Gravitation

Grawitacja

KEY TERMS AND CONCEPTS

Kluczowe terminy i pojęcia

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

Newton’s law of gravitation links falling objects, the Moon’s orbit, and planetary motion. Before Newton, many people thought the sky and Earth followed different laws. Kepler later described the three laws of planetary motion, but their connection to a deeper force was still unclear. Isaac Newton made that connection in 1665 to 1666. He showed that the same phenomenon explains why objects fall to Earth and why the Moon moves around Earth. He also found that the Moon’s centripetal acceleration...

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newton's law of gravitation

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Newton's gravity links the fall of objects on Earth with the motion of the Moon and planets. People once thought the downward pull seen near Earth was unique to our planet. Later, Nicolaus Copernicus proposed the heliocentric theory, which placed Earth and the other planets around the Sun, with the Moon orbiting Earth. Isaac Newton connected these motions in the 17th century. He reasoned that the same attraction that keeps the Moon on its path around Earth also makes an apple fall from a tree.

gravitation between spherically symmetric masses

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Gravitational Potential Inside a Shell

Gravitational potential energy for spherically symmetric bodies depends on their masses and the distance between them. The idea is to treat each body as if its mass were concentrated at its center of mass. This makes the calculation easier for high school physics. A spherically symmetric mass can be thought of as many concentric spherical shells. Now place a point mass at a distance r from the center of one shell. Every particle on a given spherical ring is the same distance from that point...

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gravity between spherical bodies

Point-Mass Approximation in Spherical Gravity

Newton’s law of gravitation gives the force between two point masses. For large spherical objects, such as the Earth, the Moon, and other planets, the same law can be used by treating each body as if its mass were concentrated at its center. This idea can be proven by comparing gravitational potential energy. The transcript shows that a hollow sphere of mass M and a point mass m give the same energy result as two extended solid spherical objects. To set up the proof, a tiny ring on the shell...

reduced mass coordinates: isolated two-body problem

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Center of Mass in the Two-Body Problem

Reduced mass coordinates help simplify the classical two-body problem. In classical mechanics, this problem describes the motion of two interacting bodies under gravity or another central force. The key idea is that the system can be treated like a single-body problem when reduced mass is used. For this description, one body is imagined to move with the reduced mass while the other body is fixed in place with infinite mass. The reduced mass is a property of the two-body system and is defined...

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acceleration due to gravity on earth

Gravity on Earth: Mass, Direction, and Value

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acceleration due to gravity on other planets

acceleration-due-to-gravity-on-other-planets

Measuring Gravity on Distant Planets

Gravity near Earth’s surface is measured as acceleration due to gravity. On Earth, students can test it with simple experiments. On other planets, that kind of direct experiment is not possible. Instead, astronomers use observations to find a planet’s gravity. They study how the planet affects nearby objects, especially its satellite. The key idea is that the acceleration depends on the planet’s gravity, not on the object’s mass. To make the calculation, scientists track a satellite’s path...

apparent weight and the earth's rotation

Earth's Spin and Apparent Weight

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

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How Earth’s Shape Affects Apparent Weight

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potential energy due to gravitation

Gravitational Binding and Potential Energy

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the principle of superposition and the gravitational field

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Adding Gravitational Forces with Superposition

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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

circular-orbits-and-critical-velocity-for-satellites

Orbital Speed and Period in Circular Paths

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energy of a satellite in a circular orbit

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kepler's first law of planetary motion

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kepler's second law of planetary motion

Kepler's Equal-Area Planet Motion

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kepler's third law of planetary motion

Kepler’s Planet Distance Law

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tidal forces

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schwarzschild radius and event horizon

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detection of black holes

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principle of equivalence

Weightlessness and Einstein’s Equivalence

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space-time curvature and the general theory of relativity

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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.

The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a ...

The escape velocity of an object is defined as the minimum initial velocity that it requires to escape the surface of another object to which it is ...

Circular Orbits and Critical Velocity for Satellites

The Moon orbits around the Earth. In turn, the Earth (and other planets) orbit the Sun. The space directly above our atmosphere is filled with artificial ...

Energy of a Satellite in a Circular Orbit

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Kepler's First Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. He ...

Kepler's Second Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. His ...

Kepler's Third Law of Planetary Motion

In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. In ...

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Nuclear Fission

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Radioactivity and Nuclear Equations

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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

When solids, liquids, or condensed gases are heated sufficiently, they radiate some of the excess energy as light. Photons produced in this manner have a ...

atomic-spectroscopy-absorption-emission-and-fluorescence

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

No object with a finite mass can travel faster than the speed of light in a vacuum. This fact has an interesting consequence in the domain of extremely ...

The Wave Nature of Light

The nature of light has been a subject of inquiry since antiquity. In the seventeenth century, Isaac Newton performed experiments with lenses and prisms ...

Nuclear Binding Energy

The difference between the calculated and experimentally measured masses is known as the mass defect of the atom. In the case of helium-4, the mass defect ...

Metal Flame Emission

Metal Flame Emission
Flame testing is an analytical technique where a sample is placed in a flame, and the characteristic flame color is used to identify ...

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 ...

atomic-absorption-spectroscopy-radiation-and-light-sources

Atomic Absorption Spectroscopy: Radiation and Light Sources

Atomic absorption spectroscopy (AAS) relies on the Beer-Lambert law, which requires that the radiation source emits a narrow range of wavelengths to match ...

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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