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

Рассмотрим объект с массой m на расстоянии r1 от центра Земли. Поскольку этот объект находится в гравитационном поле Земли, он обладает гравитационной потенциальной энергией U1.
Предположим, что объект удаляется дальше на расстояние r2 такое, что его потенциальная энергия изменяется на U2.
Следуя закону сохранения энергии, сумма кинетической и потенциальной энергии в точке r1 равна сумме кинетической энергии и потенциальной энергии в точке r2.
Рассматривая расстояние r2 как бесконечность, где гравитационное поле Земли пренебрежимо мало, потенциальная энергия U2 равна нулю.
Теоретически K2 равно нулю, так как объект становится неподвижным на бесконечности. Заменяя кинетическую энергию и потенциальную энергию в точке r1, скорость объекта в точке r1 равна квадратному корню из двух произведений гравитационной постоянной и массы Земли, деленной на r1.
Аппроксимируя r1 равным радиусу Земли, получаем выражение для скорости убегания объекта. Это минимальная начальная скорость, которая требуется объекту, чтобы покинуть гравитационное поле Земли.

Скорость покидания

Скорость покидания объекта определяется как минимальная начальная скорость, которая ему требуется, чтобы покинуть поверхность другого объекта, к которому он гравитационно привязан, и больше никогда не вернуться. Например, какая была бы минимальная скорость, с которой спутник должен был быть запущен с поверхности Земли, чтобы он только-только покинул земное гравитационное поле?

Для расчёта скорости покидания предполагается, что к любым силам трения энергия не теряется. На практике спутник, запущенный с поверхности Земли, должен не только покинуть земное гравитационное поле, но и замедляться из-за земной атмосферы. Таким образом, скорость покидания, рассчитанная исключительно на основе энергии гравитации, меньше фактической скорости покидания.

Также предполагается, что у спутника нулевая скорость на бесконечности, где сила притяжения Земли равна нулю. Поскольку скорость покидания не зависит от массы спутника, она будет такой же для любого объекта, будь то спутник или шар.

Если объект находится на большем расстоянии от поверхности Земли, например, Луна, ему потребуется меньшая скорость для покидания земного гравитационного поля. Это, конечно, только если масса не направляется прямо к Земле и не сталкивается с ней, что вызывает силы, которые не являются только гравитационными.

В альтернативу, скорость покидания может быть рассчитана, приравнивая полную энергию системы, такой как Земля и спутник, к нулю. Гравитационная потенциальная энергия на больших расстояниях обычно считается равной нулю. Поскольку скорость покидания определена как минимальная скорость, с которой спутник должен быть запущен с поверхности, его кинетическая энергия на бесконечности также может быть принята равной нулю. Конечно, если она положительна, он улетит ещё дальше и не вернётся. Таким образом, интересующая нас скорость - это скорость, при которой кинетическая энергия на бесконечности равна нулю, то есть тело только-только покидает гравитационное поле.

Расчёты показывают, что скорость покидания с поверхности Земли, если нет атмосферы, составляет около 11 км/с. В сравнении, скорость покидания из гравитационного поля Солнца составляет около 42 км/с на расстоянии Земли. Космический корабль, запущенный с Земли для покидания Солнечной системы, должен учитывать оба фактора.

Этот текст адаптирован из <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>

Скорость убегания объекта определяется как минимальная начальная скорость, необходимая ему для отрыва от поверхности другого объекта, к которому он ...

Education

JoVE Core

Physics

Физика

Unit 1

Раздел 1

Gravitation

Гравитация

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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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gravitation between spherically symmetric masses

gravitation-between-spherically-symmetric-masses

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

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acceleration due to gravity on earth

Gravity on Earth: Mass, Direction, and Value

Gravity near Earth gives objects a downward acceleration called acceleration due to gravity. It comes from Earth’s gravitational pull and points toward the center of the Earth. Newton’s law of gravitation shows that the gravitational force depends on mass, while Newton’s second law shows that the acceleration produced by a force depends on how that force acts on the object. Because of this relationship, the acceleration due to gravity is independent of an object’s mass. An object and the Earth...

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

the-principle-of-superposition-and-the-gravitational-field

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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Satellite Orbits and Energy in Space

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kepler's first law of planetary motion

Planet Orbits as Ellipses

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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

How the Moon Shapes Earth’s Tides

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schwarzschild radius and event horizon

Escape Velocity Limits and Black Holes

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detection of black holes

How Astronomers Find Black Holes

Black holes are found through indirect astronomical evidence. They were first proposed in the 1920s, but they stayed outside observational astronomy until the 1970s. Their closest known cousins are neutron stars, which are made mostly of packed neutrons and are extremely dense. A neutron star can have the mass of the Sun while its diameter is only a few kilometers. Because of that, the escape velocity from its surface is close to the speed of light. Interest in black holes rose again in the...

principle of equivalence

Weightlessness and Einstein’s Equivalence

Weightlessness and Einstein’s equivalence principle are closely connected in gravity. A person in free fall under gravity, such as someone diving toward Earth from an airplane, would feel completely weightless. A person descending in a lift may feel partially weightless. The principle of equivalence says that an object in a uniform gravitational field and an object accelerating at a constant rate with no gravity are under the same experimental conditions. In other words, no experiment can tell...

space-time curvature and the general theory of relativity

space-time-curvature-and-the-general-theory-of-relativity

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 First Law of Planetary Motion

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Kepler's Second Law of Planetary Motion

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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 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.

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Detection of Black Holes

Although black holes were theoretically postulated in the 1920s, they remained outside the domain of observational astronomy until the 1970s.
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Space-Time Curvature and the General Theory of Relativity

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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

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molecular-spectroscopy-absorption-and-emission

Molecular Spectroscopy: Absorption and Emission

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Atomic Emission Spectroscopy: Instrumentation

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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

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3.3.9-12.C

Launching a Satellite Beyond Gravity

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