Difference between revisions of "Escape speed"
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| − | The '''escape speed''' associated with any celestial body, such as a [[star]], a [[planet]], a [[dwarf planet]], or a [[moon]], is the speed at which an object, starting from the body surface, would have to travel to be able to escape the body's gravity. An object in motion has escaped the gravity of a celestial body if the body cannot cause the object to fall back toward it or to remain in a closed [[orbit]] around it. | + | The '''escape speed''' (also known as escape velocity) associated with any celestial body, such as a [[star]], a [[planet]], a [[dwarf planet]], or a [[moon]], is the speed at which an object, starting from the body surface, would have to travel to be able to escape the body's gravity. An object in motion has escaped the gravity of a celestial body if the body cannot cause the object to fall back toward it or to remain in a closed [[orbit]] around it. |
== Concept == | == Concept == | ||
| − | + | Escape speed is a scalar, not a vector. An object in motion with respect to a celestial body need not be moving directly away from that body in order to escape it. | |
When an object is traveling exactly at escape speed, it is in a parabolic orbit with the body at the focus of the parabola. A parabolic path is open; hence an object following such a path will never return to the body. | When an object is traveling exactly at escape speed, it is in a parabolic orbit with the body at the focus of the parabola. A parabolic path is open; hence an object following such a path will never return to the body. | ||
| − | An object traveling at a speed slower than the escape speed normally moves in an ellipse, which is a closed path. So long as that path does not intersect the body surface, the path is an orbit. | + | An object traveling at a speed slower than the escape speed normally moves in an ellipse, which is a closed path. So long as that path does not intersect the body surface, the path is an [[orbit]]. It will follow [[Kepler's laws of planetary motion]]. |
An object traveling ''faster'' than escape speed is following a ''hyperbolic'' orbit, which is also open. The object will always be traveling faster than a minimum speed even after infinite time. | An object traveling ''faster'' than escape speed is following a ''hyperbolic'' orbit, which is also open. The object will always be traveling faster than a minimum speed even after infinite time. | ||
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The formula that depends on [[surface gravity]] is simpler: | The formula that depends on [[surface gravity]] is simpler: | ||
| − | <math>v_{escape} = \sqrt{2gR}</math> where | + | <math>v_{escape} = \sqrt{2gR}</math> where <math>g</math> is the [[acceleration]] due to gravity at the surface. |
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| + | ===Derivation=== | ||
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| + | Consider an object of mass <math>m</math> trying to escape from a [[planet]] of mass <math>M</math>. The energy required to do this is in the form of [[kinetic energy]], and is converted completely into gravitational potential energy. Hence: | ||
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| + | <math>\frac{1}{2} m v^2 = \int^{\infty}_{R} \frac{GMm}{r^2} dr</math> | ||
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| + | Performing the integration gives: | ||
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| + | <math>\frac{1}{2} v^2 = \frac{GM}{R}</math> | ||
| + | |||
| + | Rearranging gives the result above. | ||
== See also == | == See also == | ||
* [[Surface gravity]] | * [[Surface gravity]] | ||
| + | * [[Velocity]] - as compared with [[speed]] | ||
[[Category:Astronomy]] | [[Category:Astronomy]] | ||
Latest revision as of 17:33, April 17, 2017
The escape speed (also known as escape velocity) associated with any celestial body, such as a star, a planet, a dwarf planet, or a moon, is the speed at which an object, starting from the body surface, would have to travel to be able to escape the body's gravity. An object in motion has escaped the gravity of a celestial body if the body cannot cause the object to fall back toward it or to remain in a closed orbit around it.
Concept
Escape speed is a scalar, not a vector. An object in motion with respect to a celestial body need not be moving directly away from that body in order to escape it.
When an object is traveling exactly at escape speed, it is in a parabolic orbit with the body at the focus of the parabola. A parabolic path is open; hence an object following such a path will never return to the body.
An object traveling at a speed slower than the escape speed normally moves in an ellipse, which is a closed path. So long as that path does not intersect the body surface, the path is an orbit. It will follow Kepler's laws of planetary motion.
An object traveling faster than escape speed is following a hyperbolic orbit, which is also open. The object will always be traveling faster than a minimum speed even after infinite time.
Calculation
Escape speed depends on two properties of the body to be escaped from: its mass and its radius. A body has one escape speed from its surface and a different escape speed from any altitude above that surface. The classical "escape speed" of any body is the escape speed from its surface. Once an object is in a closed orbit, the escape speed required to break out of orbit is considerably less than the escape speed from the surface.
The formula for escape speed, depending on body mass and radius, is:
<math>v_{escape} = \sqrt{\frac{2GM}{R}}</math>
The formula that depends on surface gravity is simpler:
<math>v_{escape} = \sqrt{2gR}</math> where <math>g</math> is the acceleration due to gravity at the surface.
Derivation
Consider an object of mass <math>m</math> trying to escape from a planet of mass <math>M</math>. The energy required to do this is in the form of kinetic energy, and is converted completely into gravitational potential energy. Hence:
<math>\frac{1}{2} m v^2 = \int^{\infty}_{R} \frac{GMm}{r^2} dr</math>
Performing the integration gives:
<math>\frac{1}{2} v^2 = \frac{GM}{R}</math>
Rearranging gives the result above.
See also
- Surface gravity
- Velocity - as compared with speed