Isaac Newton's analysis of escape velocity. Projectiles A and B fall back to earth. Projectile C achieves a circular orbit, D an elliptical one. Projectile E escapes.

In physics, escape velocity is the speed where the kinetic energy of an object is equal to the magnitude of its gravitational potential energy, as calculated by the equation U_g = − Gm₁m₂ / r. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The kinetic energy of an object is the extra Energy which it possesses due to its motion Potential energy can be thought of as Energy stored within a physical system It is commonly described as the speed needed to "break free" from a gravitational field (without any additional impulse). The term escape velocity can be considered a misnomer because it is actually a speed rather than a velocity, i. e. it specifies how fast the object must move but the direction of movement is irrelevant. In more technical terms, escape velocity is a scalar (and not a vector).

Overview

For a given gravitational field and a given position, the escape velocity is the minimum speed an object without propulsion needs to attain in order to "escape" from gravity, i. A gravitational field is a model used within Physics to explain how gravity exists in the universe Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of Spacecraft propulsion is any method used to change the velocity of Spacecraft and artificial Satellites There are many different methods e. , so that gravity will never manage to pull it back. For the sake of simplicity, unless stated otherwise, we will assume that the scenario we are dealing with is that an object is attempting to escape from a uniform spherical planet by moving straight up (along a radial line away from the center of the planet), and that the only significant force acting on the moving object is the planet's gravity.

Escape velocity is actually a speed (not a velocity) because it does not specify a direction: no matter what the direction of travel is, the object can escape the gravitational field. The simplest way of deriving the formula for escape velocity is to use conservation of energy. In Physics, the law of conservation of energy states that the total amount of Energy in an isolated system remains constant and cannot be created although it may Imagine that a spaceship of mass m is at a distance r from the center of mass of the planet, whose mass is M. Its initial speed is equal to its escape velocity, v_e. At its final state, it will be an infinite distance away from the planet, and its speed will be negligibly small and assumed to be 0. Kinetic energy K and gravitational potential energy U_g are the only types of energy that we will deal with, so by the conservation of energy,

K_f = 0 because final velocity is zero, and U_gf = 0 because its final distance is infinity, so

Defined a little more formally, "escape velocity" is the initial speed required to go from an initial point in a gravitational potential field to infinity with a residual velocity of zero, with all speeds and velocities measured with respect to the field. Additionally, the escape velocity at a point in space is equal to the speed that an object would have if it started at rest from an infinite distance and was pulled by gravity to that point. In common usage, the initial point is on the surface of a planet or moon. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is A natural satellite or moon is a Celestial body that Orbits a Planet or smaller body which is called the primary. On the surface of the Earth, the escape velocity is about 11. 2 kilometers per second (~6. 96 mi/s), which is approximately 34 times the speed of sound (mach 34) and at least 10 times the speed of a rifle bullet. However, at 9,000 km altitude in "space", it is slightly less than 7. 1 km/s.

The escape velocity relative to the surface of a rotating body depends on direction in which the escaping body travels. For example, as the Earth's rotational velocity is 465 m/s at the equator, a rocket launched tangentially from the Earth's equator to the east requires an initial velocity of about 10. 735 km/s relative to Earth to escape whereas a rocket launched tangentially from the Earth's equator to the west requires an initial velocity of about 11. 665 km/s relative to Earth. The surface velocity decreases with the cosine of the geographic latitude, so space launch facilities are often located as close to the equator as feasible, e. g. the American Cape Canaveral in Florida and the European Guiana Space Centre, only 5 degrees from the equator in French Guiana. Cape Canaveral from the Spanish Cabo Cañaveral, is a headland in Brevard County Florida, United States, near the center of that Florida ( is a state located in the southeastern region of the United States, bordering Alabama to the northwest and Georgia to the The Guiana Space Centre, or more commonly Centre Spatial Guyanais (CSG is a French Spaceport near Kourou in French Guiana. French Guiana (Guyane française officially fr ''Guyane'' is an Overseas department (French département d'outre-mer, or DOM) of France

Escape velocity is independent of the mass of the escaping object. It does not matter if the mass is 1 kg or 1000 kg, escape velocity from the same point in the same gravitational field is always the same. What differs is the amount of energy needed to accelerate the mass to achieve escape velocity: the energy needed for an object of mass m to escape the Earth's gravitational field is GMm / r, a function of the object's mass (where r is the radius of the Earth, G is the gravitational constant, and M is the mass of the Earth). The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass More massive objects require more energy to reach escape velocity. All of this, of course, assumes we are neglecting air resistance.

Misconceptions

Planetary or lunar escape velocity is sometimes misunderstood to be the speed a powered vehicle (such as a rocket) must reach to leave orbit; however, this is not the case, as the quoted number is typically the surface escape velocity, and vehicles never achieve that speed direct from the surface.

In fact a vehicle can leave the Earth's gravity at any speed. At higher altitude, the local escape velocity is lower. But at the instant the propulsion stops, the vehicle can only escape if its speed is greater than or equal to the local escape velocity at that position. At sufficiently high altitude this speed can approach 0 m/s.

Orbit

If an object attains escape velocity, but is not directed straight away from the planet, then it will follow a curved path. Even though this path will not form a closed shape, it is still considered an orbit. Assuming that gravity is the only significant force in the system, this object's speed at any point in the orbit will be equal to the escape velocity at that point (due to the conservation of energy, its total energy must always be 0, which implies that it always has escape velocity; see the derivation above). The shape of the orbit will be a parabola whose focus is located at the center of mass of the planet. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular An actual escape requires of course that the orbit not intersect the planet, since this would cause the object to crash. When moving away from the source, this path is called an escape orbit; when moving closer to the source, a capture orbit. An escape orbit (also known as C 3 = 0 orbit is a high-energy Parabolic orbit around the central body A capture orbit is a reverse Escape orbit. It is a Parabolic orbit with as special case a straight line in the direction of the center of the central body Both are known as C3 = 0 orbits (where C3 = - μ/a, and a is the semi-major axis). In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae

Remember that in reality there are many gravitating bodies in space, so that, for instance, a rocket that travels at escape velocity from Earth will not escape to an infinite distance away because it needs an even higher speed to escape the Sun's gravity. In other words, near the Earth, the rocket's orbit will appear parabolic, but eventually its orbit will become an ellipse around the Sun.

List of escape velocities

To leave planet Earth an escape velocity of 11. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 2 km/s is required, however a speed of 42. 1 km/s is required to escape the Sun's gravity (and exit the solar system) from the same position

Because of the atmosphere it is not useful and hardly possible to give an object near the surface of the Earth a speed of 11. The Solar System consists of the Sun and those celestial objects bound to it by Gravity. The Milky Way (a translation of the Latin Via Lactea, in turn derived from the Greek Γαλαξίας (Galaxias sometimes referred to simply 2 km/s, as these speeds are too far in the hypersonic regime for most practical propulsion systems and would cause most objects to burn up on account of atmospheric friction. In Aerodynamics, hypersonic speeds are speeds that are highly Supersonic. For an actual escape orbit a spacecraft is first placed in low Earth orbit and then accelerated to the escape velocity at that altitude, which is a little less — about 10. A Low Earth Orbit (LEO is generally defined as an Orbit within the locus extending from the Earth’s surface up to an altitude of 2000 km 9 km/s. The required acceleration, however, is generally even less because from that sort of an orbit the spacecraft already has a speed of 8 km/s.

Calculating an escape velocity

To expand upon the derivation given in the Overview,

where v_e is the escape velocity, G is the gravitational constant, M is the mass of the body being escaped from, m is the mass of the escaping body, r is the distance between the center of the body and the point at which escape velocity is being calculated, g is the gravitational acceleration at that distance, and μ is the standard gravitational parameter. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Distance is a numerical description of how far apart objects are In Physics, gravitational acceleration is the Acceleration of an object caused by the Force of gravity from another object Small body orbiting a central body Under Standard assumptions in astrodynamics we have m where m \ is the mass ^[2]

The escape velocity at a given height is times the speed in a circular orbit at the same height (compare this with equation (14) in circular motion). In Physics, circular motion is Rotation along a Circle: a circular path or a circular Orbit. This corresponds to the fact that the potential energy with respect to infinity of an object in such an orbit is minus two times its kinetic energy, while to escape the sum of potential and kinetic energy needs to be at least zero.

For a body with a spherically-symmetric distribution of mass, the escape velocity v_e from the surface (in m/s) is approximately 2. 364×10⁻⁵ m^{1. 5}kg^{−0. 5}s⁻¹ times the radius r (in meters) times the square root of the average density ρ (in kg/m³), or:

Deriving escape velocity using calculus

These derivations use calculus, Newton's laws of motion and Newton's law of universal gravitation. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass

Derivation using only g and r

the Earth's escape speed can be derived from "g," the acceleration due to gravity at the Earth's surface. Standard gravity, usually denoted by g 0 or g n is the nominal acceleration due to gravity at the Earth's surface at sea level It is not necessary to know the gravitational constant G or the mass M of the Earth. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass Let

r = the Earth's radius, and

g = the acceleration of gravity at the Earth's surface.

Above the Earth's surface, the acceleration of gravity is governed by Newton's inverse-square law of universal gravitation. In Physics, an inverse-square law is any Physical law stating that some physical Quantity or strength is inversely proportional Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass Accordingly, the acceleration of gravity at height s above the center of the Earth (where s > r ) is g(r / s)². The weight of an object of mass m at the surface is g m, and its weight at height s above the center of the Earth is gm (r / s)². Consequently the energy needed to lift an object of mass m from height s above the Earth's center to height s + ds (where ds is an infinitesimal increment of s) is gm (r / s)² ds. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have Since this decreases sufficiently fast as s increases, the total energy needed to lift the object to infinite height does not diverge to infinity, but converges to a finite amount. That amount is the integral of the expression above:

That is how much kinetic energy the object of mass m needs in order to escape. The kinetic energy of an object of mass m moving at speed v is (1/2)mv². Thus we need

The factor m cancels out, and solving for v we get

If we take the radius of the Earth to be r = 6400 kilometers and the acceleration of gravity at the surface to be g = 9. 8 m/s², we get

This is just a bit over 11 kilometers per second, or a bit under 7 miles per second, as Isaac Newton calculated. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements

Derivation using G and M

Let G be the gravitational constant and let M be the mass of the earth or other body to be escaped. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass

By applying the chain rule, we get:

Because

Since we want escape velocity

and

v₀ is the escape velocity and r₀ is the radius of the planet. In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions. Note that the above derivation relies on the equivalence of inertial mass and gravitational mass. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object

The derivations are consistent

The gravitational acceleration can be obtained from the gravitational constant G and the mass of Earth M:

where r is the radius of Earth. Thus

so the two derivations given above are consistent.

Multiple sources

The escape velocity from a position in a field with multiple sources is derived from the total potential energy per kg at that position, relative to infinity. The potential energies for all sources can simply be added. For the escape velocity this results in the square root of the sum of the squares of the escape velocities of all sources separately.

For example, at the Earth's surface the escape velocity for the combination Earth and Sun is . As a result, to leave the solar system requires a speed of 13. 6 km/s relative to Earth in the direction of the Earth's orbital motion, since the speed is then added to the speed of 30 km/s of that orbital motion

Gravity well

In the hypothetical case of uniform density, the velocity that an object would achieve when dropped in a hypothetical vacuum hole from the surface of the Earth to the center of the Earth is the escape velocity divided by , i. In Physics, a gravity well is the Gravitational potential field around a massive body (a particular kind of Potential well) e. the speed in a circular orbit at a low height. Correspondingly, the escape velocity from the center of the Earth would be times that from the surface.

A refined calculation would take into account the fact that the Earth's mass is not uniformly distributed as the center is approached. This gives higher speeds.

See also

• Gravitational potential energy
• Delta-v budget - speed needed to perform manoeuvres. Potential energy can be thought of as Energy stored within a physical system Delta- v budget (or velocity change budget) is a term used in Astrodynamics and Aerospace industry for velocity change (or delta-''v''
• Gravitational slingshot - 3 body technique for gaining energy
• Gravity well
• Two-body problem
• Black hole - have an escape velocity greater than the speed of light
• Oberth effect - burning fuel deep in a gravity field gives higher velocity

References

• Roger R. In Orbital mechanics and Aerospace engineering, a gravitational slingshot, gravity assist or swing-by is the use of the relative movement and In Physics, a gravity well is the Gravitational potential field around a massive body (a particular kind of Potential well) In Classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e The Oberth effect is a feature of Astronautics where using a Rocket engine close to a gravitational body gives a higher final speed than the same burn executed further Bate, Donald D. Mueller, and Jerry E. White (1971). Fundamentals of astrodynamics. New York: Dover Publications. ISBN 0-486-60061-0.  

1. ^ ^{a b} Solar System Data. Georgia State University. Retrieved on 2007-01-21. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 1189 - Philip II of France and Richard I of England begin to assemble troops to wage the Third Crusade.
2. ^ Bate, Mueller and White, p. 35

External links

Web-based numerical escape velocity calculator