Escape velocity Totally Explained

Everything about Escape Velocity totally explained

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 = -Gm_1m_2/r

. 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's actually a speed rather than a velocity, for example 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, for example, so that gravity will never manage to pull it back. For the sake of simplicity, unless stated otherwise, we'll assume that the scenario we're 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 doesn't 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. 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'll 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'll deal with, so by the conservation of energy, ť

(K + U_g)_i = (K + U_g)_f. ,

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

frac

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. 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's 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, for example the American Cape Canaveral in Florida and the European Guiana Space Centre, only 5 degrees from the equator in French Guiana.
Escape velocity is independent of the mass of the escaping object. It doesn't 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). More massive objects require more energy to reach escape velocity.

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 isn't 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 isn't directed straight away from the planet, then it'll follow a curved path. Even though this path won't form a closed shape, it's 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. 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. Both are known as C3 = 0 orbits (where C3 = - μ/a, and a is the semi-major axis).
Remember that in reality there are many gravitating bodies in space, so that, for instance, a rocket that travels at escape velocity from Earth won't 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

Further Information

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