NationMaster - Encyclopedia: Velocity

In physics, velocity is defined as the rate of change of position. It is a vector physical quantity; both speed and direction are required to define it. In the SI (metric) system, it is measured in metres per second: (m/s) or ms^-1. The scalar absolute value (magnitude) of velocity is speed. For example, "5 metres per second" is a scalar and not a vector, whereas "5 metres per second east" is a vector. The average velocity v of an object moving through a displacement $( Delta mathbf{x})$ during a time interval $(Δ t)$ is described by the formula: Velocity can refer to: Velocity in physics Velocity of money Velocity (software), a Java template engine Velocity (company), a publisher of video games WWE Velocity, a professional wrestling television show Velocity (novel), a novel by US author Dean Koontz Velocity (ministry), a Junior High ministry run by Pastor Hank Sanford... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... This article is about derivatives and differentiation in mathematical calculus. ... Look up position in Wiktionary, the free dictionary. ... This article is about vectors that have a particular relation to the spatial coordinates. ... A physical quantity is either a quantity within physics that can be measured (e. ... â€œSIâ€ redirects here. ... Metre per second (U.S. spelling: meter per second) is an SI derived unit of both speed (scalar) and velocity (vector), defined by distance in metres divided by time in seconds. ... See scalar for an account of the broader concept also used in mathematics and computer science. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs. ... This article does not cite any references or sources. ... See scalar for an account of the broader concept also used in mathematics and computer science. ...

$bar{mathbf{v}} = frac{Delta mathbf{x}}{Delta t}$

The rate of change of velocity is referred to as acceleration. Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

1 Equation of motion
2 Relative velocity
- 2.1 Scalar velocities
3 Polar coordinates
4 See also
5 References
6 External links

Equation of motion

Main article: Equation of motion

The instant velocity vector v of an object that has positions x(t) at time t and x(t+ $Δ t$ ) at time t+ $Δ t$ , can be computed as the derivative of position: It has been suggested that SUVAT equations be merged into this article or section. ... This article is about derivatives and differentiation in mathematical calculus. ...

$mathbf{v} = lim_{Delta t to 0}{{mathbf{x}(t+Delta t)-mathbf{x}(t)} over Delta t}={mathrm{d}mathbf{x} over mathrm{d}t}$

The equation for an object's velocity can be obtained mathematically by taking the integral of the equation for its acceleration beginning from some initial period time $t 0$ to some point in time later $t n$ . This article is about the concept of integrals in calculus. ...

The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time $(Δ t)$ is:

$mathbf{v} = mathbf{u} + mathbf{a} Delta t$

The average velocity of an object undergoing constant acceleration is $begin{matrix} frac {(mathbf{u} + mathbf{v})}{2} ; end{matrix}$ , where u is the initial velocity and v is the final velocity. To find the displacement, x, of such an accelerating object during a time interval, $Δ t$ , then: Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ...

$Delta mathbf{x} = frac {( mathbf{u} + mathbf{v} )}{2}Delta t$

When only the object's initial velocity is known, the expression,

$Delta mathbf{x} = mathbf{u} Delta t + frac{1}{2}mathbf{a} Delta t^2,$

can be used.

This can be expanded to give the position at any time t in the following way:

$mathbf{x}(t) = mathbf{x}(0) + Delta mathbf{x} = mathbf{x}(0) + mathbf{u} Delta t + frac{1}{2}mathbf{a} Delta t^2,$

These basic equations for final velocity and displacement can be combined to form an equation that is independent of time, also known as Torricelli's equation: This equation was created by Evangelista Torricelli to find the final velocity of a moving object without having a known time interval and was named after him. ...

$v^2 = u^2 + 2aDelta x.,$

The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words only relative velocity can be calculated. It has been suggested that this article or section be merged with Classical mechanics. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Relative velocity is a measurement of velocity between two objects moving in different frames of reference. ...

In Newtonian mechanics, the kinetic energy (energy of motion), $E K$ , of a moving object is linear with both its mass and the square of its velocity: The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... For other uses, see Mass (disambiguation). ...

$E_{K} = begin{matrix} frac{1}{2} end{matrix} mv^2.$

The kinetic energy is a scalar quantity. See scalar for an account of the broader concept also used in mathematics and computer science. ...

Escape velocity is the minimum velocity a body must have in order to escape from the gravitational field of the earth. To escape from the earth's gravitational field an object must have greater kinetic energy than its gravitational potential energy. The value of the escape velocity from Earth is approximately 11100 m/s Space Shuttle Atlantis launches on mission STS-71. ...

Relative velocity

Full article: Relative velocity Relative velocity is a measurement of velocity between two objects moving in different frames of reference. ...

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame. For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ...

If an object A is moving with velocity vector v and an object B with velocity vector w , then the velocity of object A relative to object B is defined as the difference of the two velocity vectors: This article is about vectors that have a particular relation to the spatial coordinates. ...

$mathbf{v}_{Arelative toB} = mathbf{v} - mathbf{w}$

Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is:

$mathbf{v}_{Brelative toA} = mathbf{w} - mathbf{v}$

Usually the inertial frame is chosen in which the latter of the two mentioned objects is in rest.

Scalar velocities

In the one dimensional case^[1], the velocities are scalars and the equation is either:

v r e l = v - ( - w)

, if the two objects are moving in opposite directions, or:

v r e l = v - ( + w)

, if the two objects are moving in the same direction.

Polar coordinates

In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an angular velocity, which is the rate of rotation about the origin (with positive quantities representing counter-clockwise rotation and negative quantities representing clockwise rotation, in a right-handed coordinate system). A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ... Angular velocity describes the speed of rotation and the orientation of the instantaneous axis about which the rotation occurs. ...

The radial and angular velocities can be derived from the Cartesian velocity and displacement vectors by decomposing the velocity vector into radial and transverse components. The transverse velocity is the component of velocity along a circle centered at the origin. The term transverse means side-to-side, as opposed to longitudinal, which means front-to-back. In automotive engineering, the term transverse refers to an engine in which the crankshaft is oriented side-to-side relative to the wheels of the vehicle. ...

$mathbf{v}=mathbf{v}_T+mathbf{v}_R$

where

$mathbf{v}_T$ is the transverse velocity

$mathbf{v}_R$ is the radial velocity

The magnitude of the radial velocity is the dot product of the velocity vector and the unit vector in the direction of the displacement.

$v_R=frac{mathbf{v} cdot mathbf{r}}{left|mathbf{r}right|}$

where

$mathbf{r}$ is displacement

The magnitude of the transverse velocity is that of the cross product of the unit vector in the direction of the displacement and the velocity vector. It is also the product of the angular speed ( $ω$ ) and the magnitude of the displacement.

$v_T=frac{|mathbf{r}timesmathbf{v}|}{|mathbf{r}|}=omega|mathbf{r}|$

such that

$omega=frac{|mathbf{r}timesmathbf{v}|}{|mathbf{x}|^2}$

Angular momentum in scalar form is the mass times the distance to the origin times the transverse velocity, or equivalently, the mass times the distance squared times the angular speed. The sign convention for angular momentum is the same as that for angular velocity. This gyroscope remains upright while spinning due to its angular momentum. ...

$L=mrv_T=mr^2omega,$

where

$m,$ is mass

$r=|mathbf{r}|$

If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion Two bodies with a slight difference in mass orbiting around a common barycenter. ... Johannes Keplers primary contributions to astronomy/astrophysics were his three laws of planetary motion. ...

References

^ Basic principle

Halliday, David, Robert Resnick and Jearl Walker, Fundamentals of Physics, Wiley; 7 Sub edition (June 16, 2004). ISBN 0471232319.

External links

Speed and Velocity (The Physics Classroom)
Introduction to Mechanisms (Carnegie Mellon University)

Kinematics Kinematics (Greek ÎºÎ¹Î½ÎµÎ¹Î½,kinein, to move) is a branch of mechanics which describes the motion of objects without the consideration of the masses or forces that bring about the motion. ...

← Integrate ... Differentiate →
Displacement (Distance) | Velocity (Speed) | Acceleration | Jerk | Snap This article is about the concept of integrals in calculus. ... This article is about derivatives and differentiation in mathematical calculus. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... This article does not cite any references or sources. ... Acceleration is the time rate of change of velocity and/or direction, and at any point on a velocity-time graph, it is given by the slope of the tangent to the curve at that point. ... This article is about the physics concept of jerk. ... Look up jounce in Wiktionary, the free dictionary. ...

Categories: Dynamics | Physical quantity

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	It turns out that whether a person is at home or at work, they expect their search tools to not only find what they are looking for quickly, but to be intuitive and user-friendly, too—traits not often found in software designed specifically for the enterprise, including search.
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The Physics Classroom (1187 words)
	velocity vector is the same as the direction in which an object is moving.
	Remember that displacement refers to the change in position and that velocity is based upon this position change.
	Velocity, a vector quantity, is the rate at which the position changes.

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Contents

Relative velocity

Scalar velocities

Polar coordinates

See also

References

External links