Velocity

This article is about velocity in physics. For other uses, see Velocity (disambiguation).

In physics, velocity is defined as the rate of change of position. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change It is a vector physical quantity; both speed and direction are required to define it. A physical Quantity is a physical property that can be quantified 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. In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or In Mathematics, the absolute value (or modulus) 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 Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of For example, "5 metres per second" is a scalar and not a vector, whereas "5 metres per second east" is a vector. In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or 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:

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

The rate of change of velocity is referred to as acceleration.

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 $\, x(t + {\Delta t})$ at time $\, t +{\Delta t}$ , can be computed as the derivative of position:

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

The equation for an object's velocity can be obtained mathematically by evaluating the integral of the equation for its acceleration beginning from some initial period time $\, t_0$ to some point in time later $\, t_n$ . In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space

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

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

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:

$v^2 = u^2 + 2a\Delta x.\,$

The above equations are valid for both Newtonian mechanics and special relativity. Torricelli's equation is an equation created by Evangelista Torricelli to find the final Velocity of an object moving with a constant acceleration without having a Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial 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. In Kinematics, relative velocity is the vector difference between the velocities of two objects as evaluated in terms of a single Coordinate

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:

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

The kinetic energy is a scalar quantity. The kinetic energy of an object is the extra Energy which it possesses due to its motion In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics, a scalar is a simple Physical quantity that is not changed by Coordinate system rotations or translations (in Newtonian mechanics or

Escape velocity is the minimum velocity a body must have in order to escape from the gravitational field of the earth. In Physics, escape velocity is the speed where the Kinetic energy of an object is equal to the magnitude of its Gravitational potential energy 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

Relative velocity

Main article: Relative velocity

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. In Kinematics, relative velocity is the vector difference between the velocities of two objects as evaluated in terms of a single Coordinate 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. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

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:

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

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:

$\, v_{rel} = v - (-w)$ , if the two objects are moving in opposite directions, or:

$\, v_{rel} = v -(+w)$ , 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). In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by Do not confuse with Angular frequency The unit for angular velocity is rad/s

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.

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

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.

$v_T=\frac{|\mathbf{r}\times\mathbf{v}|}{|\mathbf{r}|}=\omega|\mathbf{r}|$

such that

$\omega=\frac{|\mathbf{r}\times\mathbf{v}|}{|\mathbf{r}|^2}$

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. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position The sign convention for angular momentum is the same as that for angular velocity.

$L=mrv_T=mr^2\omega\,$

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. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star These relations are known as Kepler's laws of planetary motion

References

^ Basic principle

Halliday, David, Robert Resnick and Jearl Walker, Fundamentals of Physics, Wiley; 7 Sub edition (June 16, 2004). In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without In Kinematics, relative velocity is the vector difference between the velocities of two objects as evaluated in terms of a single Coordinate A free falling object achieves its terminal velocity when the downward force of gravity ( Fg)equals the upward force of drag ( Fd) The term hypervelocity usually refers to a very high Velocity, typically over 3000 meters per second (6700 mph 11000 km/h 10000 ft/s or Mach In Physics, in particular in Special relativity and General relativity, the four-velocity of an object is a Four-vector (vector in four-dimensional In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity The Lorentz factor or Lorentz term appears in several equations in Special relativity, including Time dilation, Length contraction, and the Proper-velocity, the distance traveled per unit time elapsed on the clocks of a traveling object equals coordinate velocity at low speeds ISBN 0471232319.

External links

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

Kinematics

← Integrate . Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space . . Differentiate →
Displacement (Distance) | Velocity (Speed) | Acceleration | Jerk | Snap

In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Physics, displacement is the vector that specifies the position of a point or a particle in reference to a previous position or to the origin of the chosen Distance is a numerical description of how far apart objects are Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of In Physics, jerk, jolt (especially in British English) surge or lurch, is the rate of change of Acceleration; that is

Dictionary

-noun

(physics) A vector quantity that denotes the time rate of change of position, or a speed with the directional component.
Rapidity of motion.
The rate of occurrence.
(economics) The number of times that an average unit of currency is spent during a specific period of time..

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Contents

Equation of motion

Relative velocity

Scalar velocities

Polar coordinates

See also

References

External links

Dictionary

velocity

-noun