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# acceleration: Wikis

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# Encyclopedia

Classical mechanics
$\mathbf\left\{F\right\} = \frac\left\{\mathrm\left\{d$
{\mathrm{d}t}(m \mathbf{v})
Newton's Second Law

|KeyItems = History of classical mechanicsTemplate:·wTimeline of classical mechanics |Topic1 = Branches |Items1 = StaticsTemplate:·wDynamics / KineticsTemplate:·wKinematicsTemplate:·wApplied mechanicsTemplate:·wCelestial mechanicsTemplate:·wContinuum mechanicsTemplate:·wStatistical mechanics |Topic2 = Formulations |Items2 =

|Topic3 = Fundamental concepts |Items3 = SpaceTemplate:·wTimeTemplate:·wVelocityTemplate:·wSpeedTemplate:·wMassTemplate:·wAccelerationTemplate:·wGravityTemplate:·wForceTemplate:·wImpulseTemplate:·wTorque / Moment / CoupleTemplate:·wMomentumTemplate:·wAngular momentumTemplate:·wInertiaTemplate:·wMoment of inertiaTemplate:·wReference frameTemplate:·wEnergyTemplate:·wKinetic energyTemplate:·wPotential energyTemplate:·wMechanical workTemplate:·wVirtual workTemplate:·wD'Alembert's principle |Topic4 = Core topics |Items4 = Rigid bodyTemplate:·wRigid body dynamicsTemplate:·wEuler's equations (rigid body dynamics)Template:·wMotionTemplate:·wNewton's laws of motionTemplate:·wNewton's law of universal gravitationTemplate:·wEquations of motionTemplate:·wInertial frame of referenceTemplate:·wNon-inertial reference frameTemplate:·wRotating reference frameTemplate:·wFictitious forceTemplate:·wLinear motionTemplate:·wMechanics of planar particle motionTemplate:·wDisplacement (vector)Template:·wRelative velocityTemplate:·wFrictionTemplate:·wSimple harmonic motionTemplate:·wHarmonic oscillatorTemplate:·wVibrationTemplate:·wDampingTemplate:·wDamping ratioTemplate:·wRotational motionTemplate:·wCircular motionTemplate:·wUniform circular motionTemplate:·wNon-uniform circular motionTemplate:·wCentripetal forceTemplate:·wCentrifugal forceTemplate:·wCentrifugal force (rotating reference frame)Template:·wReactive centrifugal forceTemplate:·wCoriolis forceTemplate:·wPendulumTemplate:·wRotational speedTemplate:·wAngular accelerationTemplate:·wAngular velocityTemplate:·wAngular frequencyTemplate:·wAngular displacement |Topic5 = Scientists |Items5 = Isaac NewtonTemplate:·wJeremiah HorrocksTemplate:·wLeonhard EulerTemplate:·wJean le Rond d'AlembertTemplate:·wAlexis ClairautTemplate:·wJoseph Louis LagrangeTemplate:·wPierre-Simon LaplaceTemplate:·wWilliam Rowan HamiltonTemplate:·wSiméon-Denis Poisson |cTopic=Fundamental concepts }} [[File:|250px|right|thumb|Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time t is found in the limit as time interval Δt → 0.]]

File:Acceleration
Components of acceleration for a planar curved motion. The tangential component at is due to the change in speed of traversal, and points along the curve in the direction of the velocity vector. The centripetal component ac is due to the change in direction of the velocity vector and is normal to the trajectory, pointing toward the center of curvature of the path.

In physics, acceleration is the rate of change of velocity over time.[1] In one dimension, acceleration is the rate at which something speeds up or slows down. However, since velocity is a vector, acceleration describes the rate of change of both the magnitude and the direction of velocity.[2][3] Acceleration has the dimensions L T −2. In SI units, acceleration is measured in meters per second per second (m/s2).

In common speech, the term acceleration is used for an increase in speed (the magnitude of velocity); a decrease in speed is called deceleration. In physics, a change in the direction of velocity also is an acceleration: for rotary motion, the change in direction of velocity results in centripetal (toward the center) acceleration; where as the rate of change of speed is a tangential acceleration.

In classical mechanics, for a body with constant mass, the acceleration of the body is proportional to the net force acting on it (Newton's second law):

$\mathbf\left\{F\right\} = m\mathbf\left\{a\right\} \quad \to \quad \mathbf\left\{a\right\} = \mathbf\left\{F\right\}/m$

where F is the resultant force acting on the body, m is the mass of the body, and a is its acceleration.

## Average and instantaneous acceleration

Average acceleration is the change in velocity (Δv) divided by the change in time (Δt). Instantaneous acceleration is the acceleration at a specific point in time which is for a very short interval of time as Δt approaches zero.

## Tangential and centripetal acceleration

The velocity of a particle moving on a curved path as a function of time can be written as:

$\mathbf\left\{v\right\} \left(t\right) =v\left(t\right) \frac \left\{\mathbf\left\{v\right\}\left(t\right)\right\}\left\{v\left(t\right)\right\} = v\left(t\right) \mathbf\left\{u\right\}_\mathrm\left\{t\right\}\left(t\right) ,$

with v(t) equal to the speed of travel along the path, and

$\mathbf\left\{u\right\}_\mathrm\left\{t\right\} = \frac \left\{\mathbf\left\{v\right\}\left(t\right)\right\}\left\{v\left(t\right)\right\} \ ,$

a unit vector tangent to the path pointing in the direction of motion at the chosen moment in time. Taking into account both the changing speed v(t) and the changing direction of ut, the acceleration of a particle moving on a curved path on a planar surface can be written using the chain rule of differentiation as:

\begin\left\{alignat\right\}\left\{3\right\}

\mathbf{a} & = \frac{d \mathbf{v}}{dt} \\

          & =  \frac{\mathrm{d}v }{\mathrm{d}t} \mathbf{u}_\mathrm{t} +v(t)\frac{d \mathbf{u}_\mathrm{t}}{dt} \\
& = \frac{\mathrm{d}v }{\mathrm{d}t} \mathbf{u}_\mathrm{t}+ \frac{v^2}{R}\mathbf{u}_\mathrm{n}\ , \\


\end{alignat}

where un is the unit (inward) normal vector to the particle's trajectory, and R is its instantaneous radius of curvature based upon the osculating circle at time t. These components are called the tangential acceleration and the radial acceleration or centripetal acceleration (see also circular motion and centripetal force).

Extension of this approach to three-dimensional space curves that cannot be contained on a planar surface leads to the Frenet-Serret formulas.[4][5]

## Relation to relativity

After completing his theory of special relativity, Albert Einstein realized that forces felt by objects undergoing constant proper acceleration are actually feeling themselves being accelerated, so that, for example, a car's acceleration forwards would result in the driver feeling a slight pressure between himself and his seat. In the case of gravity, which Einstein concluded is not actually a force, this is not the case; acceleration due to gravity is not felt by an object in free-fall. This was the basis for his development of general relativity, a relativistic theory of gravity.

## Notes

1. ^ Crew, Henry (2008). The Principles of Mechanics. BiblioBazaar, LLC. pp. 43. ISBN 0559368712.
2. ^ Bondi, Hermann (1980). Relativity and Common Sense. Courier Dover Publications. pp. 3. ISBN 0486240215.
3. ^ Lehrman, Robert L. (1998). Physics the Easy Way. Barron's Educational Series. pp. 27. ISBN 0764102362.
4. ^ Larry C. Andrews & Ronald L. Phillips (2003). Mathematical Techniques for Engineers and Scientists. SPIE Press. p. 164. ISBN 0819445061.
5. ^ Ch V Ramana Murthy & NC Srinivas (2001). Applied Mathematics. New Delhi: S. Chand & Co.. p. 337. ISBN 81-219-2082-5.

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# Wiktionary

Up to date as of January 15, 2010

## English

### Etymology

From Latin acceleratio; compare French accélération

### Pronunciation

• enPR: əksĕlərā'shən, IPA: /ək.sɛl.əˈreɪ.ʃən/, SAMPA: /@ksEl@"reIS@n/
• help, file
Rhymes: -eɪʃǝn

### Noun

 Singular acceleration Plural countable and uncountable; plural accelerations

acceleration (countable and uncountable; plural accelerations)

1. (uncountable) The act of accelerating, or the state of being accelerated; increase of motion or action; as opposed to retardation or deceleration.
a falling body moves toward the earth with an acceleration of velocity
2. (countable) The amount by which a speed or velocity increases (and so a scalar quantity or a vector quantity).
The boosters produce an acceleration of 20 metres per second per second.
• A period of social improvement, or of intellectual advancement, contains within itself a principle of accelerationIsaac Taylor
3. (physics) The change of velocity with respect to time (can include deceleration or changing direction).

#### Usage notes

Acceleration in SI units is measured in metres per second per second (m/s2), or in imperial units in feet per second per second (ft/s2).

#### Translations

The translations below need to be checked and inserted above into the appropriate translation tables, removing any numbers. Numbers do not necessarily match those in definitions. See instructions at Help:How to check translations.

## Interlingua

### Noun

acceleration

1. acceleration

## Swedish

### Noun

Inflection for acceleration Singular Plural
common Indefinite Definite Indefinite Definite
Base form acceleration accelerationen accelerationer accelerationerna
Possessive form accelerations accelerationens accelerationers accelerationernas

acceleration c.

1. acceleration; a change in velocity

# Simple English

Acceleration is a measure of how fast velocity changes. Acceleration is the change of velocity divided by the change of time. Acceleration is a vector, and therefore includes both a size and a direction.

## = Examples

=

• An object moving north at 10 meters per second. The object speeds up and now is moving north at 15 meters per second. The object has accelerated.
• An apple falls down. It starts falling at 0 meters per second. At the end of the first second, the apple is moving at 9.8 meters per second. The apple has accelerated. At the end of the second second, the apple is moving down at 19.6 meters per second. The apple has accelerated again.
• Jane walks east at 3 kilometers per hour. Jane's velocity does not change. Jane's acceleration is zero.
• Tom walks east at 3 kilometers per hour. Tom turns and walks south at 3 kilometers per hour. Tom has had a nonzero acceleration.
• Sally walks east at 3 kilometers per hour. Sally slows down. After, Sally walks east at 1.5 kilometers per hour. Sally has had a nonzero acceleration.

## Finding acceleration

Acceleration is the rate of change of the velocity of an object. Acceleration $\mathbfTemplate:A$ can be found by using:



\mathbfTemplate:A = {\mathbf{v_1} - \mathbf{v_0} \over { t_1 - t_0 }}

where

$\mathbf\left\{v_0\right\}$ is the velocity at the start
$\mathbf\left\{v_1\right\}$ is the velocity at the end
$t_0$ is the time at the start
$t_1$ is the time at the end

Sometimes the change in velocity $\mathbf\left\{v_1\right\} - \mathbf\left\{v_0\right\}$ is written as Δ$\mathbf\left\{v\right\}$. Sometimes the change in time $\left\{ t_1 - t_0 \right\}$ is written as Δt.

In difficult situations, the acceleration can be calculated using mathematics: in calculus, acceleration is the derivative of the velocity (with respect to time), $\mathbf\left\{a\right\} = \frac\left\{\mathrm\left\{d\right\}\mathbf\left\{v\right\}\right\}\left\{\mathrm\left\{d\right\}t\right\}$.

### Units of measurement

Acceleration has its own units of measurement. For example, if velocity is measured in meters per second, and if time is measured in seconds, then acceleration is measured in meters per second squared (m/s2).

### Other words

Acceleration can be positive or negative. When the acceleration is negative (but the velocity does not change direction), it is sometimes called deceleration. For example, when a car brakes it decelerates. Physicists usually only use the word "acceleration".

## Newton's second law of motion

There are rules for how things move. These rules are called "laws of motion". Isaac Newton is the scientist who first wrote down the main laws of motion. According to Newton's Second Law of Motion, the force something needs to accelerate an object depends on the object's mass (the amount of "stuff" the object is made from or how "heavy" it is). The formula of Newton's Second Law of Motion is $\mathbf\left\{F\right\} = m \mathbf\left\{a\right\}$, where $\mathbf\left\{a\right\}$ is the acceleration, $\mathbf\left\{F\right\}$ is the force, and $m$ the mass. This formula is very well-known, and it is very important in physics. Newton's Second Law of Motion, in short "Newton's Second Law", is often one of the first things that physics students learn.krc:Терклениу