# Acceleration: Wikis

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# Encyclopedia

For the waltz composed by Johann Strauss, see Accelerationen.
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.
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, and more specifically kinematics, acceleration is the change in velocity over time.[1] Because velocity is a vector, it can change in two ways: a change in magnitude and/or a change in direction. In one dimension, i.e. a line, acceleration is the rate at which something speeds up or slows down. However, as a vector quantity, acceleration is also the rate at which direction changes.[2][3] Acceleration has the dimensions L T−2. In SI units, acceleration is measured in metres per second squared (m/s2).

In common speech, the term acceleration commonly 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 resultant (total) force acting on it (Newton's second law):

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

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.

## Tangential and centripetal acceleration

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

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

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

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

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{alignat}{3} \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 (outward) 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, respectively. The negative of the radial acceleration is the centripetal acceleration, which points inward, toward the center of curvature.

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 herself and her 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.

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

ACCELERATION (from Lat. accelerare, to hasten, celer, quick), hastening or quickening; in mechanics, a term employed to denote the rate at which the velocity of a body, whose motion is not uniform, either increases or decreases. (See MECHANICS and HODOGRAPH.)

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# 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