- For the waltz composed by Johann Strauss, see Accelerationen.

In kinematics, acceleration is defined as the first derivative of velocity with respect to time (that is, the rate of change of velocity), or equivalently as the second derivative of position. It is a vector quantity with dimension L T^{−2}. In SI units, acceleration is measured in metres per second squared (m/s^{2}).

In common speech, the term acceleration is only used for an increase in speed (the magnitude of velocity); a decrease in speed is called deceleration. In physics, any increase or decrease in speed is referred to as acceleration, and also a change in the direction of velocity is an acceleration (the centripetal acceleration; whereas the rate of change of speed is the tangential acceleration).

In classical mechanics, the acceleration of a body is proportional to the resultant (total) force acting on it (Newton's second law):

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

where

F is the resultant force acting on the body,

m is the

mass of the body, and

a is its acceleration.

## Tangential and centripetal acceleration

The acceleration of a particle can be written as:

- $mathbf\{a\}\; =\; frac\{mathrm\{d\}v\}\{mathrm\{d\}t\}\; mathbf\{u\}\_mathrm\{t\}\; +\; frac\{v^2\}\{R\}mathbf\{u\}\_mathrm\{n\}$

where

u_{t} and

u_{n} are (respectively) the unit

tangent vector and the unit

normal vector to the particle's trajectory, and

R is its radius of

curvature. These components are called the

tangential acceleration and the

centripetal acceleration, respectively.

## Relation to relativity

After completing his theory of

special relativity,

Albert Einstein realized that forces felt by objects undergoing constant

proper acceleration are indistinguishable from those in a gravitational field. This was the basis for his development of

general relativity, a relativistic theory of

gravity. This is also the basis for the popular

twin paradox, which asks why one twin ages less when moving away from his sibling at near light-speed and then returning, since the non-aging twin can say that it is the other twin that was moving.

General relativity solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). In

special relativity, only

inertial frames of reference (non-accelerated frames) can be used and are equivalent;

general relativity considers

all frames, even accelerated ones, to be equivalent. (The path from these considerations to the full theory of general relativity is traced in the

introduction to general relativity.)

## See also

## External links