Definitions

# Acceleration

[ak-sel-uh-rey-shuhn]
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/s2).

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):

$mathbf\left\{F\right\} = mmathbf\left\{a\right\} quad to quad mathbf\left\{a\right\} = mathbf\left\{F\right\}/m$
where F is the resultant force acting on the body, m is the mass of the body, and a is its acceleration.

## Tangential and centripetal acceleration

The acceleration of a particle can be written as:
$mathbf\left\{a\right\} = frac\left\{mathrm\left\{d\right\}v\right\}\left\{mathrm\left\{d\right\}t\right\} mathbf\left\{u\right\}_mathrm\left\{t\right\} + frac\left\{v^2\right\}\left\{R\right\}mathbf\left\{u\right\}_mathrm\left\{n\right\}$
where ut and un 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.)