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# acceleration

[ak-sel-uh-rey-shuhn]
acceleration, change in the velocity of a body with respect to time. Since velocity is a vector quantity, involving both magnitude and direction, acceleration is also a vector. In order to produce an acceleration, a force must be applied to the body. The magnitude of the force F must be directly proportional to both the mass of the body m and the desired acceleration a, according to Newton's second law of motion, F=ma. The exact nature of the acceleration produced depends on the relative directions of the original velocity and the force. A force acting in the same direction as the velocity changes only the speed of the body. An appropriate force acting always at right angles to the velocity changes the direction of the velocity but not the speed. An example of such an accelerating force is the gravitational force exerted by a planet on a satellite moving in a circular orbit. A force may also act in the opposite direction from the original velocity. In this case the speed of the body is decreased. Such an acceleration is often referred to as a deceleration. If the acceleration is constant, as for a body falling near the earth, the following formulas may be used to compute the acceleration a of a body from knowledge of the elapsed time t, the distance s through which the body moves in that time, the initial velocity vi, and the final velocity vf:
a=(vf2-vi2)/2s
a=2(s-vit)/t2
a=(vf-vi)/t

Property of the motion of an object traveling in a circular path. Centripetal describes the force on the object, directed toward the centre of the circle, which causes a constant change in the object's direction and thus its acceleration. The magnitude of centripetal acceleration math.a is equal to the square of the object's velocity math.v along the curved path divided by the object's distance math.r from the centre of the circle, or math.a = math.v2/math.r.

Rate of change of velocity. Acceleration, like velocity, is a vector quantity: it has both magnitude and direction. The velocity of an object moving on a straight path can change in magnitude only, so its acceleration is the rate of change of its speed. On a curved path, the velocity may or may not change in magnitude, but it will always change in direction, which means that the acceleration of an object moving on a curved path can never be zero. If velocity is stated in metres per second (m/s) and the time interval in seconds (s), then the units of acceleration are metres per second per second (m/s/s, or m/s2). Seealso centripetal acceleration.

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.)