The **acceleration** (also known as *linear acceleration* ) is the physical quantity characterizing the __speed__ with which varies the __speed__ of a particle. It is a *vector magnitude* , so it is totally determined by the properties that identify it: its magnitude, direction and direction.

Summary

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- 1 Denotation
- 2 Equation
- 3 Acceleration in Kinematics
- 4 Signs of acceleration
- 5 Acceleration in Dynamics
- 6 Examples of the concept of acceleration
- 7 Measurement of acceleration
- 8 Acceleration in relativistic mechanics
- 9 Sources
- 10 Bibliographies

Denotation

It is denoted by the letter (a). In the __International System of Units__ (SI) it is expressed in meters / second squared (m / s²), while in the __Cegesimal System of Units__ (CGS) it is measured in centimeters / second squared (cm / s²). Acceleration are units of speed per unit of time, or the dimensions of acceleration are L / T², in other words, units of __length__ (L) per unit of time (T) raised to the second power in the denominator. Examples of frequently used acceleration units are: m / s², km / h² and cm / s².

In summary, the units of acceleration are:

International system:

1 m / s2

Cegesimal system:

1 cm / s2 = 1 Gal

The *Gal* is the name assigned to the acceleration unit in the Cegesimal System, that is, to the centimeter per second at -2 (cm.s-2). The symbol for this unit is Gal. This name was given in honor of __Galileo Galilei__ , who was the first to measure the *acceleration of gravity* . It is an unusual unit, because it does not belong to the International System of Units. The *gravitational acceleration* of the Earth varies between 976 and 983 Gal.

By definition:

1 Gal = 1 cm.s-2.

Its equivalence with the SI unit is:

1 Gal = 0.01 ms-2

Equation

The following formula is used to determine the value of **acceleration** :

a = (velocity variation) / time interval

that is to say,

a = (ΔV / Δt) = (V – Vo) / t

In this formula, Vo means the initial velocity of the particle and V its final velocity reached at time t. When the particle’s initial velocity is zero (that is, the particle starts from the state of rest Vo = 0), then the acceleration formula is reduced to:

a = V / t

Acceleration in Kinematics

- If the particle is moving with constant speed in a straight line (
__Uniform Rectilinear Motion,__or*MRU*), then its acceleration is zero. - If the acceleration of the particle moving in a straight line is constant and different from zero, then said particle is said to be animated by a
*Uniformly*__Varied Rectilinear Movement__or*MRUV*(in this movement the acceleration coincides with the direction of the velocity of the particle at each point of the path). - The acceleration of a particle can take values greater than zero (a> 0, that is, positive in relation to the speed direction), less than zero (a <0, negative in relation to the speed direction) and even equal to zero. When the particle moves with constant acceleration greater than zero, the particle’s
*motion*is said to be*uniformly accelerated*and the particle’s velocity increases over time. If the particle moves with constant acceleration less than zero, then the particle’s*motion*is*uniformly retarded*and the particle’s speed decreases as time passes. Particles with acceleration equal to zero are at rest or moving with constant speed. - In the case of a
*curvilinear movement*, the acceleration produces a variation of the__modulus__and the direction of the velocity vector, that is, the acceleration represents for the velocity vector the same as the velocity for the__position vector__.

Signs of acceleration

If the speed increases in module we say that the movement is *accelerated* , on the other hand if the speed decreases in module we say that the movement is *decelerated* . In *accelerated motion,* acceleration and velocity have the same direction. On the other hand, if the movement is decelerated, the acceleration has the opposite direction (opposite direction) to the speed. In the __Vertical Free Fall Movement__ : when the body ascends it decelerates. When the body descends it accelerates.

Acceleration in Dynamics

In Newtonian mechanics, for a body with constant __mass__ , the acceleration of the body is directly __proportional__ to the __force__ acting on itself and inversely proportional to its mass ( __Newton’s Second Law__ ):

(F) is the *resulting force* acting on the body, (m) is the mass of the body, and (a) is the acceleration.

The previous relation is valid in any __inertial reference system__ .

According to Newtonian mechanics, a particle cannot follow a curved path unless a certain acceleration acts on it as a consequence of the action of a force, since if it did not exist, its movement would be rectilinear. Also, a particle in *rectilinear motion* can only change its speed under the action of an acceleration in the same direction as its speed (directed in the same direction if it accelerates; or in the opposite direction if it decelerates). In short: an object is only accelerated if a force is applied to it. According *to Newton’s Second Law of Motion* , the change in velocity is directly proportional to the applied force. Example: A falling body is accelerated by the __Force of gravity__ .

Examples of the concept of acceleration

- The
*acceleration of gravity*(g) on Earth is the acceleration produced by the Earth’s__gravitational force__; its value on the Earth’s surface, in a Free Fall Movement, is the same for all bodies, whatever their mass when it is possible to neglect the resistance of the air, it is approximately 9.8 m / s². At the poles: g = 9.83 m / s² (Maximum) and in Ecuador: g = 9.78 m / s² (Minimum) - The
*average acceleration*is defined as the__quotient__of a = V / t. - The
*instantaneous acceleration*, is defined as the limit to which the incremental ratio tends Dv / Dt where Dt → 0; this is the__derivative__of the velocity vector with respect to time. - The
__centripetal acceleration__. - The
*angular acceleration*, is defined as the change of the__angular velocity__, that is, a change in the rate of rotation or the direction of the axis.

Acceleration measurement

Acceleration measurement can be done with a data acquisition system and a simple __accelerometer__ .

Acceleration in relativistic mechanics

**Special relativity:**

The analogue of acceleration in relativistic mechanics is called quadriaceleration and is a *quadrivector* whose three spatial components for small speeds coincide with those of **Newtonian acceleration** (the temporal component for small speeds is proportional to the power of the force divided by the __speed of the light__ and the mass of the particle).

A Quadrivector is the mathematical representation in the form of a four-dimensional vector of a magnitude.

**General relativity**:

In the general theory of relativity the case of acceleration is more complicated, since because __space-time itself__ is curved (see image below: curvature of space-time), a particle on which no force acts it can follow a curved path, in fact the curved line that follows a particle on which no external force acts is called the __Geodetic Line__ .