TheInfoList

In
mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximately 10.7 million ... , acceleration is the rate of change of the
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ... of an object with respect to time. Accelerations are
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
quantities (in that they have
magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (mathematics), the relative size of an object *Norm (mathematics), a term for the size or length of a vector *Order of ...
and
direction Direction may refer to: *Relative direction, for instance left, right, forward, backwards, up, and down ** Anatomical terms of location for those used in anatomy *Cardinal direction Mathematics and science *Direction vector, a unit vector that ...
). The orientation of an object's acceleration is given by the orientation of the ''net''
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ... acting on that object. The magnitude of an object's acceleration, as described by
Newton's Second Law In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which a ...
, is the combined effect of two causes: * the net balance of all external
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ... s acting onto that object — magnitude is
directly proportional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to this net resulting force; * that object's
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
, depending on the materials out of which it is made — magnitude is
inversely proportional In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
to the object's mass. The SI unit for acceleration is
metre per second squared The metre per second squared is the unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatr ...
(, ). For example, when a
vehicle A vehicle (from la, vehiculum) is a machine A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine A molecular machine, nanite, or ... starts from a standstill (zero velocity, in an
inertial frame of reference In classical physics and special relativity, an inertial frame of reference is a frame of reference that is not undergoing acceleration. In an inertial frame of reference, a physical object with zero net force acting on it moves with a const ...
) and travels in a straight line at increasing speeds, it is ''accelerating'' in the direction of travel. If the vehicle turns, an acceleration occurs toward the new direction and changes its motion vector. The acceleration of the vehicle in its current direction of motion is called a ''linear'' (or ''tangential'' during
circular motion In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ... s) acceleration, the
reaction Reaction may refer to a process or to a response to an action, event, or exposure: Physics and chemistry *Chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic transformations, chemical transformat ...
to which the passengers on board experience as a force pushing them back into their seats. When changing direction, the effecting acceleration is called ''radial'' (or ''orthogonal'' during circular motions) acceleration, the reaction to which the passengers experience as a
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force A fictitious force (also called a pseudo force, d'Alembert force, or inertial force) is a force that appears to act on a mass whose motion is described using a non-inertial ref ...
. If the speed of the vehicle decreases, this is an acceleration in the opposite direction and mathematically a negative, sometimes called ''deceleration'', and passengers experience the reaction to deceleration as an
inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an o ... l force pushing them forward. Such negative accelerations are often achieved by
retrorocket A retrorocket (short for ''retrograde rocket'') is a rocket engine A rocket engine uses stored rocket propellant Rocket propellant is the reaction mass of a rocket. This reaction mass is ejected at the highest achievable velocity from ...
burning in
spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite alt=, A full-size model of the Earth observation satellite ERS 2 ">ERS_2.html" ;"title="Earth observation satellite ERS 2">Earth obse ... . Both acceleration and deceleration are treated the same, they are both changes in velocity. Each of these accelerations (tangential, radial, deceleration) is felt by passengers until their relative (differential) velocity are neutralized in
reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. It is called a ''name ... to the vehicle.

# Definition and properties ## Average acceleration An object's average acceleration over a period of
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
is its change in
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ... $\left( \Delta \mathbf\right)$ divided by the duration of the period $\left( \Delta t\right)$. Mathematically, :$\bar = \frac.$

## Instantaneous acceleration Instantaneous acceleration, meanwhile, is the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ... of the average acceleration over an
infinitesimal In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
interval of time. In the terms of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ... , instantaneous acceleration is the
derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... of the velocity vector with respect to time: :$\mathbf = \lim_ \frac = \frac$ As acceleration is defined as the derivative of velocity, , with respect to time and velocity is defined as the derivative of position, , with respect to time, acceleration can be thought of as the
second derivative In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study o ... of with respect to : :$\mathbf = \frac = \frac$ (Here and elsewhere, if motion is in a straight line,
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
quantities can be substituted by scalars in the equations.) By the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
, it can be seen that the
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ... of the acceleration function is the velocity function ; that is, the area under the curve of an acceleration vs. time ( vs. ) graph corresponds to velocity. :$\mathbf = \int \mathbf \ dt$ Likewise, the integral of the jerk function , the derivative of the acceleration function, can be used to find acceleration at a certain time: :$\mathbf = \int \mathbf \ dt$

## Units

Acceleration has the
dimensions In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
of velocity (L/T) divided by time, i.e.
L T −2. The SI unit of acceleration is the
metre per second squared The metre per second squared is the unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatr ...
(m s−2); or "metre per second per second", as the velocity in metres per second changes by the acceleration value, every second.

## Other forms

An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, although its speed may be constant. In this case it is said to be undergoing ''centripetal'' (directed towards the center) acceleration.
Proper acceleration In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at r ...
, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an
accelerometer An accelerometer is a tool that measures proper acceleration In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative ... . In classical mechanics, for a body with constant mass, the (vector) acceleration of the body's center of mass is proportional to the net
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ... vector (i.e. sum of all forces) acting on it (
Newton's second law Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
): :$\mathbf = m\mathbf \quad \to \quad \mathbf = \frac$ where F is the net force acting on the body, ''m'' is the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
of the body, and a is the center-of-mass acceleration. As speeds approach the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
,
relativistic effects Relativistic quantum chemistry combines relativistic mechanics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanics, quantum mechanical desc ...
become increasingly large.

# Tangential and centripetal acceleration  The velocity of a particle moving on a curved path as a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
of time can be written as: :$\mathbf \left(t\right) =v\left(t\right) \frac = v\left(t\right) \mathbf_\mathrm\left(t\right) ,$ with ''v''(''t'') equal to the speed of travel along the path, and :$\mathbf_\mathrm = \frac \ ,$ 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 u''t'', the acceleration of a particle moving on a curved path can be written using the
chain rule In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...
of differentiation for the product of two functions of time as: :$\begin \mathbf & = \frac \\ & = \frac \mathbf_\mathrm +v\left(t\right)\frac \\ & = \frac \mathbf_\mathrm+ \frac\mathbf_\mathrm\ , \\ \end$ where un is the unit (inward)
normal vector In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
to the particle's trajectory (also called ''the principal normal''), and r is its instantaneous
radius of curvature In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a co ...
based upon the
osculating circle In differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point ''p'' on the curve has been traditionally defined as the circle passing through ''p'' and a pair of additional points on the curve in ...
at time ''t''. These components are called the tangential acceleration and the normal or radial acceleration (or centripetal acceleration in circular motion, see also
circular motion In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ... and
centripetal force A centripetal force (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power ... ). Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the
Frenet–Serret formulas spanned by T and N In differential geometry Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geomet ...
.

# Special cases

## Uniform acceleration ''Uniform'' or ''constant'' acceleration is a type of motion in which the
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ... of an object changes by an equal amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in
free fall #REDIRECT Free fall #REDIRECT Free fall In Newtonian physics, free fall is any motion of a body where gravity Gravity (), or gravitation, is a list of natural phenomena, natural phenomenon by which all things with mass or energy—inc ... in a uniform gravitational field. The acceleration of a falling body in the absence of resistances to motion is dependent only on the
gravitational field In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ... strength '' g'' (also called ''acceleration due to gravity''). By
Newton's Second Law In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which a ...
the
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ... $\mathbf$ acting on a body is given by: :$\mathbf = m \mathbf$ Because of the simple analytic properties of the case of constant acceleration, there are simple formulas relating the
displacement Displacement may refer to: Physical sciences Mathematics and Physics *Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path c ...
, initial and time-dependent
velocities The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction o ... , and acceleration to the time elapsed: :$\mathbf\left(t\right) = \mathbf_0 + \mathbf_0 t+ \tfrac \mathbft^2 = \mathbf_0 + \fract$ :$\mathbf\left(t\right) = \mathbf_0 + \mathbf t$ : where * $t$ is the elapsed time, * $\mathbf_0$ is the initial displacement from the origin, * $\mathbf\left(t\right)$ is the displacement from the origin at time $t$, * $\mathbf_0$ is the initial velocity, * $\mathbf\left(t\right)$ is the velocity at time $t$, and * $\mathbf$ is the uniform rate of acceleration. In particular, the motion can be resolved into two orthogonal parts, one of constant velocity and the other according to the above equations. As
Galileo Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a in the field of who focuses their studies on a specific question or field o ... showed, the net result is parabolic motion, which describes, e. g., the trajectory of a projectile in a vacuum near the surface of Earth.

## Circular motion

In uniform
circular motion In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ... , that is moving with constant ''speed'' along a circular path, a particle experiences an acceleration resulting from the change of the direction of the velocity vector, while its magnitude remains constant. The derivative of the location of a point on a curve with respect to time, i.e. its velocity, turns out to be always exactly tangential to the curve, respectively orthogonal to the radius in this point. Since in uniform motion the velocity in the tangential direction does not change, the acceleration must be in radial direction, pointing to the center of the circle. This acceleration constantly changes the direction of the velocity to be tangent in the neighboring point, thereby rotating the velocity vector along the circle. • For a given speed $v$, the magnitude of this geometrically caused acceleration (centripetal acceleration) is inversely proportional to the radius $r$ of the circle, and increases as the square of this speed: :$a_c = \frac \;.$ • Note that, for a given
angular velocity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ... $\omega$, the centripetal acceleration is directly proportional to radius $r$. This is due to the dependence of velocity $v$ on the radius $r$. :$v = \omega r.$ Expressing centripetal acceleration vector in polar components, where $\mathbf$ is a vector from the centre of the circle to the particle with magnitude equal to this distance, and considering the orientation of the acceleration towards the center, yields :$\mathbf = -\frac\cdot \frac\;.$ As usual in rotations, the speed $v$ of a particle may be expressed as an
''angular speed'' with respect to a point at the distance $r$ as :$\omega = \frac .$ Thus $\mathbf = -\omega^2 \mathbf \;.$ This acceleration and the mass of the particle determine the necessary
centripetal force A centripetal force (from Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power ... , directed ''toward'' the centre of the circle, as the net force acting on this particle to keep it in this uniform circular motion. The so-called '
centrifugal force In Newtonian mechanics, the centrifugal force is an inertial force A fictitious force (also called a pseudo force, d'Alembert force, or inertial force) is a force that appears to act on a mass whose motion is described using a non-inertial ref ...
', appearing to act outward on the body, is a so-called pseudo force experienced in the
frame of reference In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ... of the body in circular motion, due to the body's
linear momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum ( pl. momenta) is the product of the mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what ...
, a vector tangent to the circle of motion. In a nonuniform circular motion, i.e., the speed along the curved path is changing, the acceleration has a non-zero component tangential to the curve, and is not confined to the principal normal, which directs to the center of the osculating circle, that determines the radius $r$ for the centripetal acceleration. The tangential component is given by the angular acceleration $\alpha$, i.e., the rate of change $\alpha = \dot\omega$ of the angular speed $\omega$ times the radius $r$. That is, :$a_t = r \alpha.$ The sign of the tangential component of the acceleration is determined by the sign of the
angular acceleration In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
($\alpha$), and the tangent is always directed at right angles to the radius vector.

# Relation to relativity

## Special relativity

The special theory of relativity describes the behavior of objects traveling relative to other objects at speeds approaching that of light in a vacuum.
Newtonian mechanics Newton's laws of motion are three Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows: ''Law 1''. A body continues ...
is exactly revealed to be an approximation to reality, valid to great accuracy at lower speeds. As the relevant speeds increase toward the speed of light, acceleration no longer follows classical equations. As speeds approach that of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached; an object with mass can approach this speed
asymptotically 250px, A curve intersecting an asymptote infinitely many times. In analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc ...
, but never reach it.

## General relativity

Unless the state of motion of an object is known, it is impossible to distinguish whether an observed force is due to
gravity Gravity (), or gravitation, is a by which all things with or —including s, s, , and even —are attracted to (or ''gravitate'' toward) one another. , gravity gives to s, and the causes the s of the oceans. The gravitational attracti ... or to acceleration—gravity and inertial acceleration have identical effects.
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest physicists of all time. Einstein is known for developing the theory of relativity The theo ... called this the
equivalence principle In the theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (fr ...
, and said that only observers who feel no force at all—including the force of gravity—are justified in concluding that they are not accelerating.Brian Greene, '' The Fabric of the Cosmos: Space, Time, and the Texture of Reality'', page 67. Vintage

# Conversions

* Acceleration (differential geometry) *
Four-vector In special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in oth ...
: making the connection between space and time explicit *
Gravitational acceleration In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
*
Inertia Inertia is the resistance of any physical object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an o ... * Orders of magnitude (acceleration) *
Shock (mechanics) A mechanical or physical shock is a sudden acceleration caused, for example, by impact (mechanics), impact, drop, kick, earthquake, or explosion. Shock is a transient physical excitation. Shock describes matter subject to extreme rates of force ...
* Shock and vibration data logger
measuring 3-axis acceleration * Space travel using constant acceleration *
Specific force Specific force is defined as the non-gravitational force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. movi ...