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mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objects r ...
, acceleration is the rate of change of the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of an object with respect to time. Accelerations are
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
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). 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 of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
acting on that object. The magnitude of an object's acceleration, as described by
Newton's Second Law Newton's laws of motion are three basic 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: # A body remains at rest, or in motion ...
, 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 of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
s acting onto that object — magnitude is
directly proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
to this net resulting force; * that object's
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
, depending on the materials out of which it is made — magnitude is
inversely proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
to the object's mass. The SI unit for acceleration is
metre per second squared The metre per second squared is the unit of acceleration in the International System of Units (SI). As a derived unit, it is composed from the SI base units of length, the metre, and time, the second. Its symbol is written in several forms as m/s ...
(, \mathrm). For example, when a
vehicle A vehicle (from la, vehiculum) is a machine that transports people or cargo. Vehicles include wagons, bicycles, motor vehicles (motorcycles, cars, trucks, buses, mobility scooters for disabled people), railed vehicles (trains, trams), wa ...
starts from a standstill (zero velocity, in an
inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
) 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, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of ro ...
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 *Nuclear reaction *Reaction (physics), as defined by Newton's third law *Chain reaction (disambiguation). Biology and me ...
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 centripetal 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 (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
. If the speed of the vehicle decreases, this is an acceleration in the opposite direction and mathematically a negative, sometimes called deceleration or retardation, and passengers experience the reaction to deceleration as an
inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
l force pushing them forward. Such negative accelerations are often achieved by
retrorocket A retrorocket (short for ''retrograde rocket'') is a rocket engine providing thrust opposing the motion of a vehicle, thereby causing it to decelerate. They have mostly been used in spacecraft, with more limited use in short-runway aircraft land ...
burning in
spacecraft A spacecraft is a vehicle or machine designed to fly in outer space. A type of artificial satellite, spacecraft are used for a variety of purposes, including communications, Earth observation, meteorology, navigation, space colonization, p ...
. Both acceleration and deceleration are treated the same, as 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 acceleration due to change in speed.


Definition and properties


Average acceleration

An object's average acceleration over a period of
time Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to ...
is its change in
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, \Delta \mathbf, divided by the duration of the period, \Delta t. Mathematically, \bar = \frac.


Instantaneous acceleration

Instantaneous acceleration, meanwhile, is the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of the average acceleration over an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
interval of time. In the terms of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, instantaneous acceleration is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
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, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
of with respect to : \mathbf = \frac = \frac (Here and elsewhere, if motion is in a straight line,
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
quantities can be substituted by scalars in the equations.) By the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
, it can be seen that the
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of the acceleration function is the velocity function ; that is, the area under the curve of an acceleration vs. time ( vs. ) graph corresponds to the change of velocity. \mathbf = \int \mathbf \, dt Likewise, the integral of the jerk function , the derivative of the acceleration function, can be used to find the change of acceleration at a certain time: \mathbf = \int \mathbf \, dt


Units

Acceleration has the
dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordina ...
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 of acceleration in the International System of Units (SI). As a derived unit, it is composed from the SI base units of length, the metre, and time, the second. Its symbol is written in several forms as m/s ...
(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 ...
, 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. Proper acceleration is the acceleration (the rate of change of velocity) of a body in its own instantaneous rest frame; this is different from coordinate acceleration, which is accele ...
. In
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classical ...
, 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 of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
vector (i.e. sum of all forces) acting on it ( Newton’s second law): \mathbf = m\mathbf \quad \implies \quad \mathbf = \frac where is the net force acting on the body, is the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementar ...
of the body, and is the center-of-mass acceleration. As speeds approach the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
,
relativistic effects Relativistic quantum chemistry combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is an explanation for the color of ...
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 type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
of time can be written as: \mathbf(t) = v(t) \frac = v(t) \mathbf_\mathrm(t) , with 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 and the changing direction of , the acceleration of a particle moving on a curved path can be written using the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
of differentiation for the product of two functions of time as: \begin \mathbf & = \frac \\ & = \frac \mathbf_\mathrm +v(t)\frac \\ & = \frac \mathbf_\mathrm+ \frac\mathbf_\mathrm\ , \end where is the unit (inward)
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
to the particle's trajectory (also called ''the principal normal''), and is its instantaneous
radius of curvature In differential geometry, the radius of curvature, , is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius o ...
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 i ...
at time . These components are called the
tangential acceleration In mechanics, acceleration is the Rate (mathematics), rate of change of the velocity of an object with respect to time. Accelerations are Euclidean vector, vector quantities (in that they have Magnitude (mathematics), magnitude and Direction ...
and the normal or radial acceleration (or centripetal acceleration in circular motion, see also
circular motion In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of ro ...
and
centripetal force A centripetal force (from Latin ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
). Geometrical analysis of three-dimensional space curves, which explains tangent, (principal) normal and binormal, is described by the
Frenet–Serret formulas In differential geometry, the Frenet–Serret formulas describe the kinematic properties of a particle moving along a differentiable curve in three-dimensional Euclidean space \mathbb^, or the geometric properties of the curve itself irrespective ...
.


Special cases


Uniform acceleration

''Uniform'' or ''constant'' acceleration is a type of motion in which the
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
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 In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on i ...
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, a gravitational field is a model used to explain the influences that a massive body extends into the space around itself, producing a force on another massive body. Thus, a gravitational field is used to explain gravitational phenome ...
strength (also called ''acceleration due to gravity''). By
Newton's Second Law Newton's laws of motion are three basic 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: # A body remains at rest, or in motion ...
the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a p ...
\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 ...
, initial and time-dependent
velocities Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, and acceleration to the time elapsed: \begin \mathbf(t) &= \mathbf_0 + \mathbf_0 t + \tfrac \mathbft^2 = \mathbf_0 + \tfrac \left(\mathbf_0 + \mathbf(t)\right) t \\ \mathbf(t) &= \mathbf_0 + \mathbf t \\ (t) &= ^2 + 2\mathbf mathbf(t)-\mathbf_0\end where * t is the elapsed time, * \mathbf_0 is the initial displacement from the origin, * \mathbf(t) is the displacement from the origin at time t, * \mathbf_0 is the initial velocity, * \mathbf(t) 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) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
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, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of ro ...
, 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, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
\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 ''centrum'', "center" and ''petere'', "to seek") is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous c ...
, 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 (also called a "fictitious" or "pseudo" force) that appears to act on all objects when viewed in a rotating frame of reference. It is directed away from an axis which is paralle ...
', appearing to act outward on the body, is a so-called
pseudo force A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. It is related to Newton's second law of motion, which tr ...
experienced in the
frame of reference In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both mathema ...
of the body in circular motion, due to the body's
linear momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass a ...
, 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 (\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 basic 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: # A body remains at rest, or in motion ...
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 In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
, 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 In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
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 and most influential physicists of all time. Einstein is best known for developing the theory ...
called this the
equivalence principle In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...
, 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


See also

* Acceleration (differential geometry) *
Four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
: making the connection between space and time explicit *
Gravitational acceleration In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodies ...
*
Inertia Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...
*
Orders of magnitude (acceleration) This page lists examples of the acceleration occurring in various situations. They are grouped by orders of magnitude. See also * G-force *Gravitational acceleration * Mechanical shock * Standard gravity * International System of Units (SI) * SI ...
*
Shock (mechanics) A mechanical or physical shock is a sudden acceleration caused, for example, by impact, drop, kick, earthquake, or explosion. Shock is a transient physical excitation. Shock describes matter subject to extreme rates of force with respect to ti ...
* 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 per unit mass. :\mbox = \frac Specific force (also called g-force and mass-specific force) is measured in meters/second² (m·s−2) which is the units for acceleration. Thus, specific for ...


References


External links


Acceleration Calculator
Simple acceleration unit converter
Acceleration Calculator
Acceleration Conversion calculator converts units form meter per second square, kilometer per second square, millimeter per second square & more with metric conversion. {{Authority control Dynamics (mechanics) Kinematic properties Temporal rates Vector physical quantities