Acceleration: Difference between revisions

In [[mechanics]], '''acceleration''' is the [[Rate (mathematics)|rate]] of change of the [[velocity]] of an object with respect to time. Acceleration is one of several components of [[kinematics]], the study of [[motion]]. Accelerations are [[Euclidean vector|vector]] quantities (in that they have [[Magnitude (mathematics)|magnitude]] and [[Direction (geometry)|direction]]).<ref>{{cite book |title=Relativity and Common Sense |first=Hermann |last=Bondi |pages=[https://archive.org/details/relativitycommon0000bond/page/3 3] |publisher=Courier Dover Publications |year=1980 |isbn=978-0-486-24021-3 |url=https://archive.org/details/relativitycommon0000bond/page/3 }}</ref><ref>{{cite book |title=Physics the Easy Way |pages=[https://archive.org/details/physicseasyway00lehr_0/page/27 27] |first=Robert L. |last=Lehrman |publisher=Barron's Educational Series |year=1998 |isbn=978-0-7641-0236-3 |url=https://archive.org/details/physicseasyway00lehr_0/page/27 }}</ref> The orientation of an object's acceleration is given by the orientation of the ''net'' [[force]] acting on that object. The magnitude of an object's acceleration, as described by [[Newton's second law]],<ref>{{cite book |title=The Principles of Mechanics |first=Henry |last=Crew |publisher=BiblioBazaar, LLC |year=2008 |isbn=978-0-559-36871-4 |pages=43}}</ref> is the combined effect of two causes:

* that object's [[mass]], depending on the materials out of which it is made — magnitude is [[Inverse proportionality|inversely proportional]] to the object's mass.

The [[International System of Units|SI]] unit for acceleration is [[metre per second squared]] ({{nowrap|m⋅s⁻²}}, <math>\mathrm{\tfrac{m}{s^2}}</math>).

For example, when a [[vehicle]] starts from a [[Wikt:standstill|standstill]] (zero velocity, in an [[inertial frame of reference]]) 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]]s) acceleration, the [[reaction (physics)|reaction]] 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]]. If the speed of the vehicle decreases, this is an acceleration in the opposite direction of the velocity vector (mathematically a [[negative number|negative]], if the movement is unidimensional and the velocity is positive), sometimes called '''deceleration'''<ref>{{cite book |title=Mechanics |author1=P. Smith |author2=R. C. Smith |edition=2nd, illustrated, reprinted |publisher=John Wiley & Sons |year=1991 |isbn=978-0-471-92737-2 |page=39 |url=https://books.google.com/books?id=Zzh_unG7OAsC}} [https://books.google.com/books?id=Zzh_unG7OAsC&pg=PA39 Extract of page 39]</ref><ref>{{cite book |title=Physics, Volume One: Chapters 1-17, Volume 1 |author1=John D. Cutnell |author2=Kenneth W. Johnson |edition=1st0, illustrated |publisher=John Wiley & Sons |year=2014 |isbn=978-1-118-83688-0 |page=36 |url=https://books.google.com/books?id=PJWDBgAAQBAJ}} [https://books.google.com/books?id=PJWDBgAAQBAJ&pg=PA36 Extract of page 36]</ref> or '''retardation''', and passengers experience the reaction to deceleration as an [[inertia]]l force pushing them forward.  Such negative accelerations are often achieved by [[retrorocket]] burning in [[spacecraft]].<ref>{{cite book |author1=Raymond A. Serway |author2=Chris Vuille |author3=Jerry S. Faughn |title=College Physics, Volume 10 |year=2008 |publisher=Cengage |isbn=9780495386933 |page=32 |url=https://books.google.com/books?id=CX0u0mIOZ44C&pg=PA32}}</ref> 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 neutralised in [[frame of reference|reference]] to the acceleration due to change in speed.

[[File:Acceleration as derivative of velocity along trajectory.svg|right|thumb|Acceleration is the rate of change of velocity. At any point on a trajectory, the magnitude of the acceleration is given by the rate of change of velocity in both magnitude and direction at that point. The true acceleration at time {{mvar|t}} is found in the limit as [[time interval]] {{math|Δ''t'' → 0}} of {{math|Δ'''v'''/Δ''t''}}.]]

An object's average acceleration over a period of [[time in physics|time]] is its change in [[velocity]], <math>\Delta \mathbf{v}</math>, divided by the duration of the period, <math>\Delta t</math>. Mathematically,

<math display="block">\bar{\mathbf{a}} = \frac{\Delta \mathbf{v}}{\Delta t}.</math>

Instantaneous acceleration, meanwhile, is the [[limit of a function|limit]] of the average acceleration over an [[infinitesimal]] interval of time. In the terms of [[calculus]], instantaneous acceleration is the [[derivative]] of the velocity vector with respect to time:

<math display="block">\mathbf{a} = \lim_{{\Delta t} \to 0} \frac{\Delta \mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt}.</math>

<math display="block">\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{x}}{dt^2}.</math>

 

(Here and elsewhere, if [[Rectilinear motion|motion is in a straight line]], [[Euclidean vector|vector]] quantities can be substituted by [[Scalar (physics)|scalars]] in the equations.)

<math display="block" qid=Q11465>\mathbf{\Delta v} =  \int \mathbf{a} \, dt.</math>

Likewise, the integral of the [[Jerk (physics)|jerk]] function {{math|''j''(''t'')}}, the derivative of the acceleration function, can be used to find the change of acceleration at a certain time:

<math display="block">\mathbf{\Delta a} =  \int \mathbf{j} \, dt.</math>

[[Proper acceleration]], the acceleration of a body relative to a free-fall condition, is measured by an instrument called an [[accelerometer]].

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]] vector (i.e. sum of all forces) acting on it ([[Newton's laws of motion#Newton's second law|Newton's second law]]):

<math display="block" qid=Q2397319>\mathbf{F} = m\mathbf{a} \quad \implies \quad \mathbf{a} = \frac{\mathbf{F}}{m},</math>

where {{math|'''F'''}} is the net force acting on the body, {{mvar|m}} is the [[mass]] of the body, and {{math|'''a'''}} is the center-of-mass acceleration. As speeds approach the [[speed of light]], [[Special relativity|relativistic effects]] become increasingly large.

{{See also|Centripetal force#Local coordinates|Tangential velocity}}

<math display="block">\mathbf{v}(t) = v(t) \frac{\mathbf{v}(t)}{v(t)} = v(t) \mathbf{u}_\mathrm{t}(t) , </math>

with {{math|''v''(''t'')}} equal to the speed of travel along the path, and

<math display="block">\mathbf{u}_\mathrm{t} = \frac{\mathbf{v}(t)}{v(t)} \, , </math>

a [[Differential geometry of curves#Tangent vector|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 {{math|''v''(''t'')}} and the changing direction of {{math|'''u'''_''t''}}, the acceleration of a particle moving on a curved path can be written using the [[chain rule]] of differentiation<ref>{{cite web|last1=Weisstein|first1=Eric W.|title=Chain Rule| url=http://mathworld.wolfram.com/ChainRule.html |website=Wolfram MathWorld| publisher=Wolfram Research| access-date=2 August 2016}}</ref> for the product of two functions of time as:

           & =  \frac{dv}{dt} \mathbf{u}_\mathrm{t} +v(t)\frac{d \mathbf{u}_\mathrm{t}}{dt} \\

where {{math|'''u'''_n}} is the unit (inward) [[Differential geometry of curves#Normal or curvature vector|normal vector]] to the particle's trajectory (also called ''the principal normal''), and {{math|'''r'''}} is its instantaneous [[Curvature#Curvature of plane curves|radius of curvature]] based upon the [[Osculating circle#Mathematical description|osculating circle]] at time {{mvar|t}}. The components

:<math>\mathbf{a}_\mathrm{t} = \frac{dv }{dt} \mathbf{u}_\mathrm{t} \quad\text{and}\quad \mathbf{a}_\mathrm{c} = \frac{v^2}{r}\mathbf{u}_\mathrm{n}</math>

are called the [[tangential acceleration]] and the normal or radial acceleration (or centripetal acceleration in circular motion, see also [[circular motion]] and [[centripetal force]]), respectively.

\mathbf{s}(t) &= \mathbf{s}_0 + \mathbf{v}_0 t + \tfrac{1}{2} \mathbf{a}t^2 = \mathbf{s}_0 + \tfrac{1}{2} \left(\mathbf{v}_0 + \mathbf{v}(t)\right) t \\

{v^2}(t) &= {v_0}^2 + 2\mathbf{a \cdot}[\mathbf{s}(t)-\mathbf{s}_0],

\end{align}</math>

 

where

* <math>\mathbf{s}_0</math> is the initial displacement from the origin,

* <math>\mathbf{s}(t)</math> is the displacement from the origin at time <math>t</math>,

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]] showed, the net result is parabolic motion, which describes, e.g., the trajectory of a projectile in vacuum near the surface of Earth.<ref>{{cite book |title=Understanding physics |author1=David C. Cassidy |author2=Gerald James Holton |author3=F. James Rutherford |publisher=Birkhäuser |year=2002 |isbn=978-0-387-98756-9 |page=146 |url=https://books.google.com/books?id=iPsKvL_ATygC&q=parabolic+arc+uniform-acceleration+galileo&pg=PA146}}</ref>

<math display="block"> \mathbf {a_c}= -\frac{v^2}{|\mathbf {r}|}\cdot \frac{\mathbf {r}}{|\mathbf {r}|}\,. </math>

 

As usual in rotations, the speed <math>v</math> of a particle may be expressed as an [[angular velocity|''angular speed'']] with respect to a point at the distance <math>r</math> as

<math display="block" qid=Q161635>\omega = \frac {v}{r}.</math>

 

Thus <math> \mathbf {a_c}= -\omega^2  \mathbf {r}\,. </math>

In multi-dimensional [[Cartesian coordinate system]]s, acceleration is broken up into components that correspond with each dimensional axis of the coordinate system. In a two-dimensional system, where there is an x-axis and a y-axis, corresponding acceleration components are defined as<ref>{{Cite web |title=The Feynman Lectures on Physics Vol. I Ch. 9: Newton's Laws of Dynamics |url=https://www.feynmanlectures.caltech.edu/I_09.html |access-date=2024-01-04 |website=www.feynmanlectures.caltech.edu}}</ref><math display="block">a_x=dv_x/dt=d^2x/dt^2,</math> <math display="block">a_y=dv_y/dt=d^2y/dt^2.</math>The two-dimensional acceleration vector is then defined as <math>\textbf{a}=<a_x, a_y></math>. The magnitude of this vector is found by the [[Euclidean distance|distance formula]] as<math display="block">|a|=\sqrt{a_x^2+a_y^2}.</math>In three-dimensional systems where there is an additional z-axis, the corresponding acceleration component is defined as<math display="block">a_z=dv_z/dt=d^2z/dt^2.</math>The three-dimensional acceleration vector is defined as <math>\textbf{a}=<a_x, a_y, a_z></math> with its magnitude being determined by<math display="block">|a|=\sqrt{a_x^2+a_y^2+a_z^2}.</math>