View source for Displacement (geometry)

{{short description|Vector relating the initial and the final positions of a moving point}}
[[File:Distancedisplacement.svg|thumb|Displacement versus distance travelled along a path]]
{{Classical mechanics}}
In [[Geometry|geometry]] and mechanics, a '''displacement''' is a [[Euclidean vector|vector]] whose length is the shortest [[Distance|distance]] from the initial to the final [[Position (vector)|position]] of a point P undergoing [[Physics:Motion|motion]].<ref>{{cite web| url=http://www.physicsclassroom.com/Class/1DKin/U1L1c.cfm|title=Describing Motion with Words| author=Tom Henderson| work=The Physics Classroom|access-date=2 January 2012}}</ref> It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final position of the point [[Physics:Trajectory|trajectory]]. A displacement may be identified with the [[Translation (geometry)|translation]] that maps the initial position to the final position.

A displacement may be also described as a ''relative position'' (resulting from the motion), that is, as the final position {{math|''x''<sub>f</sub>}} of a point relative to its initial position {{math|''x''<sub>i</sub>}}. The corresponding displacement vector can be defined as the [[Affine space#Subtraction and Weyl's axioms|difference]] between the final and initial positions:
<math display="block"> s = x_\textrm{f} - x_\textrm{i} = \Delta{x}</math>

In considering motions of objects over time, the instantaneous [[Physics:Velocity|velocity]] of the object is the rate of change of the displacement as a function of time. The instantaneous [[Physics:Speed|speed]], then, is distinct from velocity, or the time rate of change of the distance travelled along a specific path. The velocity may be equivalently defined as the time rate of change of the position vector. If one considers a moving initial position, or equivalently a moving origin (e.g. an initial position or origin which is fixed to a train wagon, which in turn moves on its rail track), the velocity of P (e.g. a point representing the position of a passenger walking on the train) may be referred to as a relative velocity, as opposed to an absolute velocity, which is computed with respect to a point which is considered to be 'fixed in space' (such as, for instance, a point fixed on the floor of the train station).

For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity, which is a vector, and differs thus from the average speed, which is a scalar quantity.

== Rigid body ==
In dealing with the motion of a [[Rigid body|rigid body]], the term ''displacement'' may also include the [[Physics:Rotation|rotation]]s of the body. In this case, the displacement of a particle of the body is called '''linear displacement''' (displacement along a line), while the rotation of the body is called ''[[Physics:Angular displacement|angular displacement]]''.<ref>{{cite web |url=https://www.grc.nasa.gov/www/k-12/airplane/angdva.html |title=Angular Displacement, Velocity, Acceleration |author=<!--Not stated--> |date= 13 May 2021|website= NASA Glenn Research Center|publisher= National Aeronautics and Space Administration|access-date=9 November 2023 |quote=}}</ref>

== Derivatives ==

For a position vector <math>\mathbf{s}</math> that is a function of time <math>t</math>, the derivatives can be computed with respect to <math>t</math>.  The first two derivatives are frequently encountered in physics.

;[[Physics:Velocity|Velocity]]
:<math>\mathbf{v} = \frac{d\mathbf{s}}{\mathrm{d}t}</math> 
;[[Physics:Acceleration|Acceleration]]
:<math>\mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{s}}{dt^2}</math>
;[[Physics:Jerk|Jerk]]
:<math>\mathbf{j} = \frac{d\mathbf{a}}{dt} = \frac{d^2\mathbf{v}}{dt^2}=\frac{d^3\mathbf{s}}{dt^3}</math>

These common names correspond to terminology used in basic kinematics.<ref name='stewart'>{{cite book
|last= Stewart
|first= James
|title= [[Calculus]]
|publisher= Brooks/Cole
|year= 2001
|edition= 2nd
|isbn= 0-534-37718-1
|chapter= §2.8 - The Derivative As A Function
}}
</ref>  By extension, the higher order derivatives can be computed in a similar fashion.  Study of these higher order derivatives can improve approximations of the original displacement function.  Such higher-order terms are required in order to accurately represent the displacement function as [[Taylor series|a sum of an infinite series]], enabling several analytical techniques in engineering and physics. The fourth order derivative is called [[Physics:Jounce|jounce]].

==See also==
* [[Affine space]]
* [[Physics:Deformation (mechanics)|Deformation (mechanics)]]
* [[Physics:Displacement field (mechanics)|Displacement field (mechanics)]]
* [[Equipollence (geometry)]]
* [[Motion vector]]
* Position vector
* [[Astronomy:Radial velocity|Radial velocity]]

==References==
{{Reflist}}

==External links==

{{Kinematics}}
{{Classical mechanics derived SI units}}

[[Category:Motion (physics)]]
[[Category:Length]]
[[Category:Vector physical quantities]]
[[Category:Geometric measurement]]

{{Sourceattribution|Displacement (geometry)}}