Time–frequency analysis

2013-11-18T00:36:41Z

2601:8:B280:6F2:91BD:34DB:57EF:E44: /* Ideal TF distribution function */

[[Image:Distancedisplacement.svg|thumb|336px|Displacement versus distance traveled along a path.]]
A '''displacement''' is the shortest [[distance]] from the initial to the final [[position (vector)|position]] of a point P.<ref>{{cite web
|url=http://www.physicsclassroom.com/Class/1DKin/U1L1c.cfm
|title=Describing Motion with Words
|author= Tom Henderson
|work= The Physics Classroom
|accessdate=2 January 2012}}
</ref>
Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P. A 'displacement vector' represents the length and direction of that imaginary straight path.

A [[position vector]] expresses the position of a point P in space in terms of a displacement from an arbitrary reference point O (typically the origin of a coordinate system). Namely, it indicates both the distance and direction of an imaginary motion along a straight line from the reference position to the actual position of the point.

A displacement may be also described as a 'relative position': the final position of a point ('''''R<sub>f</sub>''''') relative to its initial position ('''''R<sub>i</sub>'''''), and a displacement vector can be mathematically defined as the [[vector subtraction|difference]] between the final and initial position vectors:

:<math>
\boldsymbol{s}=\boldsymbol{R_f-R_i}=\Delta\boldsymbol{R}
</math>

In considering motions of objects over time the instantaneous [[velocity]] of the object is the rate of change of the displacement as a function of time. The velocity then is distinct from the instantaneous [[speed]] which is the time rate of change of the distance traveled 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 equivalenty a moving origin (e.g. an initial position or origin which is fixed to a train wagon, which in turn moves with respect to 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. (Note that the average [[velocity]], as a vector, differs from the [[average speed]] that is the ratio of the path length — a scalar — and the time interval.)

== Rigid body ==
In dealing with the motion of a [[rigid body]], the term ''displacement'' may also include the [[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 '''[[angular displacement]]'''.

== Derivatives ==
For a position vector '''''s''''' that is a function of time ''t'', the derivatives can be computed with respect to ''t''. These derivatives have common utility in the study of [[kinematics]], [[control theory]], and other sciences and engineering disciplines.

[[Velocity]]
:<math>\boldsymbol{v}=\frac{\text{d}\boldsymbol{s}}{\text{d}t}</math> (where d'''''s''''' is an [[Differential (infinitesimal)|infinitesimally]] small displacement)

[[Acceleration]]
:<math>\boldsymbol{a}=\frac{\text{d}\boldsymbol{v}}{\text{d}t}=\frac{\text{d}^2\boldsymbol{s}}{\text{d}t^2}</math>

[[Jerk (physics)|Jerk]]
:<math>\boldsymbol{j}=\frac{\text{d}\boldsymbol{a}}{\text{d}t}=\frac{\text{d}^2\boldsymbol{v}}{\text{d}t^2}=\frac{\text{d}^3\boldsymbol{s}}{\text{d}t^3}</math>

These common names correspond to terminology used in basic kinematics.<ref name='stewart'>{{cite book
|last= Stewart
|first= James
|authorlink=James Stewart (mathematician)
|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.

==See also==
* [[Position vector]]
* [[Affine space]]

==References==
{{reflist}}

{{Kinematics}}

[[Category:Motion]]
[[Category:Length]]

[[Category:Vectors]]

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Time–frequency analysis