# Displacement (vector)

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[math]\displaystyle{ \textbf{F} = \frac{d}{dt} (m\textbf{v}) }[/math] |

A **displacement** is a vector whose length is the shortest distance from the initial to the final position of a point P.^{[1]} It quantifies both the distance and direction of an imaginary motion along a straight line from the initial position to the final position of the point. A displacement may be identified with the translation that maps the initial position to the final position.

A displacement may be also described as a 'relative position': the final position of a point (x*_{f}*) relative to its initial position (x

*), and a displacement vector can be mathematically defined as the difference between the final and initial positions:*

_{i}- [math]\displaystyle{ \boldsymbol{s}=\boldsymbol{s_f-s_i}=\Delta\boldsymbol{s} }[/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 instantaneous speed then is distinct from velocity, or 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 equivalently 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 rotations 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, vibration sensing and other sciences and engineering disciplines.

- [math]\displaystyle{ \boldsymbol{v}=\frac{\text{d}\boldsymbol{s}}{\text{d}t} }[/math] (where d
is an infinitesimally small displacement)**s**

- [math]\displaystyle{ \boldsymbol{a}=\frac{\text{d}\boldsymbol{v}}{\text{d}t}=\frac{\text{d}^2\boldsymbol{s}}{\text{d}t^2} }[/math]

- [math]\displaystyle{ \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.^{[2]} 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 a sum of an infinite series, enabling several analytical techniques in engineering and physics. The fourth order derivative is called jounce.

## See also

- Equipollence (geometry)
- Position vector
- Affine space

## References

- ↑ Tom Henderson. "Describing Motion with Words".
*The Physics Classroom*. http://www.physicsclassroom.com/Class/1DKin/U1L1c.cfm. Retrieved 2 January 2012. - ↑ Stewart, James (2001). "§2.8 - The Derivative As A Function".
*Calculus*(2nd ed.). Brooks/Cole. ISBN 0-534-37718-1.