# Displacement (geometry)

__: Vector relating the initial and the final positions of a moving point__

**Short description**Part of a series on |

Classical mechanics |
---|

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

In geometry and mechanics, a **displacement** is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion.^{[1]} 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 trajectory. 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* (resulting from the motion), that is, as the final position *x*_{f} of a point relative to its initial position *x*_{i}. The corresponding displacement vector can be defined as the difference between the final and initial positions:
[math]\displaystyle{ s = x_\textrm{f} - x_\textrm{i} = \Delta{x} }[/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 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, 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*.^{[2]}

## Derivatives

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

- Velocity
- [math]\displaystyle{ \mathbf{v} = \frac{d\mathbf{s}}{\mathrm{d}t} }[/math]
- Acceleration
- [math]\displaystyle{ \mathbf{a} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{s}}{dt^2} }[/math]
- Jerk
- [math]\displaystyle{ \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.^{[3]} 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

- Affine space
- Deformation (mechanics)
- Displacement field (mechanics)
- Equipollence (geometry)
- Motion vector
- Position vector
- Radial velocity

## References

- ↑ Tom Henderson. "Describing Motion with Words".
*The Physics Classroom*. http://www.physicsclassroom.com/Class/1DKin/U1L1c.cfm. - ↑ "Angular Displacement, Velocity, Acceleration". National Aeronautics and Space Administration. 13 May 2021. https://www.grc.nasa.gov/www/k-12/airplane/angdva.html.
- ↑ Stewart, James (2001). "§2.8 - The Derivative As A Function".
*Calculus*(2nd ed.). Brooks/Cole. ISBN 0-534-37718-1.

## External links

Original source: https://en.wikipedia.org/wiki/Displacement (geometry).
Read more |