# Displacement (geometry)

## Vector relating the initial and the final positions of a moving point / From Wikipedia, the free encyclopedia

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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:

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.

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]}

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

- Velocity
- $\mathbf {v} ={\frac {d\mathbf {s} }{\mathrm {d} t}}$
- Acceleration
- $\mathbf {a} ={\frac {d\mathbf {v} }{dt}}={\frac {d^{2}\mathbf {s} }{dt^{2}}}$
- Jerk
- $\mathbf {j} ={\frac {d\mathbf {a} }{dt}}={\frac {d^{2}\mathbf {v} }{dt^{2}}}={\frac {d^{3}\mathbf {s} }{dt^{3}}}$

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.

- Tom Henderson. "Describing Motion with Words".
*The Physics Classroom*. Retrieved 2 January 2012. - "Angular Displacement, Velocity, Acceleration".
*NASA Glenn Research Center*. National Aeronautics and Space Administration. 13 May 2021. Retrieved 9 November 2023. - Stewart, James (2001). "§2.8 - The Derivative As A Function".
*Calculus*(2nd ed.). Brooks/Cole. ISBN 0-534-37718-1.

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