![Distancedisplacement](https://upload.wikimedia.org/wikipedia/commons/4/40/Distancedisplacement.svg)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and mechanics, a displacement is a
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
whose length is the shortest
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
from the initial to the final
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
of a point P undergoing
motion
In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and mea ...
.
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 trajectory or flight path is the path that an object with mass in motion follows through space as a function of time. In classical mechanics, a trajectory is defined by Hamiltonian mechanics via canonical coordinates; hence, a complete traj ...
. A displacement may be identified with the
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
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 of a point relative to its initial position . The corresponding displacement vector can be defined as the
difference
Difference, The Difference, Differences or Differently may refer to:
Music
* ''Difference'' (album), by Dreamtale, 2005
* ''Differently'' (album), by Cassie Davis, 2009
** "Differently" (song), by Cassie Davis, 2009
* ''The Difference'' (al ...
between the final and initial positions:
In considering motions of objects over time, the instantaneous
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of the object is the rate of change of the displacement as a function of time. The instantaneous
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
, 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
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, which is a vector, and differs thus from the
average speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
, which is a scalar quantity.
Rigid body
In dealing with the motion of a
rigid body
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external force ...
, the term ''displacement'' may also include the
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
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
Angular displacement of a body is the angle (in radians, degrees or revolutions) through which a point revolves around a centre or a specified axis in a specified sense. When a body rotates about its axis, the motion cannot simply be analyzed ...
.
Derivatives
For a position vector
that is a function of time
, the derivatives can be computed with respect to
. The first two derivatives are frequently encountered in physics.
;
Velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
:
;
Acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
:
;
Jerk
:
These common names correspond to terminology used in basic kinematics.
[
] 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
*
Displacement field (mechanics)
*
Equipollence (geometry)
*
Motion vector
*
Position vector
*
Affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
References
External links
*
{{Classical mechanics derived SI units
Motion (physics)
Length
Vector physical quantities
Geometric measurement
Kinematic properties