Displacement (vector)

Displacement versus distance traveled along a path.

A displacement is the shortest distance from the initial to the final position of a point P^[1]. 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 ($\vec r_f$) relative to its initial position ($\vec r_i$), and a displacement vector can be mathematically defined as the difference between the final and initial position vectors:

$\vec s = \vec r_f - \vec r_i =\Delta \vec r$

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.

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 is called angular displacement.

Derivatives

^[2] Velocity

$\vec v =\frac {d \vec r} {dt}$

Acceleration

$\vec a =\frac {d \vec v} {dt}=\frac {d ^2\vec r} {dt^2}$

Jolt/Jerk/Surge/Lurch

$\vec j =\frac {d \vec a} {dt}=\frac {d ^2\vec v} {dt^2}=\frac {d ^3\vec r} {dt^3}$

Snap/Jounce

$\vec s =\frac {d \vec j} {dt}=\frac {d^2 \vec a} {dt^2}=\frac {d ^3\vec v} {dt^3}=\frac {d ^4\vec r} {dt^4}$

Crackle

$\vec c=\frac {d \vec s} {dt}=\frac {d^2 \vec j} {dt^2}=\frac {d^3 \vec a} {dt^3}=\frac {d^4 \vec v} {dt^4}=\frac {d^5 \vec r} {dt^5}$

Pop/Pounce

$\vec p=\frac {d \vec c} {dt}=\frac {d^2 \vec s} {dt^2}=\frac {d^3 \vec j} {dt^3}=\frac {d^4 \vec a} {dt^4}=\frac {d^5 \vec v} {dt^5}=\frac {d^6 \vec r} {dt^6}$

Lock

$\vec l=\frac {d \vec p} {dt}=\frac {d^2 \vec c} {dt^2}=\frac {d^3 \vec s} {dt^3}=\frac {d^4 \vec j} {dt^4}=\frac {d^5 \vec a} {dt^5}=\frac {d^6 \vec v} {dt^6}=\frac {d^7 \vec r} {dt^7}$

Drop

$\vec d=\frac {d \vec l} {dt}=\frac {d^2 \vec p} {dt^2}=\frac {d^3 \vec c} {dt^3}=\frac {d^4 \vec s} {dt^4}=\frac {d^5 \vec j} {dt^5}=\frac {d^6 \vec a} {dt^6}=\frac {d^7 \vec v} {dt^7}=\frac {d^8 \vec r} {dt^8}$

Where $\vec r$ is the position vector, $\vec v$ is the velocity vector, $\vec a$ is the acceleration vector, $\vec j$ is the jerk vector, $\vec s$ is the snap vector, $\vec c$ is the crackle vector, $\vec p$ is the pop vector, $\vec l$ is the lock vector, and $\vec d$ is the drop vector.

See also

• Position vector
• Affine space

1. ^ http://www.physicsclassroom.com/Class/1DKin/U1L1c.cfm
2. ^ http://wearcam.org/absement/Derivatives_of_displacement.htm