Displacement (geometry)

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.

Displacement versus distance traveled along a path

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 relatively to its initial position x_i. The corresponding displacement vector can be defined as the difference between the final and initial positions:

${\displaystyle {s}={x_{\textrm {f}}-x_{\textrm {i}}}=\Delta {x}}$

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

Velocity

${\displaystyle {\textbf {v}}={\frac {\mathrm {d} {\textbf {s}}}{\mathrm {d} t}}}$

Acceleration

${\displaystyle {\textbf {a}}={\frac {\mathrm {d} {\textbf {v}}}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}{\boldsymbol {s}}}{\mathrm {d} t^{2}}}}$

Jerk

${\displaystyle {\textbf {j}}={\frac {\mathrm {d} {\textbf {a}}}{\mathrm {d} t}}={\frac {\mathrm {d} ^{2}{\textbf {v}}}{\mathrm {d} t^{2}}}={\frac {\mathrm {d} ^{3}{\textbf {s}}}{\mathrm {d} t^{3}}}}$

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

1. Tom Henderson. "Describing Motion with Words". The Physics Classroom. Retrieved 2 January 2012.
2. Stewart, James (2001). "§2.8 - The Derivative As A Function". Calculus (2nd ed.). Brooks/Cole. ISBN 0-534-37718-1.