Displacement (vector)
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In physics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. The vector directs from the reference point to the current position.
When the reference point is the origin of the chosen coordinate system, the displacement vector is better referred to as the position vector, which expresses position by the straight line directed from the previous position to the current position, as opposed to the scalar quantity distance which expresses only the length. This use of displacement vector can describe the complete motion as well as the path of the particle.
When the reference point is a previous position of the particle, the displacement vector indicates the sense of movement by a vector directing from the previous position to the current position. This use of displacement vector is useful for defining the velocity and acceleration vectors of the particle.
By plotting the displacement, relative to the starting point, against time on a position vs. time graph, the average velocity or the instantaneous velocity can be found by taking the slope of the graph or the derivative of the graph, respectively.
In dealing with the motion of a rigid/firm body, the term displacement may also include the rotations of the body.
[edit] Distance Traveled
If the displacement of an object is described by a vector function
- <math>\mathbf{r}(t):\R \to \mathrm{V}^n</math>,
then the distance traveled as a function of <math>t</math> is described by the integral of one with respect to arc length.
- <math>s(t)=\int_{0}^{t}1\,\mathrm{d}s</math>
where
- <math>\mathrm{d}s</math> is the arc length differential
The arc length differential is described by the following equation:
- <math>\mathrm{d}s=\left|\mathbf{r}'(t)\right|\,\mathrm{d}t=\left|\mathbf{v}(t)\right|\,\mathrm{d}t=v(t)\,\mathrm{d}(t)</math>
where
- <math>\mathbf{v}(t)</math> is velocity
- <math>v(t)\,</math> is speed
[edit] See also
- Position vector
- Affine space, which distinguishes between points in space and displacements between them
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