Displacement (vector)

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In physics, displacement is the vector that specifies the change in position of a point or a particle in reference to a previous position. When the previous point is the origin, this is better referred to as a position.

Displacement vector versus distance traveled along a path. Notice that the length of the displacement is also a (minimum) distance.

A position vector is a simplified representation of motion. Namely, it indicates both the length and direction of a hypothetical motion along a straight line from the reference position, and may be described as a sequence of very small displacements. On the other hand, a distance is typically defined as a scalar quantity and can be used to indicate both the length of a displacement (minimum distance) and the length of a curved path (traveled distance), but not the direction of the motion.

The displacement vector is the difference between the final and initial positions. This difference, divided by the time needed to perform the motion, defines the average velocity of the point or particle.

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.

Distance Travelled

If the position of an object is described by a vector function

$\mathbf{r}(t):\R \to \mathrm{V}^n$ ,

then the distance traveled as a function of $t$ is described by the integral of one with respect to arc length.

$d(t)=\int_{0}^{t}1\,\mathrm{d}s$

where

d s

is the arc length differential

The arc length differential is described by the following equation:

$\mathrm{d}s=\left|\mathbf{r}'(t)\right|\,\mathrm{d}t=\left|\mathbf{v}(t)\right|\,\mathrm{d}t=v(t)\,\mathrm{d}(t)$

where

$\mathbf{v}(t)$ is velocity

$v(t)\,$ is speed.

Displacement, position, and velocity

A position vector can be viewed as an hypothetical displacement of a point or particle from the origin of a coordinate system to the location of a point at a given time.

On a graph representing the position of a particle with respect to time (position vs. time graph), the slope of the straight line joining two points on the graph is the average velocity of the particle during the corresponding time interval, while the slope of the tangent to the graph at a given point is the instantaneous velocity at the corresponding time (first derivative of the particle position).

Displacement and the equations of motion

To calculate displacement all vectors and scalars must be taken into consideration. The following formulas can be used to calculate displacement for, $s$ , for an object undergoing constant acceleration..

$\mathbf{s} = {\mathbf{u}t+{1\over 2}\mathbf{a}t^2}$

$\mathbf{v} = \mathbf{u}+ \mathbf{a}t$

$\mathbf {v^2} = \mathbf{u^2}+2\mathbf{as}$

Where:

$\mathbf{u}$ Initial velocity

$\mathbf{v}$ Final velocity

$\mathbf{a}$ Acceleration

$\mathbf{t}$ Time

$\mathbf{s=}$ Δ $\mathbf{x}$ Displacement

$\mathbf{x}$ Position (in one dimension)

$\mathbf{r}$ Position (in three dimensions)

It should also emphasized that vector directions, negative and positive signs, are important when calculating displacement

Height displacement

Height displacement is the distance an object peaks in height vertically. If, for example, a ball was thrown up in the air and back into the owners hand the displacement would be zero, since displacement over a period of time is defined as the distance between an object's starting and finishing points.

However using the equation $s = {ut+{1\over 2}at^2}$ can be shortened to $h = {ut-{1\over2} gt^2}$ to calculate overall vertical height meaning time is an important factor in the calculation. $g$ is the acceleration caused by gravity which stays constant at approximately $9.81\ \text{m}/\text{s}^2$ , depending on the direction of the object is travelling a negative or positive sign is required since it is an equation of motion and is a vector,quantity.

Distance Travelled

Displacement, position, and velocity

Displacement and the equations of motion

Height displacement

See also