# Displacement (vector)

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.

File:Distancedisplacement.svg
Displacement vector versus distance traveled along a path

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.

## Distance Traveled

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

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

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

${\displaystyle s(t)=\int _{0}^{t}1\,\mathrm {d} s}$

where

${\displaystyle \mathrm {d} s}$ is the arc length differential

The arc length differential is described by the following equation:

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

where

${\displaystyle \mathbf {v} (t)}$ is velocity
${\displaystyle v(t)\,}$ is speed

## Calculating displacement

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To calculate displacement all vectors and scalars must be taken into consideration [1][2][3]. The following formula is used to calculate displacement , ${\displaystyle s}$[1][2].

${\displaystyle s={ut+{1 \over 2}at^{2}}}$ [1][2]

Where:[3]

${\displaystyle \mathbf {u} }$ Initial velocity
${\displaystyle \mathbf {v} }$ Final speed
${\displaystyle \mathbf {a} }$ Acceleration
${\displaystyle \mathbf {t} }$Time

## Displacement and the equations of motion

The three main equations of motion can be used to calculate displacement [1][2][3]

They are:

${\displaystyle \mathbf {S} ={ut+{1 \over 2}at^{2}}}$
${\displaystyle \mathbf {v} =u+at}$
${\displaystyle \mathbf {v^{2}} =u^{2}+2as}$
• It should also emphasized that vector directions, negative and positive signs, are important when calculating displacement [1][2][3]

### Height displacement

Height displacement is the distance an object peaks in height vertically [1][2] if for example a ball was thrown up in the air and back into the owners hand the displacement would be zero, since displacement is defined as the distance an object is from its starting point.[3]

However using the equation ${\displaystyle s={ut+{1 \over 2}at^{2}}}$[2][3] can be shortened to ${\displaystyle h={1 \over 2}gt^{2}}$[2][3] to calculate overall vertical height meaning time is an important factor in the calculation. ${\displaystyle g}$ is the acceleration caused by gravity which stays constant at approximately ${\displaystyle 9.8\ {\text{m}}/{\text{s}}^{2}}$, depending on the direction the object is travelling a negative sign or positive sign is required since it is an equation of motion and is a vector quantity.[3]