Editing Displacement

= Displacement: Definition and Mathematical Representation =

== Introduction ==
'''Displacement''' is a fundamental concept in physics and kinematics. It represents the change in the position of an object from its initial point to its final point. Displacement is a '''vector quantity''', meaning it has both magnitude and direction.

It differs from '''distance''', which only considers how much ground an object has covered, regardless of direction.

== Definition ==

Mathematically, displacement is defined as:

<math>
\vec{d} = \vec{r}_{\text{final}} - \vec{r}_{\text{initial}}
</math>

Where:
* <math>\vec{d}</math> is the displacement vector,
* <math>\vec{r}_{\text{final}}</math> is the final position vector,
* <math>\vec{r}_{\text{initial}}</math> is the initial position vector.

== SI Unit ==

The SI unit of displacement is the '''meter (m)''':

<math>
1\, \mathrm{m} = 100\, \mathrm{cm} = 0.001\, \mathrm{km}
</math>

== Characteristics ==

* It is a **vector** (includes both magnitude and direction).
* Can be **positive, negative, or zero** depending on direction.
* Independent of the actual path taken; depends only on start and end points.
* Minimum possible distance between two positions.

== One-Dimensional Example ==

If an object moves from position <math>x_1 = 2\, \mathrm{m}</math> to <math>x_2 = 5\, \mathrm{m}</math>, then:

<math>
\Delta x = x_2 - x_1 = 5\, \mathrm{m} - 2\, \mathrm{m} = 3\, \mathrm{m}
</math>

If it moves from 5 m to 2 m:

<math>
\Delta x = 2\, \mathrm{m} - 5\, \mathrm{m} = -3\, \mathrm{m}
</math>

The sign indicates direction.

== Relation to Velocity ==

Velocity is the rate of change of displacement:

<math>
\vec{v} = \frac{d\vec{d}}{dt}
</math>

This highlights the importance of displacement in determining an object’s motion.

== Graphical Representation ==

* A displacement–time graph shows how position changes over time.
* The slope of this graph gives the object's velocity.

== Displacement vs. Distance ==

{| class="wikitable"
! Quantity !! Type !! Directional? !! Value Range
|-
| Distance || Scalar || No || Always ≥ 0
|-
| Displacement || Vector || Yes || Can be positive, negative, or zero
|}

Example:
If a person walks 4 m east and 3 m west, total distance is:

<math>
d = 4 + 3 = 7\, \mathrm{m}
</math>

Displacement is:

<math>
\vec{d} = 4 - 3 = 1\, \mathrm{m}\, \text{(east)}
</math>

== Applications ==

* Describing motion in kinematics
* Calculating velocity and acceleration
* Physics simulations and animation
* GPS and navigation systems

== See Also ==
* [[Distance]]
* [[Velocity]]
* [[Vector (physics)]]
* [[Kinematics]]
* [[Position Vector]]