Editing Distance

= Distance: Definition and Mathematical Representation =

== Introduction ==
'''Distance''' is a basic concept in kinematics and everyday measurements. It refers to the total length of the path traveled by an object during motion. Distance is a '''scalar quantity''', meaning it has magnitude but no direction.

It is always a non-negative value and differs from '''displacement''', which is a vector.

== Definition ==

Mathematically, distance is represented as the total path length covered by an object:

<math>
d = \int_{t_1}^{t_2} |\vec{v}(t)|\, dt
</math>

Where:
* <math>d</math> is the distance traveled,
* <math>\vec{v}(t)</math> is the velocity as a function of time,
* <math>t_1</math> and <math>t_2</math> are the initial and final times.

For motion with constant speed:

<math>
d = v \cdot t
</math>

Where:
* <math>v</math> is the speed,
* <math>t</math> is the time duration.

== SI Unit ==

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

<math>
1\, \mathrm{m} = 100\, \mathrm{cm} = 1000\, \mathrm{mm}
</math>

== Characteristics ==

* Scalar quantity: Only magnitude, no direction.
* Always positive or zero.
* Depends on the path taken, not just start and end points.
* Greater than or equal to the magnitude of displacement.

== Distance vs. Displacement ==

* '''Distance''' measures the actual path length.
* '''Displacement''' is the straight-line change in position (a vector).

Example:
If you walk 3 m east and then 4 m west, the total distance is:

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

But the displacement is:

<math>
\Delta x = 3\, \mathrm{m} - 4\, \mathrm{m} = -1\, \mathrm{m}
</math>

== Graphical Interpretation ==

* The area under a speed–time graph gives the total distance traveled.
* The slope of a distance–time graph gives the instantaneous speed.

== Applications ==

* Measuring how far a vehicle or person has traveled.
* Navigation and GPS tracking.
* Road transportation and logistics.
* Sports and fitness tracking.

== See Also ==
* [[Displacement]]
* [[Speed]]
* [[Velocity]]
* [[Kinematics]]
* [[Scalar Quantity]]