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:

$d=\underset{t_1}{\overset{t_2}{\int }}|\overrightarrow{v}(t)|dt$

Where:

• $d$ is the distance traveled,
• $\overrightarrow{v}(t)$ is the velocity as a function of time,
• $t_1$ and $t_2$ are the initial and final times.

For motion with constant speed:

$d=v\cdot t$

Where:

• $v$ is the speed,
• $t$ is the time duration.

SI Unit

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

$1\mathrm{m}=100\mathrm{c}\mathrm{m}=1000\mathrm{m}\mathrm{m}$

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:

$d=3\mathrm{m}+4\mathrm{m}=7\mathrm{m}$

But the displacement is:

$\mathrm{\Delta }x=3\mathrm{m}-4\mathrm{m}=-1\mathrm{m}$

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