Velocity

Velocity: Definition and Mathematical Representation

Introduction

Velocity is a fundamental concept in physics that describes the rate at which an object changes its position with respect to time. Unlike speed, velocity is a vector quantity—it has both magnitude and direction.

Velocity is essential in kinematics, dynamics, and many real-world applications such as vehicle motion, projectile paths, and orbital mechanics.

Definition

The instantaneous velocity is defined as the time derivative of displacement:

$\overrightarrow{v}=\frac{d\overrightarrow{s}}{dt}$

Where:

• $\overrightarrow{v}$ is the velocity vector,
• $\overrightarrow{s}$ is the displacement vector,
• $t$ is time.

Average Velocity

Average velocity over a time interval is given by:

${\overrightarrow{v}}_{\text{avg}}=\frac{\mathrm{\Delta }\overrightarrow{s}}{\mathrm{\Delta }t}$

Where:

• $\mathrm{\Delta }\overrightarrow{s}$ is the change in displacement,
• $\mathrm{\Delta }t$ is the change in time.

SI Unit

The SI unit of velocity is:

$\mathrm{m}/\mathrm{s}\text{(meters per second)}$

Velocity vs. Speed

• Velocity includes direction; it’s a vector.
• Speed is the magnitude of velocity; it’s a scalar.

Example: An object moving in a circle at constant speed has changing velocity due to direction change.

Motion with Constant Acceleration

When acceleration is constant, the following kinematic equation relates velocity and time:

$v=u+at$

Where:

• $v$ is the final velocity,
• $u$ is the initial velocity,
• $a$ is the acceleration,
• $t$ is time.

Relative Velocity

The velocity of object A with respect to object B is:

${\overrightarrow{v}}_{A/B}={\overrightarrow{v}}_A-{\overrightarrow{v}}_B$

This concept is crucial in problems involving two or more moving observers or reference frames.

Graphical Interpretation

• The slope of a displacement–time graph gives velocity.
• The area under a velocity–time graph gives displacement.

Applications

• Vehicle dynamics and navigation
• Ballistic and projectile motion
• Fluid flow (e.g., velocity fields in aerodynamics)
• Orbital mechanics and astronomy

See Also

• Acceleration
• Displacement
• Speed
• Kinematics
• Relative Motion
• Graphical Analysis of Motion