Speed and Velocity

1. Core Idea

Both speed and velocity describe how fast an object moves, but they differ in a crucial way:

• Speed → how fast (scalar)
• Velocity → how fast and in what direction (vector)

2. Speed

Definition

Speed is the rate at which distance is traveled.

speed = $$\frac{distance}{time}$$​

Characteristics:

• scalar quantity
• always nonnegative
• ignores direction

Average Speed

Average Speed = $$\frac{total distance}{total time}$$​

3. Velocity

Definition

Velocity is the rate of change of displacement.

\(\vec{v}_{avg}=\frac{\Delta{\vec{r}}}{\Delta{t}}\)​

Characteristics:

• vector quantity
• includes direction
• can be positive, negative, or zero (in 1D)

4. Instantaneous Velocity

To describe motion at a specific moment:

\(\vec{v}=\frac{d\vec{r}}{dt}\)​

Velocity is the derivative of position with respect to time.

5. Speed vs Velocity

Example:

An object moves
10 m east, then 10 m west

• distance = 20 m
• displacement = 0

So:

• average speed ≠ 0
• average velocity = 0

6. Speed from Velocity

Speed is the magnitude of velocity:

speed = \(\vec{v}\)

If

\(\vec{v}=(v_x , v_y)\)

then

\(|\vec{v}|=\sqrt{{v_x}^2+{v_y}^2}\)​​

This follows:

7. Velocity Components

In two dimensions:

\(\vec{v}=\vec{v_x}\hat{i}+\vec{v_y}\hat{j}\)​

Each component represents motion along a specific axis.

8. Graphical Interpretation

Position–Time Graph

• slope → velocity
• steeper slope → greater velocity

Velocity–Time Graph

• slope → acceleration
• area under curve → displacement

9. Physical Meaning

• Speed tells how fast an object is moving
• Velocity tells how fast and in which direction

Direction matters because it determines how motion changes and how vectors combine.

Summary

Speed:

speed = $$\frac{distance}{time}$$​​

Velocity:

\(\vec{v}_{avg}=\frac{\Delta{\vec{r}}}{\Delta{t}}\)​ , ​\(\vec{v}=\frac{d\vec{r}}{dt}\)​

Key idea:

• speed = magnitude of velocity
• velocity = direction + magnitude

These concepts are essential for analyzing motion in one and multiple dimensions in kinematics.