Velocity Formula

The Formula

v = \frac{\Delta x}{\Delta t} (displacement divided by time)

When to use: How fast something is moving AND which way it's heading—direction is essential.

Quick Example

60 km/h north is a velocity; -10 m/s means moving in the negative direction.

Notation

\vec{v} is the velocity vector in m/s, \Delta \vec{x} is the displacement vector in metres, \Delta t is the time interval in seconds, and d\vec{x}/dt denotes the derivative of position with respect to time.

What This Formula Means

The rate of change of position with respect to time, including both magnitude and direction.

How fast something is moving AND which way it's heading—direction is essential.

Formal View

Average velocity is defined as \vec{v}_{\text{avg}} = \frac{\Delta \vec{x}}{\Delta t}, and instantaneous velocity is the limit \vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{x}}{\Delta t} = \frac{d\vec{x}}{dt}.

Worked Examples

Example 1

easy
A car travels 150 \text{ m} east in 10 \text{ s}. What is its average velocity?

Solution

  1. 1
    Average velocity is displacement divided by time.
  2. 2
    Here the displacement is 150 \text{ m} east and the time is 10 \text{ s}.
  3. 3
    v_{\text{avg}} = \frac{\Delta x}{\Delta t} = \frac{150}{10} = 15 \text{ m/s east}

Answer

v_{\text{avg}} = 15 \text{ m/s east}
Velocity is a vector quantity that includes both speed and direction. Average velocity equals the total displacement divided by the total time.

Example 2

medium
A cyclist rides 200 \text{ m} north in 40 \text{ s}, then 200 \text{ m} south in 60 \text{ s}. What is the average velocity and average speed?

Example 3

medium
A car accelerates from 10 \text{ m/s} to 30 \text{ m/s} in 5 \text{ seconds}. Find the average velocity.

Common Mistakes

  • Using total distance instead of displacement — a round trip of 10 km out and back has zero average velocity but 20 km of distance.
  • Dropping the direction and treating velocity as if it were speed — velocity is a vector quantity.
  • Confusing instantaneous velocity with average velocity — average velocity uses total displacement over total time, while instantaneous is the velocity at one specific moment.

Why This Formula Matters

Gives a complete description of how an object's position changes over time.

Frequently Asked Questions

What is the Velocity formula?

The rate of change of position with respect to time, including both magnitude and direction.

How do you use the Velocity formula?

How fast something is moving AND which way it's heading—direction is essential.

What do the symbols mean in the Velocity formula?

\vec{v} is the velocity vector in m/s, \Delta \vec{x} is the displacement vector in metres, \Delta t is the time interval in seconds, and d\vec{x}/dt denotes the derivative of position with respect to time.

Why is the Velocity formula important in Physics?

Gives a complete description of how an object's position changes over time.

What do students get wrong about Velocity?

Speed is just the magnitude of velocity; velocity also includes direction.

What should I learn before the Velocity formula?

Before studying the Velocity formula, you should understand: displacement.

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