Velocity: More Than Just Speed
The Fundamental Difference: Speed vs. Velocity
Many people use the words "speed" and "velocity" interchangeably in everyday conversation. In physics, however, they have distinct and important meanings. Understanding this difference is the first step to mastering the concept of velocity.
Speed is a scalar quantity. This means it only has magnitude (a numerical value). If you say a car is moving at 60 km/h, you are describing its speed. You are not specifying the direction it is traveling.
Velocity, on the other hand, is a vector quantity. This means it has both magnitude and direction. Saying a car is moving at 60 km/h due north is a description of its velocity. The direction is a crucial part of the information.
The average velocity $( \vec{v}_{avg} )$ is calculated as the total displacement $( \Delta \vec{s} )$ divided by the total time $( \Delta t )$.
$ \vec{v}_{avg} = \frac{\Delta \vec{s}}{\Delta t} = \frac{\vec{s}_f - \vec{s}_i}{t_f - t_i} $
Where $\vec{s}_f$ is the final position, $\vec{s}_i$ is the initial position, $t_f$ is the final time, and $t_i$ is the initial time.
Consider a person walking around a square field. They start at point A, walk to point B, then C, and finally back to A. The total distance traveled is the perimeter of the square. However, because they ended up exactly where they started, their overall displacement is zero. Therefore, their average velocity for the entire trip is also zero, even though their average speed was a positive number. This example perfectly illustrates why velocity depends on displacement (the change in position), not the total path length traveled.
Displacement: The Path That Matters for Velocity
To fully grasp velocity, you must understand displacement. Displacement is the straight-line change in an object's position from its starting point to its ending point. It is also a vector quantity, meaning it has both magnitude and direction.
Imagine you are on a treasure hunt. The map says the treasure is 10 meters east of your starting point. It doesn't matter if you walk in a zig-zag, a circle, or a straight line to get there. Your displacement when you find the treasure is 10 meters east. The actual path you took is the distance, which is a scalar.
This is why the formula for average velocity uses displacement $(\Delta \vec{s})$, not distance. Velocity is concerned with how effectively an object changes its position, not with the scenic route it might have taken.
| Aspect | Distance | Displacement |
|---|---|---|
| Definition | The total length of the path traveled. | The straight-line change in position from start to finish. |
| Quantity Type | Scalar (Magnitude only) | Vector (Magnitude and Direction) |
| Can it be zero? | Only if the object does not move. | Yes, if the object returns to its starting point. |
| Symbol | $d$ | $\vec{s}$ |
Calculating Velocity in One and Two Dimensions
Velocity calculations can be performed in one dimension (like a straight road) or two dimensions (like a map).
One-Dimensional Velocity: This is the simplest case. Movement is restricted to a straight line, often defined along the x-axis. Direction is indicated by a positive (+) or negative (-) sign. For example, a car moving east at 20 m/s has a velocity of $ +20 m/s $, while a car moving west at the same speed has a velocity of $ -20 m/s $.
Example: A train starts at a station (position $ 0 m $) and moves east. After 120 seconds, it is 3000 meters east of the station. Its displacement is $ \Delta \vec{s} = 3000 - 0 = 3000 m $ east. The time interval is $ \Delta t = 120 - 0 = 120 s $. Therefore, its average velocity is:
$ \vec{v}_{avg} = \frac{3000 m}{120 s} = 25 m/s $ east.
Two-Dimensional Velocity: When an object moves on a plane, we break down its velocity into components, usually along the x and y-axes. The overall velocity is the vector sum of these components. If a boat is moving with a velocity of $ 4 m/s $ east and $ 3 m/s $ north, its overall velocity can be found using the Pythagorean theorem.
The magnitude of the velocity (its speed) is:
$ |\vec{v}| = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 m/s $.
The direction can be found using trigonometry: $ \theta = \tan^{-1}(\frac{3}{4}) \approx 37^{\circ} $ north of east.
A Closer Look at Average and Instantaneous Velocity
We've been discussing average velocity, which gives us a big-picture view of motion over a time interval. But what about the velocity at a specific moment?
Instantaneous Velocity is the velocity of an object at a specific instant in time. If you look at the speedometer in a car, the reading it shows is your instantaneous speed. The instantaneous velocity would be that speedometer reading plus your direction of travel at that exact moment.
Mathematically, as the time interval $( \Delta t )$ in the average velocity formula approaches zero, the average velocity becomes the instantaneous velocity $( \vec{v} )$.
$ \vec{v} = \lim_{\Delta t \to 0} \frac{\Delta \vec{s}}{\Delta t} $
In calculus, this is the derivative of displacement with respect to time.
Velocity in Action: Real-World Scenarios and Calculations
Let's apply our knowledge to some concrete situations to see how velocity works in the real world.
Scenario 1: The Commuter's Journey
Maria drives from her home to her office. Her office is 15 km due east of her home. The journey takes her 30 minutes.
- Displacement: $ \Delta \vec{s} = 15 km $ East.
- Time: $ \Delta t = 0.5 h $.
- Average Velocity: $ \vec{v}_{avg} = \frac{15 km}{0.5 h} = 30 km/h $ East.
After work, she drives back home along the same route, also in 30 minutes. For the entire day (to the office and back), her total displacement is zero (she ended up where she started). Therefore, her average velocity for the entire day is 0 km/h, even though her average speed was 30 km/h.
Scenario 2: The Soccer Pass
A soccer player kicks a ball from one side of the field to the other. The field is 100 meters long and 60 meters wide. The ball is kicked from the southwest corner to the northeast corner in 4 seconds.
- Displacement: We need the straight-line distance. Using the Pythagorean theorem: $ \Delta \vec{s} = \sqrt{100^2 + 60^2} = \sqrt{10000 + 3600} = \sqrt{13600} \approx 116.6 m $ in a northeast direction.
- Time: $ \Delta t = 4 s $.
- Average Velocity: $ \vec{v}_{avg} = \frac{116.6 m}{4 s} \approx 29.15 m/s $ Northeast.
This example shows how to handle velocity in two dimensions, where the direction is not simply positive or negative.
The Link Between Velocity and Acceleration
Velocity rarely stays constant. When velocity changes, we say the object is accelerating. Acceleration is the rate of change of velocity with respect to time. It is also a vector quantity.
$ \vec{a}_{avg} = \frac{\Delta \vec{v}}{\Delta t} $
Since velocity is a vector, a change in velocity can be a change in magnitude (speeding up or slowing down), a change in direction (turning a corner at constant speed), or both. A car going around a circular track at a constant speed of 100 km/h is still accelerating because its direction is continuously changing. This is called centripetal acceleration.
| Change in Motion | Is there Acceleration? | Reason |
|---|---|---|
| Car speeds up on a straight highway | Yes | Magnitude of velocity is changing. |
| Car slows down on a straight highway | Yes | Magnitude of velocity is changing. |
| Car turns a corner at constant speed | Yes | Direction of velocity is changing. |
| Car moves at constant speed in a straight line | No | Neither magnitude nor direction is changing. |
Common Mistakes and Important Questions
A: No, absolutely not. Average speed is total distance divided by total time. In a round trip, the distance traveled is a positive number, so the average speed must also be a positive number. Only the displacement, and therefore the average velocity, is zero.
A: Yes. This happens whenever an object moves in a curved path or circle at a constant speed. The speed (magnitude) remains the same, but the direction of motion is continuously changing. Since velocity depends on direction, a change in direction means a change in velocity, and therefore the object is accelerating.
A: Metres per second (m/s) is the unit for velocity. It tells you how many meters of displacement occur every second. Metres per second squared (m/s²) is the unit for acceleration. It tells you how many meters per second the velocity changes every second.
Velocity is a powerful concept that provides a complete description of motion by incorporating both how fast an object is moving and where it is headed. By distinguishing it from speed, understanding its reliance on displacement, and learning to calculate both its average and instantaneous values, we build a strong foundation for exploring all of kinematics[1]. From a simple walk in the park to the complex orbit of a planet, velocity is the key to quantifying how things move through our universe. Remember, the next time you describe a moving object, stating its velocity—not just its speed—gives you the full picture.
Footnote
[1] Kinematics: The branch of mechanics that describes the motion of points, objects, and systems of bodies without considering the forces that cause the motion.