Physical terms can be confusing. A pair of words that very often gets mixed up is *speed* and *velocity*. Although very closely related, these two concepts refer to two different ways of describing an object’s motion.

In the following article we present three different approaches to explaining what the difference between speed and velocity really is: the first one, a very concise one; the second one, more in detail; and the third one, through an easy example. Let’s get started!

## Speed vs. velocity

While speed refers to the distance covered by a moving object per unit time, velocity tells us not just that, but also the direction in which the object is moving. Velocity is therefore expressed mathematically as a vector, while speed is represented by a scalar.

## The difference between speed and velocity explained in detail

First, let’s talk about movement. You can be sure an object has moved when its position, measured with respect to a reference point, changes over time. Imagine you hit a cue ball on a pool table, making it displace from point *A* to point *B*, as shown in the following image.

If you fix your reference point, also called *origin*, at the initial position of the ball (*A*), then its initial location can be mathematically expressed as:

This is because the distance between the initial position of the ball and our point of reference is zero. Now, to find the final position of the ball with respect to the origin, you would only have to measure the distance between *A* and *B*. Let’s say it is equal to 50 cm. So, the final position of the cue ball can be expressed as:

In this example, the initial position of the ball with respect to *A* was zero, and its final position was 50 cm. The change in position indicates the object effectively moved and covered a distance of 50 cm. Physicists call this a *displacement*.

An object’s displacement is defined as a **vector**, meaning it comprises both the magnitude and the direction of the displacement. If you feel unsure about this definition, go ahead and read more about vectors in this article.

A vector’s magnitude is expressed by a number, which in this case represents the difference between the object’s final and initial positions. In our previous example it equates to 50 – 0 cm = 50 cm. The displacement magnitude, as that of any vector, has the same units as the phenomenon being measured, in this case units of length, like meters, kilometers or miles.

Now, the displacement direction can be expressed in various ways. For example, if we use a two-dimensional coordinate system where *x* represents the horizontal length and *y* the vertical length on the surface of our imaginary pool table, the cue ball’s initial position can be written as:

Here, the first zero in the parenthesis represents the ball’s position along the *x*-axis, while the second zero corresponds to its position along the *y*-axis. The cue ball’s final position is then:

The following image exemplifies this:

The displacement vector direction can be then expressed as the angle formed by its components along the *x*– and *y*-axes. Every component is defined as the difference between the final and initial positions along each axis. So, the displacement’s component along the *x*-axis is 50 cm – 0 = 50 cm, while its component along the *y*-axis equates to 0 – 0 cm = 0 cm. Since in our example the ball’s movement occurred solely horizontally, its *y* component is zero, as expected.

Now, the angle formed by both components can be calculated like this:

Where P sub *y* represents the displacement’s *y* component, and P sub *x* its *x* component. In our example, the fraction in equation 5 is zero, and so the angle between both components is zero degrees. This coincides with our experiment since movement only occurs horizontally, or with a 0° angle with respect to the *x*-axis. Thanks to our coordinate system, we have expressed *mathematically* the direction of movement of the cue ball.

The displacement vector tells us not only how much an object has moved, but also in which direction. Now, keep in mind that its definition can be extended to three-dimensional movement. Just by adding a z-axis to the previous procedure we can describe the displacement of planes, cars, planets, or every other moving body in the universe. In general, a displacement vector in space can be written as follows:

Where *x*, *y* and *z* represent the changes in position along the respective axes.

Now, by hitting the cue ball on our imaginary pool table we have displaced it. The transition from its initial to its final position had to occur within a certain time. If it didn’t, we would have to accept the cue ball’s motion occurred *instantly*, which would imply it *teleported* instead of displaced. The time rate at which a body displaces, meaning the distance it is able to cover per time unit, is what we call its ** speed**. The faster an object moves, the higher is its speed, meaning the more distance it travels per unit time (second, minute, hour, etc.).

As you probably concluded already, speed tells us how fast an object displaces in space, but it does not tell us in which direction. Nevertheless, since we have expressed the body’s displacement as a vector, we could describe its rate of change simply by dividing equation 6 by the time needed to execute said displacement. Let’s assume the cue ball moved from A to B in 2 seconds. In this case:

Notice that, in this case, since we divide a vector (the displacement) by a number (or scalar), we get another vector. This vector expresses the rate at which a body displaces similar to its speed, but it also provides information about the direction in which it moves. By looking at equation 7 we can conclude that the ball moved at 25 cm/s in the direction of the x-axis. We call this vectorial expression of speed the object’s ** velocity**. So, a general expression of a body’s velocity is:

Where *t* represents the time needed to complete displacement *s*. If you want to learn more about velocity, its different types and how to calculate it, we recommend you read our article on how to calculate velocity.

## Difference between speed and velocity with an example

Imagine a migrating goose is flying at a constant altitude of 300 m above sea level with a velocity given by the following vector:

Here *N* indicates the direction to the north, *E* to the east and *A* the altitude above sea level. Keep in mind that the goose is flying at a *constant* height of 300 m, which means its velocity along the vertical axis is zero, as shown above.

Now, to illustrate the difference between speed and velocity, we want to answer the following questions:

- What is the total speed of the goose?
- How much time will the goose need to advance 300 km to the north?

Let’s start with the first one. The total speed simply refers to how fast the goose is moving altogether. Since the vector given above provides the specific rates of movement (how many kilometers are covered per hour of flight) in two perpendicular directions (N and E) we can calculate the total rate of displacement i.e., the goose’s speed, using the Pythagorean theorem:

Note that in this equation we also represent speed with a *v*, but in this case, there is no arrow above it, indicating that it is a scalar, not a vector. This means the total speed of the goose is 32,3 km/h. That’s it. If we are provided with this number instead of the vector shown in equation 9, we will never know precisely in which direction the goose is flying. If that were the case, we would not be able to answer the second question.

Now, to determine the time the goose needs to fly 300 km to the north, we need to take a close look at its velocity. We see that, in that specific direction, the goose is able to cover 30 km in one hour of flight. So, in order to find the answer to the second question, we only need to do a cross multiplication:

Thanks to the velocity vector, we are able to calculate how much time the goose will need to cover some distance in a specific direction. This would not have been possible if we only had its flying speed to begin with!

## Sources

Young, Hugh; Freedman, Roger A. *University Physics. Chapter 8: linear momentum, impulse, and collisions*. Pearson, 2009.