In everyday language, speed and velocity are treated as if they mean exactly the same thing. If you heard someone comment that “the car’s velocity is 25 miles per hour,” you wouldn’t bat an eyelid. But in physics, that everyday comment about the velocity of an object contains a critical error.

If you were to write 25 miles per hour (or 11 meters per second) as the answer to a question that asked you for a *velocity*, you would be wrong. But if that same question asked you for the *speed* of the car, you’d be right. Why?

Understanding the difference between the speed of an object and its velocity tells you the answer, sets you up for future problems involving circular motion and introduces you to the important concept of a *vector quantity*.

#### TL;DR (Too Long; Didn't Read)

Speed is a scalar quantity (only having a magnitude), but velocity is a vector quantity (with a magnitude and a direction). Velocity is speed **with a direction**.

## Speed vs. Velocity

The key difference between speed and velocity is that speed is a *scalar quantity* and velocity is a *vector quantity*.

Scalar quantities are things like temperature, pressure and energy, which are completely described by their “size” or *magnitude*. So if the temperature of some water is 20 degrees Celsius, you don’t need any more information to tell you everything about that value – the number and its unit completely defines the temperature of the water.

Vectors, like velocity, acceleration and force, have a magnitude but also have a *direction*, and without information about the direction, they aren’t complete.

The definition of speed is simply the rate of change of distance traveled, or the distance traveled per unit of time. So if you told somebody about a car driving 10 m/s, that would be a speed, and you can remember this easily because that would be what showed on a speedometer (although probably in a non-SI unit). However, if you say it’s traveling at 10 m/s *to the right*, you’ve added information about the direction of the motion and described the vector quantity that is the car’s velocity. In mathematical terms, speed is the *magnitude of the velocity* and has an absolute value.

This distinction opens up the possibility that the velocity of an object can be constantly changing even when it has a constant speed, and thus you can have acceleration (another vector quantity – the rate of change of velocity) despite a constant speed. Consider that same car driving at a constant speed of 15 m/s around a circular race course. The amount of distance it covers per unit of time (its speed) isn’t changing, but **the direction is continuously changing**, so it does not have a constant velocity.

## Velocity, Speed and Acceleration Equations

The difference in the definition of speed vs. that of velocity shows up in the equations for both, as well as an implicit recognition that velocity is a vector quantity.

For speed *v*, the definition is simply the distance *d* traveled over the time interval *t* in question:

For velocity ** v**, the symbol is bolded (or displayed with an arrow over the top of the

*v*, useful in hand-written equations) to signify that it’s a vector and it relates the displacement

**** (a vector describing the final location relative to a chosen starting location, in one, two or three dimensions) to the interval of time in which the displacement took place.

*s*The instantaneous velocity is given by the derivative of displacement with respect to time:

The unit of velocity is simply a unit of distance over a unit of time, such as meters per second (m/s) or kilometers per hour (km/h).

Acceleration ** a** is another vector, and it’s defined as the rate of change of velocity

**** with respect to time:

*v*## The Importance of Noting Opposite Directions

The distinction between speed and velocity is important because of things like opposite directions and the relationship between velocity and other vectors like acceleration.

As well as cars driving around a track, another example is a merry-go-round horse traveling at the constant speed of 2 m/s. Because it travels in a circle, its linear direction is continuously changing, and therefore its velocity is constantly changing and it has an acceleration (for circular motion, this is called centripetal acceleration).

Another example shows the importance of looking at velocity vs. simply considering speed. Imagine two carts on a track are hurtling towards each other and set to collide. When they do, one of them *must* change direction. If you don’t set up a common frame of reference that allows you to show the difference in the direction of motion as well as their speeds (i.e., the difference in velocity), this information will be lost – and it wouldn’t even be clear they were on a collision course!

The fact that velocity is a vector quantity is crucial for the process of adding together velocities – if they are both in the same direction, they add together, but if they’re in opposite directions (say, *x* and -*x*) the result is a subtraction. To find the net velocity of an object – for example, a bowling ball rolling across a travelator (the moving walkways often found in airports) moving in the opposite direction – you *need* the directional information about each to calculate whether the ball will end up moving forwards or backwards after a period of time.

In this case, you’d define one velocity as in the *x* direction (say, the direction of motion of the bowling ball) and the other (the travelator’s motion) as in the *-x* direction, then add the vector quantities, which in practice would mean subtracting the speed of the travelator from that of the bowling ball because they’re moving in opposite directions.

## Average vs. Instantaneous Velocity

The difference between average and instantaneous velocity is crucial when the motion is not linear (i.e., in a straight line), such as a runner traversing an athletics track. At any given moment, her *instantaneous velocity* is her speed and the direction in which she is traveling at that exact time, for example, 7 m/s due east. But her average velocity is her total *displacement* over the complete time interval her movement took place in, say, 60 seconds. This means that if she does a complete 400-meter lap, returning to her original location, her total displacement is 0 m, and so her average velocity would be 0 m/s.

This seems absurd because it’s obvious that her *average* *speed* was definitely not 0 m/s. This is defined as her total *distance* traveled over the period of time, so if she ran the 400-meter track in 60 seconds, her average speed would be 400 m / 60 s = 6.67 m/s. Her *instantaneous speed* is simply her speed at a specific moment in time – for example, if you paused a video of her run, her speed at that exact moment – in other words, the number of meters she was traveling per unit of time at that moment.

This shows how careful you need to be with the measure you choose. Instantaneous velocity is much more useful than average velocity on a looped (or any non-linear) track, while there are benefits to finding both instantaneous and average speed if you don’t need to know her direction of motion.