Speed is a number that measures of how fast something is moving. Velocity is a vector that indicates how fast something is moving and in which direction. Under this framework, the magnitude of velocity is the speed, and speed is never a negative value. In common usage, speed and velocity are often interchangeable. However, from the definitions you can see that they are not interchangable in physics courses. So just be aware of the distinction and do your best to practice being precise in your language.

Although we have a lot of experience with velocity, it takes a bit of thought to translate those experiences into something formal. In order for there to be velocity, there must be some amount of motion. And that motion will require an amount of time. Velocity is made up of the ratio of those two quantities. Specifically, we define velocity as \(v = \frac{\Delta s}{\Delta t}\text{,}\) where \(\Delta s\) is the change in position during the observed movement, and \(\Delta t\) is the amount of time the movement took. Notice that this creates a unit of \(\frac{\text{Length}}{\text{Time}}\text{,}\) such as meters per second or miles per hour.

This formula is often rewritten as \(\Delta s = v \cdot \Delta t\text{,}\) or more simply as \(d = vt\text{.}\)

Technically, this formula gives us the average velocity. The reason is that this calculation doesn’t take into account all of the nuances of what happens between the start and end of the motion. In the example, it’s probably the case that the car stopped somewhere along the way (at a stop sign or traffic light), and at that moment the car was not moving at 60 miles per hour. Simiarly, it’s likely that the car was driving faster than 60 mph on the freway. But because the elapsed time is so large, all of those details are lost and we are only left with the overall picture.

Another way to think about the velocity is to try to keep track of all of those details. Rather than just thinking about the starting and stopping times and positions, we want to know what was happening at every moment of the trip. We would call that the instantaneous velocity. It’s not hard to imagine creating a chart of values for velocity at specific times along the trip, similar to how we can create a chart of the position at any time. However, there’s an interesting question that arises from this. What do we even mean by "velocity at a specific time"? How would we measure it? Velocity requires movement, and movement requires time, so we will always need both a start time and a stop time for any measurement, and they can’t be the same. This brings us back to the arrow paradox! The Arrow Paradox

We will say more about this in a later section, but the if we practically want to try to get instantaneous velocity, we need to collect data in small enough increments that any change in velocity between the start and stop time is "small enough" to be ignored. So if we imagine two consecutive frames of a movie, we can think about how much the object moved over that period of time, and then use that with the framerate to get a good estimate of the instantaneous velocity.