Velocity vs. Speed

In my previous post, I introduced the concepts of position, displacement, and total distance traveled. I also talked about how position and displacement are vectors (they have both magnitude and direction), while total distance traveled is a scalar (it’s just a number with no direction attached).

Today we’re going to talk about another vector, velocity, and a related scalar, speed. You’ve probably used these words interchangeably in your everyday speech, but in physics, they mean two very different things. Average velocity () is the change in position () over the change in time (), while average speed () is the total distance traveled () over the change in time:

In these expressions, we call the time interval that we are considering. “Time interval” is our fancy physics way of saying “time span,” or, “Some time has passed.” So the average velocity formula is really telling us, “Some time () has passed, and during that time our object experienced a displacement of .” In terms of the SI units involved, we would say that our object has changed its position by some number of meters (in some direction) over some number of seconds. In math and physics speak, “over” literally means “divide by,” so we’ve got units of meters divided by seconds (m/s), which we say as “meters per second” (“per” being our other “divide by” indicator word*).

Average speed also has units of m/s, but as we saw above, it is equal to . Since speed and velocity have the same units, they must be related in some way, and it turns out that it all boils down to how big our time interval is. When is large, average speed and average velocity are definitely not guaranteed to be the same magnitude (size). For example, I might travel once around a circle. My average speed would be the distance traveled (the circumference, ) divided by my travel time, which is definitely a nonzero number. On the other hand, my average velocity would be my displacement (how far from start to stop), which after going around once would be , divided by my travel time. Thus the average velocity is zero, quite a different number from my average speed, which is greater than zero.

This all starts to change if we go to smaller and smaller (shorter and shorter time intervals). Imagine going down to a time interval of 1 nanosecond ( seconds); 1 picosecond ( seconds); 1 femtosecond ( seconds); etc. If you can imagine getting smaller and smaller until you’ve got seconds, then you’ve got the right idea. At this point, we’ve collapsed our time interval down to a single instant of time: the starting and stopping times are right on top of each other. When we consider the speed and velocity of an object during this instant in time, we say that we are thinking about the instantaneous speed and instantaneous velocity of the object. I like to think of instantaneous speed and instantaneous velocity as answers to the question, “What is the speed(velocity) of the object right NOW?” where I snap my fingers at the “NOW,” just to really emphasize how quick the moment is.

Remember above how we said that average speed and average velocity are not guaranteed to have the same magnitude? Well, it turns out that when we shrink our time interval down to an instant of time so that we’re considering instantaneous speed and instantaneous velocity, we can guarantee that they have the same magnitude. After all, if only an instant of time has passed, the displacement and the distance traveled must be the same size; there simply hasn’t been enough time for them to become noticeably different (or to get a bit fancy language-wise, for them to diverge from each other).

At this point, you may be feeling like this is all just semantics and wondering why I’m harping on it so much (for an entire blog post?!). Here’s the thing: physics is all about whether or not some aspect of an object is changing. Is its position changing?; is its velocity changing? – and perhaps most importantly, WHY are these things changing? (We’re not quite at the WHY yet, but we’ll get there.) The point is that physics is about change, so we had better have a good understanding of what that actually means and how we can start to think about it in a formal (mathematical) sense**.

Note that in all future blog posts, since at heart physics is concerned with instantaneous change, when I use the word “velocity” on its own, it is implied that I mean “instantaneous velocity.” If I am talking about “average velocity,” I will always include the word “average” in the term.

I didn’t have any practice problems embedded in this post, which means no pictures of Cheyenne today…or does it?

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*One of the challenging things about learning physics is that you need to not only learn the definitions of various new concepts, but that you also need to understand how physicists use the English language. I’ll do my best to point out the peculiar words and phrases found in physics lingo as I write this blog, but if you’re ever unclear about something, just ask!

**Physics being all about change is why it is so entwined with calculus, which is the mathematics of change over very very very small (“infinitesimal”) intervals. (If we’re thinking about the change in time, we use the word “instantaneous” rather than the word “infinitesimal,” but you could similarly think about an infinitesimal change in position, or in volume, or in…whatever you’re studying that’s changing.) That being said, you can understand some basic aspects of physics without the use of calculus, which is where we’re at in this blog right now (“algebra-based physics”).