Today, we’re going to talk about average speed vs average velocity, and how average speed isn’t just average velocity with no direction. We will be talking about distance and displacement very often, so if you don’t know the difference between them exactly, go check out the previous article. 

The definition of speed and its equation is much well known.

Speed is equal to distance over time. In simpler mathematical terms we sometimes say that v is equal to ∆x over ∆t — or so you thought. Many people think that that is the equation for speed. Yet ∆x in no way represents distance. Hear me out.

∆ the the capitalized form of ∂, the greek letter delta. If you look up Wikipedia, you will find only one relevant entry:

Change of any changeable quantity, in mathematics and the sciences …

Delta is the initial letter of the Greek word διαφορά diaphorá, “difference”.

More specifically, the average change (once we talk about instantaneous velocities, you will see "d" as opposed to "∆" for very small changes). In fact, Wikipedia also cites explicitly that

is the definition of the symbol ∆. Therefore, it follows that

(Instead of writing numbers as subscripts, Physicists tend to use i and t as subscripts for initial and terminal.) The bar over v stands for “average”, which you’ll use very often if you do statistics.

Notably, ∆x here is terminal position minus initial position. Not distance, displacement. But we have already argued that distance is not necessarily equal to displacement. What happened here?

Well this “v = ∆x / ∆t” was never an equation for speed. v here stands for velocity. In fact, the definition of velocity is that

And this is the difference between average speed and average velocity. Really, physicists only use speed instead of velocity when the object only moves in one direction, when speed is equal to velocity.