(Notice: Vectors are kind of like dimensions. The equation defining average velocity in symbols then becomes (thinking about a velocity in 2D or 3D): (In some texts, an average is indicated by putting a bar over the variable, but since we are already putting a vector arrow over a variable to indicate it has direction, this would get too messy.) As discussed in the page Values, change, and rates of change, we will use the symbol Δ to mark when we mean a change in something. We express this in symbols by putting an angle brackets around the velocity to indicate "average" - like this: $\langle v \rangle$. Thus, in the Garfield cartoon below (keeping the "dog" metaphor), the fact that Garfield kicked Odie back to his starting point means that his average velocity was 0 - despite having moved in the middle, because the two motions (to the right and to the left) cancelled each other out. We call this the average velocity because it only pays attention to the beginning and the end - how big the change was in your position - and not how did you get from your starting point to the finishing point. We mean velocity to be a vector with "how far did you move" really standing for "what was your vector displacement"? This allows us to do much more with velocity than the word equation does. Warning: Although the word equation helps with making conceptual sense of what's going on with velocity, it doesn't capture everything we are thinking about when we talk about velocity.
(We show the icon of the dog here since we want you to "make sense" of the velocity equation, pulling the picture of the spotted dog out of a picture with lots of spots. That way, doing either a bigger distance in the same time or the same distance in a shorter time both give a larger number to the velocity and agree with our sense of what it means to move faster.
To make sense out of this, let's first write it as a words equation:Īverage velocity = (How far did you move?) / (How long did it take you?)
Velocity is the answer to the question: How fast are you changing your position? It's basically asking for a comparison of where you are at two different times and makes the rate of change quantitative.
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