Distance and displacement in one dimension | One-dimensional motion | AP Physics 1 | Khan Academy

Previous videos we've talked a little bit about distance traveled versus displacement. What I'm going to do in this video is discuss it on a one-dimensional number line, and we'll get a little bit more mathy in this video. So here is my number line, and let's say that this is 0, 1, 2, 3, 4, and it keeps going on, and then in the negative direction: negative 1, negative 2, and negative 3.

Let's say that I start off with a lemon. Let's say my lemon starts off right over here at zero on my number line. And let's say it first moves two to the right. So it first moves two to the right; I'll denote that by plus two. Then from there, it moves three to the left. So then it moves three to the left, and I will use negative for the left, so it moves three to the left.

And then let's say that it then moves another one to the left, so then it goes another one to the left, and I'll denote negative one as moving one to the left. So based on what we know about distance traveled and displacement, what is the distance traveled for this dot? Distance traveled! Pause the video and see if you can figure that out.

Well, remember, distance traveled is the entire path length or the entire length of the dot's journey. So this is going to be equal to 2 to the right, so plus 2, and then 3 to the left. Now this is an important notion when we talk about distances: we wouldn't say positive or negative; we just care about the absolute value of the amount that we are traveling.

So we won't specify a direction. Now, you might say, "Hey, where's the direction being specified?" Well, implicitly, whether something is positive or negative on this number line is giving a direction. But if we're talking about distances, we wouldn't pay attention to the direction; we only care about the magnitude.

So this would be two plus three plus one. It doesn't matter if this is one to the left or one to the right; it doesn't matter if it's positive one or negative one. We care about its absolute value; we care about its magnitude. So the distance traveled in this example is going to be 6 units. Whatever the units are on my number line right over here – if these are in meters, then this would be 6 meters.

Now, what is the displacement? And remember, displacement is the net change in position. Displacement! What is that going to be? Pause the video and see if you can figure it out.

Well, displacement is going to be – you could view this as equal to your final position, and we'll use x. Let's say this is the x-axis, so we'll say x final, your final position, minus your initial position. It's really just your change in position. So what is your change in position here?

Well, your final position is you are at negative two at x equals negative two, and then what was your initial position? Your initial position you started at zero, so negative two minus zero is equal to negative two. So how would we visualize that on our drawing here? Well, we started here. Just think about what is your net change in position. You started here, and regardless of what your path was, you ended up 2 to the left. So your displacement is negative 2.

Now, displacement – we care not just about the magnitude; we care about the magnitude and the direction. And now, so you might be saying, "Well, where is the direction specified?" If I just say negative 2, well, the sign in a one-dimensional case is giving us our direction. So the sign is giving us a direction. I start off implicitly with this notion that negative is to the left and positive is to the right.

And we're in this one-dimensional world, and those are the only two directions that I can travel in. So if I'm in this one-dimensional world, or if I'm thinking about just one dimension, the sign gives me my direction. So that's why displacement – where I care about the magnitude and the direction – I do care about the sign, while distance, where I only care about the magnitude, I don't care about the sign.

So I just keep adding up the magnitudes. While over here, another way you could think about it: you first get displaced by 2 to the right, so that's plus 2; the plus says to the right. Then you get displaced by 3 to the left, so that is minus 3, and then you get displaced by one to the left again, so that's minus 1. That's why we're talking about displacement; that's why we care about the sign.

And if you were to add all of these together, you are going to get a net displacement of negative 2. But an easier way was just: what's your final position minus your initial position?