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## Distance, displacement, and coordinate systems

# Worked example: distance and displacement from position-time graphs

## Video transcript

- [Instructor] In other videos,
we've already talked about the difference between
distance and displacement. And we also saw what it meant to plot position versus time. What we're gonna do in this video is use all of those skills. We're going to look at
position versus time graphs, and use them in order to
figure out displacement and distance traveled. So this first question says, a 3.2 kilogram iguana runs back and forth along the ground. The following graph shows
the horizontal position of the iguana in meters over time. The first question is what is the displacement
of the iguana between zero seconds and five seconds? So, be very careful. It's not saying between zero
seconds and six seconds. It's saying between zero
seconds and five seconds. So pause the video, and see
if you can figure that out. Well, displacement is just
your change in position. And sign matters. We care about direction. So it's going to be your final position minus your initial position. Well, what is your final position? Well, we're not talking about the final, the last time that we have on the graph. Here our final is going
to be at 5 seconds. So our final position is
at a positive six meters. So our final position
is positive six meters. And from that you're gonna
subtract our initial position, where our initial position
was at negative two meters. Negative two meters. And so this is going to be equal to six minus negative two
is positive eight meters. So one way to think about it is net this iguana, shifts eight meters in the positive direction. And if we think positive
direction is, say, to the right, it would be eight meters to the right. We could draw a number line here. So if we, it's sometimes confusing 'cause we're talking about
the horizontal direction, but we're plotting
position vertically here. But we could take the same number line and make it horizontal. And you'd have negative
six, and negative four, negative two, zero, two, four, six, eight. And so what's happening here is the iguana is starting at negative two, and then over the next three seconds, it goes to positive six. It goes over here, and so it
shifts to the right by eight. And that's what we saw right over there. Six minus negative two is eight. Now what about the distance
traveled by the iguana over that same time period? Pause the video, and see
if you can figure it out. Well, the distance is the
total length traveled. The total path length. But once again, it went from negative two all the way to positive six
over the first three seconds, and then it just stays there. So if we think about distance, we're actually going
to get the same result. Even though we don't care
about direction here, we just care about the magnitude, we still get the same thing. It traveled eight meters. So it travels eight meters. So we're gonna get the exact same result. Now what would be interesting is the thing what would be different if
instead of five seconds, if this said six seconds? If this was between zero and six seconds? Well, then for displacement, we'd say, hey, look, we went plus
eight, but then we're gonna go another negative six. So this right over here, lemme make that, you're gonna go another negative six, and so you actually have, you'd have the plus eight minus six. So if you did it for all
the way to the sixth second, then your displacement is
going to be plus two meters. You have the eight, and
then you subtract the six. Another way top think about it is you would finish at zero. You started at negative two. Zero minus negative two is two. If you wanted to think about the distance between zero seconds and six seconds, well, you would have this eight meters. And then, you would go, even though you're going back to zero. So you're going back to
zero right over here. Six meters. You wouldn't subtract it because the total path,
you traveled eight meters to the right and then
six meters to the left. And so you would add 'em. So if you said over the first six seconds, instead of the first five seconds, this would be 14 meters. Let me make that clear again. Displacement and distance, in both cases, you have plus
eight over that first leg, I guess you could say. And then when we talk about displacement, we subtract it 'cause we're
now moving to the left. We move to the left by six. So in the case of displacement,
you subtract the six, and you have a net displacement of plus two. But distance, the total path traveled, you have the eight to the right, and then six to the left. Which gives you a total
path traveled of 14. Let's do one more example here. A 2.7 kilogram armadillo rolls in a straight line in the desert. The following graph shows
the horizontal position in the... The horizontal position of the armadillo in meters over time. And so let's think about the same thing. Over the first 24 seconds, let's go all the way to the 24th second. Let's think about what the displacement is and what the distance traveled is. So, first pause this video, and see if you can figure
out the displacement over the first 24 seconds. Well, this is going to
be our final position minus our starting position. Our final position, we're at zero meters. Our starting position,
we were at time zero, we were at six meters. All right, lemme just
write the numbers down. So it is negative six. Let me say that one more time. At time 24, notice our
vertical coordinate. We are in a position of zero. That's where that came from. When we started, at time zero, our position was right over here. So our final, minus our
starting, is negative six. And you could also see that. If you just look along this line, we shifted from positive six to zero, which would be a shift of six to the left if you made this horizontal. Or a displacement of negative six. Lemme draw this on a horizontal line just to make this a little bit more clear. So if you have zero, four, oh... Zero, three, six, nine, 12, 15, so on and so forth. We are starting at six,
do that purple color. We're starting at six, we do a bunch of stuff in between, but then we end up after
24 seconds at zero. So our shift, we went six to the left. Or we have a displacement of negative six. Zero minus six is negative six. So now let's try to figure out distance. Pause the video, and
figure out the distance that this armadillo travels
over this 24 seconds. So, this is interesting. So it's right here. It starts off at position of six. So let me do it right here. It starts off at position of six. It stays there for the
first eight seconds. Then, from the eighth
second to the 16th second, it's position increases
by nine to get to 15. So it does this. It goes to 15. So this is going to be plus. This is going to be plus nine. And then, on the 16th second, it goes from 15 back to zero. So it goes from 15 back to zero. If we're thinking about displacement, we would write a minus 15 here, and then we would net these
out to get to a negative six. But we're thinking about distance. So we want, we think
about the absolute value of the various parts of the path traveled. So all of these are gonna be positive. We just say, hey, what
is the total journey? So this is gonna be plus 15. We just care about the
length of these arrows, not the direction. And so, nine plus 15 is 24. So this is interesting. Even though the armadillo
traveled a total of 24 meters, its entire path was 24 meters long. Its net shift, its displacement, is six meters to the left. And this is another thing to emphasize. This negative number, this is implying direction. It's saying, if we're
look at this number line, it's saying to the left. Even if it was a positive six,
because we're talking about displacement, it would imply, positive would mean to the right. Distance doesn't tell you about direction. It just tells you the absolute magnitude of the total distance traveled, or the length of the path.