Distance, displacement, and coordinate systems
- [Instructor] What we're going to do in this video is think about different ways to represent how position can change over time. So one of the more basic ways is through a table. For example right over here in the left column I have time, maybe it's in seconds, and in the right column I have position and this could be in some units, let's say it's in meters. So at time zero we're at three, after one second, we are still at three, after two seconds we're at negative one then after three seconds, we're at zero, after four seconds we're at zero, still at zero, after five seconds we are at two, maybe two meters. Now this is somewhat useful, but it's a little bit difficult to visualize. And it also doesn't tell us what's happening in between these moments, what's happening at time half of a second. Did we just not move, did our position just not change, or did it change and then it got back to where it originally was after one second? We don't know when we look at a table like this. But another way to think about it would be some type of animation. For example, let's say we have our number line, and let's say the object that's moving is a lemon. And so at time zero, it starts at position three, so that's where it is right now, and let's see if we can animate it. I'm just gonna try to count off five seconds and move the lemon accordingly to what we see on this position timetable or time position table. Zero one two three four five. So that was somewhat useful, but maybe even more useful thing would be to graph this somehow so that we don't have to keep looking at animation so that we can just look at with our eyes what happens over time. So for that, we can construct what's known as a position time graph. Typically, time is on your horizontal axis and position is on your vertical axis. So let's think about this a little bit. So at time equals zero, our position is at three. So at time zero, our position is at three, and then at time equal one, we're at three again, at time two, we are at negative one, at time two, our position is negative one, at time three, our position is zero, so our position is zero. Remember, even though we're thinking about left right here, here position is up down. So here our position is zero at time three, and then at time four, our position is still zero, and then at time five, our position is at two. Our position is at two. So for the first second, I don't have a change in position or at least that's what I assumed when I animated the lemon, and then as I go from the first second to the second second, my position went from three to negative one, from three to negative one, and if we do that at a constant rate we would have a line that looks something like this, I'm trying, that's supposed to be a straight line, and then from time two to three, we go from position negative one to zero, from negative one to zero. Here, it would've been going from negative one to zero moving one to the right, but over here, since we're plotting our position on the vertical axis, it looks like we went up but this is just really going from position negative one to position zero from time two seconds to three seconds. Now from three to four, at least the way I depicted it, our position does not change, and then from time four to five, our position goes from zero to two, from zero to two. And so what I have constructed here is known as a position time graph, and from this, without an animation, you can immediately get an understanding of how the thing's position has changed over time. So let's do the animation one more time, and just try to follow along on the position time graph, and maybe I'll slow it down a little bit. So for the first second we're gonna be stationary, so we can just count off one Mississippi. And then we go to, our position goes to negative one over the next second, so then we would go two Mississippi. And then we would go three Mississippi, four Mississippi, and then five Mississippi. But hopefully you get an appreciation that this is just the way of immediately glancing and seeing what's happening.