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Mr. Andersen shows you how to interpret a position vs. time graph for an object with constant velocity. The slope of the line is used to find the velocity. A phet simulation is also included.
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Transcript Provided by YouTube:
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Hi. It’s Mr. Andersen and today I’m going to do position versus time part 2. And so
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if you haven’t seen part 1 make sure you go back and watch part 1. Unless this seems easy.
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And then just sit back and relax. In part 1 of position versus time what I showed you
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how to do was how to interpret a position versus time graph. And eventually make a velocity
00:27
versus time graph. And so in the first video I showed you that the slope of the line is
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always going to tell you the velocity. And so if I say this is the velocity versus time,
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and we set this equal to 0 here in the middle, this would be a negative slope. And so that
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would be a negative velocity. So from time 1, 0 to 2, we would have a negative velocity.
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But that negative velocity would be constant. I’ll put it like right there. And maybe we’ll
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change the color so that you can see it a little better. So this would be that negative
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velocity. Right here however we have a positive velocity. And the slope is about the same.
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Not quite. But pretty close to the same. So it would be a positive slope. And so that’s
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going to be over here. Now this would be a straight vertical line between the 2. I also
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showed you that if you ever have no slope, like we would right here, then that means
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that we’re going to have a velocity of 0. Or for the next 3 seconds we’re going to have
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no velocity. But the one thing I didn’t show you is what to do when you get a curved line.
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And so the best way to attack a curved line is to look for any point where it actually
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is 0. And so if this is time 0, we know that that’s going to be zero at that point. So
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we could actually mark it on there. But what goes on in this middle part we’ll have to
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deal with in this podcast. And so let me show you how that works. Again we’re going to use
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The Moving Man. So The Moving Man is a simulation from the University of Colorado. It’s a way
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to move a man around and actually see how position versus time and a velocity versus
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time will actually change. So let’s go to The Moving Man. What I’ve done is positioned
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Moving Man down here at negative 10. And now I have the position versus time graph. But
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also the velocity versus time graph. And so I have control of that. My hand is a little
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bit shaky but what I’m going to be looking at is the man uphere. And then I’m going to
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try to get him to kind of speed up. Go over. Visit the house. And then come back. That’s
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my goal. So let met get it started. So I’m going to start Mr. Man moving. So he’s kind
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of going slow. Now I’m really going to speed him up. But he’s getting too fast so I want
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to slow down. And now let’s have him turn around. And go back. Now he’s going fast.
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And so he better slow down before we get to the other side. And let me pause it. Okay.
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Now let’s play that back and show you what happened. And so as we play it you can see
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that down here along the bottom my hand is a little bit shaky. But if I were to kind
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of fill this in it should be a fairly straight line. That’s what I was trying to do. And
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this should be a fairly straight line as well. And a fairly straight line there. So I kind
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of cheated a little but. But let’s play it out and see what happens. As we play it we
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notice that with the position versus time again we learn that as the slope of this line
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starts to increase then the velocity of the man starts to increase. And so what I really
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want you to watch is what happens right up here and the top. Where the slope flattens
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out. And so let’s do that. So right there. What’s the man doing? Well at this point,
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at that top point on here the man actually stops for a moment. And so when is the man
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stopped? The man is always stopped where this slope is 0. Because the man was also stopped
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remember just sitting right back here. And so as the slope increases as we move across,
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then the man actually speeds up. Let’s watch what happens the rest of the way. He momentarily
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stops. And then he’s going to go in the opposite direction. And he’s going the opposite direction.
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He’s going to start speeding up. And then he slows down as we get towards the end. And
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so the neat thing about The Moving Man is it shows you if we ever get a curved line
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on a position versus time graph it indicates that we are either speeding up or slowing
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down. And so last time we dealt with straight lines on position versus time. And that was
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always constant velocity. But here it gets a little bit more complex. And so you should
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be able to look at a graph like this and figure out exactly what The Moving Man is actually
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doing. And so what would The Moving Man be doing here? He would be stopped. And what
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would the moving man be doing right here? He’d be stopped. And right here? He’d be stopped.
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And right there he’d be stopped. And right there he’d be stopped. And so how do I know
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that? Well at this point there’s a line in math, in geometry we call that the tangent
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line. And the line that is perfectly parallel to this graph at that one point is called
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the tangent line. And at this point that tangent line would be 0. And at this point it would
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0. At this point it would be 0 as well. And so the fun part is figuring out what happens
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from here, where it is 0 to here where it is 0 as well. And so the way that I’d use
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this. This is a little magic pen. And what do I mean by magic pen? What I’ll do is I’ll
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actually hold this pen up to the screen or the graph or whatever I’m doing. And if the
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pen, as I follow the line and it goes up and down and up and down. As I follow the line
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anytime the pen starts to increase that tells me that the speed is increasing or the velocity
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is increasing. Anytime it starts to turn this way that means it’s slowing down. And anytime
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it moves like this as I move across, that means that the velocity is actually increasing.
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But it’s increasing in the negative. And so I think of this almost like a throttle. And
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I can look at the pen and it tells me how fast that object is actually moving. So let’s
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do some problems as they might be presented to you. On the left side here we have a position
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versus time graph. And the first thing you should always look at is, is it a position
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versus time graph? Are they giving me a velocity versus time graph? And so this is a position
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versus time graph. And so what we can do from that is we can always go to the velocity versus
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time graph. And so the first thing I should do look at is at any point which it is 0.
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And so here’s the point at which the object is actually going 0. And so I could even note
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that on here. So if I’m going to draw a line right across this side like that, at time
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2 it’s actually going to be going 0. And so I could mark that on my graph. Is there another
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place where it is actually going zero? Yeah we could see it right up at the top. So at
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time 6 it’s going to be going 0 as well. And where else is it going 0. It’s going 0 from
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8 in. So from 8 in it’s going 0 as well. Okay. So those are the points where I know the speed
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is going to be 0. Now I’ve just got to figure out what’s going on from here, time 0 to time
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2. And so using my magic pen I kind of hold it up to the graph and I see that my magic
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pen is pointed down. And as I follow the graph, it will actually flatten out. And since it’s
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pointed down that means that we have a negative velocity. And since it’s really pointed down
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far that means we have a negative velocity that’s really big. And so I know that it starts
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with a negative velocity that’s very large. And we’re not going to have to calculate velocities
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because it’s tricky to do that when it’s actually curving. We could do it at one point but I
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know that between here and here it is actually going like that. Okay. What happens after
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it hits the bottom? Well as it hits the bottom then it starts to actually go up again. And
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I know that I have to somehow get over to here where it’s at 0 again. And so what I
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can look at is that right here it’s actually going to start to increase. And then it’s
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going to slow down again. And so I know that what it’s going to do is it’s going to go
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up like that and then it’s going to come back down again. So now I’m right here. We’re at
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time 6. And it’s actually not moving. So what happens from here down to 8? Well it’s going
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to go from a flat velocity to an increase in the negative velocity. And so we’re going
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to see an increasing in the negative velocity as well. So it’s going to go like that. And
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then the whole this is going to kind of come up and then it’s going to be 0 again. And
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so that would be the graph for this position versus time. Velocity is increasing. Velocity
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is decreasing or increasing in the negative. And then the velocity stays at 0. Let’s try
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an even cooler one. Now when you look at one like this the first temptation is to say this
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is a ball that is bouncing. And you start to have this idea that it’s maybe bouncing
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from the left to the right. Remember in any of these graphs, just like The Moving Man,
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we’re just looking at an object that moves along one dimension. So it’s moving back and
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forth. But this actually could be a bouncing ball. And so the first thing I want to do
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is figure out where it’s not moving. And so we would say that it has a slope of 0 here.
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So at time 2. Let’s put this in. At time 2 it’s going to be right here on the line. Let
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me actually change color. So at time 2 it’s going to be right here. Where else is it not
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moving? Right here at time 6. It’s got a slope of 0. And where else? I would say at 10 it’s
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got a slope of 0 as well. So it’s like that. Okay. Now I just use my magic pen if that
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makes sense again. And so at time 0 I hold my pen up to the graph and I see that it’s
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a positive slope and it’s a really really steep slope. And so between time 0 and time
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2 it’s actually going to slow down. And so I know that it’s going to start here. And
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then it’s going to end up at 0. Where does it go from there? Well from here down to here
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it’s just going to keep increasing. But it’s increasing in the negative. And so it’s going
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to increase in the negative. It’s going to keep going like that. All the way until we
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get to point 4. So it’s going to increase in the negative until we get to point 4. Now
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something weird happens. Right here when I get down to the bottom, it’s really going
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fast in the negative and then it instantly goes up and is really large in the positive.
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And so what I would do on my velocity versus time graph is I have to go way back here again.
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In other words I’m going to have that vertical line. Goes just like that. And then it’s going
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to go down to 0 again. And then it’s going to go all the way down to 8. Like that. And
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then the velocity is going to go up again. So the velocity is going to go up like that.
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And then it’s going to come down to 0. Which is kind of a crazy line. So if you think about
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it the velocity this whole time is actually going from positive to large negative and
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stopping off at 0. And then it’s doing the same thing and the same thing. Now in some
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of the next videos what I’m going to show you how to do is actually to go from a velocity
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versus time graph. Let me choose a good color. From a velocity versus time graph to an acceleration
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versus time graph. Now the cool thing about that is that the rules that we followed to
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go from a position versus time graph to a velocity versus time graph, again using the
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slope of this line got us the velocity versus time graph. We do the same thing when we go
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to an acceleration versus time graph. If I were to follow my magic pen along this line,
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what I’ll notice is that the acceleration versus time graph is always going to be pointed
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down. And it’s always going to be negative. And so when we talk about, in science, the
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acceleration due to gravity being negative 9.8 meters per second squared, that’s why.
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In other words we always have an acceleration towards the center of the earth. And even
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though the ball is bouncing and bouncing and bouncing and bouncing, it always has a negative
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acceleration. And that’s constant. So we’ll get into that in a little more detail in the
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next few podcasts. But I hope that’s helpful.
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This post was previously published on YouTube.
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Photo credit: Screenshot from video.
