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