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- [Instructor] Let's talk about
position versus time graphs.

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These are tricky.

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If you've never seen these
these can be really tricky.

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Physicists love these.

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Teachers love these.

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They're on lots of tests.

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Why do so many people love these things?

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Because you could compact
a ton of information

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about the motion of an object

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into the small little space right here.

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Basically specify the
entire motion of the object,

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and you didn't even have
to write an equation

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or say a bunch of words.

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It's all just right here.

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So these are actually really handy.

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You should know how to deal with these.

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So this graph represents
the motion of an object.

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And instead of just saying object,

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let's make it specific.

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Let's say it's a turtle.

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A turtle, not just any turtle.

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A turtle with a jet pack
on this turtle's back.

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And I don't want a sternly worded letter.

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I don't want a bunch of nasty comments.

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Let's put a helmet on this turtle.

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It's a pink helmet.

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She's pretty.

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And now we got turtle safety.

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You always gotta use rocket safety.

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Alright so let's say this
turtle's moving around.

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And this graph represents the motion

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of this particular turtle.

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The first mistake a lot of people make

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is they think that well maybe

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the shape of this graph

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is the same as the shape the
turtle takes through space.

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So maybe the turtle went forward
and then down and then up,

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but that's not right.

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In fact it turns out
that's not even close.

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To figure out what this
graph actually says

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let me lay down a horizontal access here.

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This access is gonna represent
the horizontal position.

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So I'm gonna label this x

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and it's gonna be measured in meters.

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And I'm doing that because look over here

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we're graphing.

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In this case I wrote is as x,

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so this is gonna be
the horizontal position

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of the turtle.

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So the horizontal position is
what we're actually graphing.

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What that means is if you find the turtle

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at some point over here at x equals two,

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then the graph should represent

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that the turtle is at x equals two

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by showing the value is two.

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So somewhere at like two and four seconds

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this turtle was at two meters.

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And that's what this
graph's gonna tell you.

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So let's just read this graph

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and figure out what this
particular turtle did.

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If this turtle didn't go
forward, down, and up,

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what did this turtle do?

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We'll start at t equals zero.

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We'll go up from there.

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And at t equals zero

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the value of this graph is three.

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And the value of this graph

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is representing the horizontal position,

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so the value of the graph

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is giving you the horizontal position.

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So at t equals zero,

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the turtle is at three meters.

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So let's put her over at three meters.

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She starts over here.

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Three meters, that's equal zero.

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Now what happens?

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So at t equals one second,

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same thing.

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We read our graph by going up,

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hit the graph,

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then we go left to figure
out where we're at.

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Again turtle's still at three.

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At two seconds we come up,

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hit the graph.

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We come over to the left to
figure out where we're at.

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This turtle's still at
three, that's awkward.

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This turtle didn't even move.

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For the first two seconds this
turtle's just sitting here.

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So a straight line a horizontal
line on a position graph

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represents no motion whatsoever.

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There was no motion.

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This is awkward.

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Turtle was probably trying to figure out

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how to turn on her jet pack.

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Should've read the instructions.

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Sorry.

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Alright.

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Now what happens?

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Turtle at some later time, four seconds,

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is at negative five meters.

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That's all the way back here.

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So between two seconds and four seconds,

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this turtle rocketed back this way.

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That's also awkward.

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Turn down the reverse booster.

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What a noob, ah turtle.

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Here we go.

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Made it all the way back to here.

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Then what does the turtle do?

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After that point, turtle rockets forward.

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Makes it back to zero at this point.

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And then all the way back to three meters,

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so this turtle rockets
forward back to three meters.

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That's what the turtle did.

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That's what this graph is representing,

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and that's how you read it.

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But there's more than that in here.

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I told you there was a lot
of information and there is.

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So one piece of information you can get

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is the displacement of the turtle.

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And the displacement
I'm gonna represent this

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with a delta x.

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And remember the displacement
is the final position.

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Minus the initial position.

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You can find the displacement
between any two times here,

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we're just gonna find
it for simplicity's sake

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for the total time shown on the graph.

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But I could've found it between
zero and like four seconds.

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Let's just do zero to 10, the whole thing.

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So what's the final position?

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The final position would be
the position the turtle has.

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At 10 seconds she was at three meters.

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At 10 seconds 'cause I
read the graph right there.

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Minus initially, 'cause we're
considering the total time,

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at zero seconds, the
turtle was also at three,

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that means the total
displacement was zero.

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And that makes sense 'cause
this turtle started at three.

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Rocketed back to five, well
actually started at three,

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stood there for a second or two,

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rocketed back to five,

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rocketed back to three,

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ended at the same place she started,

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no total displacement.

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What else can we find?

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We can figure out the total distance.

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For the total distance traveled

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remember distance is the sum
of all the path links traveled.

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So for this first path link,

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there was no distance traveled.

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That was the awkward part.

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We're not gonna talk about that.

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'Cause it might hurt hrr feelings.

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Then, so this is zero meters,

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plus between two seconds and four seconds,

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the turtle went from three to five.

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That's a distance
traveled of eight meters.

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And should we make that negative?

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Nope.

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Distance is always positive.

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We make all these path links positive,

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we round them all up.

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So eight meters.

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Because the turtle went from three

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all the way back to five.

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That's the total distance
of eight meters traveled.

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Plus between four seconds and 10 seconds,

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the turtle made it from
negative five meters

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all the way back to three meters.

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That means she traveled
another eight meters.

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That means the total distance
traveled was 16 meters

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for the whole trip.

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Again you could have
found this for two points

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any two points on here.

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Alright what else can you figure out?

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You can figure out the
say average velocity,

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sometimes people
represent that with a bar.

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Sometimes they just say the AVG.

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Oops, AVG.

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What does this mean?

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Remember average velocity is
the displacement per time.

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And let's find the total.

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So we're finding the total values here.

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So the total average velocity,

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I need the total displacement

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already found that.

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Total displacement was
zero for the whole trip.

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So this is zero meters,

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divided by it doesn't really matter now.

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But 10 seconds was the time it took

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for that entire displacement.

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Not meters, 10 seconds.

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So this equals zero.

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There's no total average velocity.

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The average velocity for
the entire trip was zero.

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Because the turtle had
no total displacement.

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How about average speed?

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So the average speed,

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I'm just gonna write it as average speed.

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Maybe you'll see it as an s with a bar

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maybe an s with an AVG.

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I don't know.

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Physicists use all kinds of letters.

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You don't know what you're gonna get.

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But the average speed is defined

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to be the distance per time.

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And again let's try to find
the total average speed

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for the whole 10 seconds.

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That's not too bad 'cause I
already found the total distance

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that was 16 meters.

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So 16 meters divided by the total time.

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It took 10 seconds for that entire trip.

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This turtle, she was going
1.6 meters per second,

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on average.

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That was her average speed.

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Probably would've been a little higher

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if she didn't have that
technical difficulty

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here at the beginning.

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Alright we can figure out
more than this though.

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We can figure out the
instantaneous velocity.

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Maybe you'll see it as VINST.

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Maybe you'll just see it as V.

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'Cause that's usually
what we're talking about

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when we're talking about velocity.

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We're talking about the
instantaneous value a lot.

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What is this?

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Here's the key idea.

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In fact this is maybe
the most important idea

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of this whole video.

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To find the instantaneous velocity,

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when giving a position versus time graph,

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you look at the slope.

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Because it turns out the slope

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of a position versus time graph

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is the velocity in that direction.

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So since we had a horizontal
position graph versus time,

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this slope is gonna give us the velocity

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in the ex direction.

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And not only that if we
find the average slope,

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we get the average velocity.

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And if we find the instantaneous slope,

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we're gonna get the
instantaneous velocity.

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So how do I do that?

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How do I find the instantaneous slope?

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Well in general.

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If you gotta curve the graph.

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You're gonna have to use calculus.

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But we're in luck here.

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Because look at these
lines they're all straight.

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And what that means

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is that the average slope
between any two points

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on one of these lines

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is gonna equal the instantaneous slope

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at any point on the line.

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So let's make this specific.

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Let's saw we wanna find
the instantaneous velocity

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at three seconds,

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pick any point, three seconds.

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How do we do that?

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Well we gotta figure out what we mean.

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By instantaneous velocity we mean

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the velocity at three seconds

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slope here but I gotta go to the graph

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so I take my three,

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I go down to the graph

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I wanna know what the
instantaneous slope was

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at that point right there.

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Let me draw on top of this thing here.

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I wanna know what the
slope was right there.

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How do I do that?

270
00:09:09,657 --> 00:09:11,364
Well I told you the key is that

271
00:09:11,364 --> 00:09:15,431
the average slope between
any two points on this line

272
00:09:15,431 --> 00:09:17,341
so I can pick these two if I want,

273
00:09:17,341 --> 00:09:19,050
the average slope between these two points

274
00:09:19,050 --> 00:09:22,550
is gonna equal the instantaneous
slope at any point,

275
00:09:22,550 --> 00:09:24,906
because look this slope isn't changing.

276
00:09:24,906 --> 00:09:26,251
Slope's the same the whole way.

277
00:09:26,251 --> 00:09:29,058
And if you take the average
of a bunch of quantities,

278
00:09:29,058 --> 00:09:30,603
they're exactly the same.

279
00:09:30,603 --> 00:09:32,468
You're just gonna get the same value

280
00:09:32,468 --> 00:09:33,971
as any one of these quantities.

281
00:09:33,971 --> 00:09:35,315
That was a complicated way of saying

282
00:09:35,315 --> 00:09:38,643
if you took the average of eight and eight

283
00:09:38,643 --> 00:09:40,349
and eight and eight,

284
00:09:40,349 --> 00:09:41,182
what are you gonna get?

285
00:09:41,182 --> 00:09:42,992
The average value of those is eight

286
00:09:42,992 --> 00:09:45,022
which is the same as
any one of these values.

287
00:09:45,022 --> 00:09:47,374
So if you ever have a graph
that's a straight line

288
00:09:47,374 --> 00:09:48,207
you're in luck.

289
00:09:48,207 --> 00:09:49,411
You don't need calculus.

290
00:09:49,411 --> 00:09:52,299
You find the average velocity by taking,

291
00:09:52,299 --> 00:09:54,845
sorry you can find the
instantaneous slope at any point

292
00:09:54,845 --> 00:09:57,842
by taking the average velocity
between any two points.

293
00:09:57,842 --> 00:09:58,675
I'm picking these two.

294
00:09:58,675 --> 00:09:59,508
Why these two?

295
00:09:59,508 --> 00:10:00,341
'Cause they're convenient look

296
00:10:00,341 --> 00:10:01,174
I know exactly where they're at.

297
00:10:01,174 --> 00:10:02,854
That's three and two,

298
00:10:02,854 --> 00:10:06,067
and this one's negative five and four.

299
00:10:06,067 --> 00:10:07,935
You might wonder why, why is this true?

300
00:10:07,935 --> 00:10:10,740
Why is the velocity equal to the slope?

301
00:10:10,740 --> 00:10:12,366
Well remember from math class?

302
00:10:12,366 --> 00:10:15,033
Slope was the rise over the run.

303
00:10:15,985 --> 00:10:17,944
And you might have seen that as okay

304
00:10:17,944 --> 00:10:19,944
this is math class here,

305
00:10:20,794 --> 00:10:23,774
y one over x two minus x one

306
00:10:23,774 --> 00:10:24,607
but you saw it like that

307
00:10:24,607 --> 00:10:27,190
because in math class
typically the vertical access

308
00:10:27,190 --> 00:10:29,058
was always y.

309
00:10:29,058 --> 00:10:31,130
In a horizontal axis it was always x.

310
00:10:31,130 --> 00:10:32,228
This is physics.

311
00:10:32,228 --> 00:10:34,264
Our horizontal access isn't x.

312
00:10:34,264 --> 00:10:35,930
Our horizontal access is t.

313
00:10:35,930 --> 00:10:38,254
And our vertical access
is what we're calling x.

314
00:10:38,254 --> 00:10:41,546
So for physics class
the slope of this graph

315
00:10:41,546 --> 00:10:45,133
particularly the rise in
this case is this axis

316
00:10:45,133 --> 00:10:47,005
so it's gonna be x two

317
00:10:47,005 --> 00:10:48,266
minus x one

318
00:10:48,266 --> 00:10:49,163
over the run.

319
00:10:49,163 --> 00:10:53,069
Well that's gonna be t two minus t one.

320
00:10:53,069 --> 00:10:54,861
Alright so how do we do this?

321
00:10:54,861 --> 00:10:57,177
Well this is .2 this is .1.

322
00:10:57,177 --> 00:10:58,010
How do you know?

323
00:10:58,010 --> 00:10:59,415
How come this isn't two and that's one?

324
00:10:59,415 --> 00:11:01,815
The point further in time
is the one you choose

325
00:11:01,815 --> 00:11:03,365
as the second point.

326
00:11:03,365 --> 00:11:05,890
So at four seconds and
negative five meters

327
00:11:05,890 --> 00:11:07,057
that's our .2.

328
00:11:08,163 --> 00:11:09,957
Alright so x two,

329
00:11:09,957 --> 00:11:11,179
that would be negative five,

330
00:11:11,179 --> 00:11:12,943
'cause I'm just reading my graph, that .2.

331
00:11:12,943 --> 00:11:14,576
That's negative five.

332
00:11:14,576 --> 00:11:16,362
So I got negative five meters

333
00:11:16,362 --> 00:11:18,612
minus x one that's this.

334
00:11:19,496 --> 00:11:20,753
Don't make x one four.

335
00:11:20,753 --> 00:11:21,654
That's a time,

336
00:11:21,654 --> 00:11:23,487
that's not a position.

337
00:11:24,747 --> 00:11:25,642
So point one,

338
00:11:25,642 --> 00:11:28,405
the horizontal position was three.

339
00:11:28,405 --> 00:11:29,952
So positive three.

340
00:11:29,952 --> 00:11:33,009
Put the negative here 'cause
the negative's in the formula.

341
00:11:33,009 --> 00:11:34,637
And then divide it by time two,

342
00:11:34,637 --> 00:11:36,714
that was four seconds.

343
00:11:36,714 --> 00:11:39,381
And minus t one was two seconds.

344
00:11:42,816 --> 00:11:44,562
And if you saw this thing,

345
00:11:44,562 --> 00:11:46,887
negative five and negative
three and negative eight

346
00:11:46,887 --> 00:11:48,837
divided by two seconds.

347
00:11:48,837 --> 00:11:50,303
Oops can't figure out my units.

348
00:11:50,303 --> 00:11:51,136
Oh look at that.

349
00:11:51,136 --> 00:11:53,429
I got negative four meters per second.

350
00:11:53,429 --> 00:11:55,141
That was the instantaneous velocity

351
00:11:55,141 --> 00:11:56,848
at three seconds.

352
00:11:56,848 --> 00:11:59,333
Negative four meters per second.

353
00:11:59,333 --> 00:12:01,531
Negative because the
turtle was going backwards.

354
00:12:01,531 --> 00:12:03,038
Remember that was the awkward?

355
00:12:03,038 --> 00:12:04,789
She turned on the reverse booster

356
00:12:04,789 --> 00:12:06,293
instead of the forward booster.

357
00:12:06,293 --> 00:12:08,281
Negative and four because look it

358
00:12:08,281 --> 00:12:10,921
going four meters every second.

359
00:12:10,921 --> 00:12:13,120
Made it eight meters and two seconds,

360
00:12:13,120 --> 00:12:16,695
that means she was going four
meters per second on average.

361
00:12:16,695 --> 00:12:18,771
And since it's a straight line,

362
00:12:18,771 --> 00:12:21,575
that was the rate she
was going at any moment.

363
00:12:21,575 --> 00:12:22,554
Beautiful.

364
00:12:22,554 --> 00:12:25,569
Alright that would've been
if the follow up question

365
00:12:25,569 --> 00:12:27,986
is what is it at 2.4 seconds?

366
00:12:29,595 --> 00:12:30,489
Don't get concerned.

367
00:12:30,489 --> 00:12:31,668
Look it's the same everywhere.

368
00:12:31,668 --> 00:12:32,806
It'd be the same answer.

369
00:12:32,806 --> 00:12:33,826
Negative four meters per second.

370
00:12:33,826 --> 00:12:35,537
For this whole line.

371
00:12:35,537 --> 00:12:36,921
What else can we figure
out one last thing.

372
00:12:36,921 --> 00:12:40,989
Let's say you were asked
what's the instantaneous speed

373
00:12:40,989 --> 00:12:41,906
at a point?

374
00:12:42,823 --> 00:12:45,134
So I'm gonna write that as SINST

375
00:12:45,134 --> 00:12:46,643
instantaneous speed,

376
00:12:46,643 --> 00:12:47,476
or just s.

377
00:12:47,476 --> 00:12:49,405
'Cause that's usually
what we mean by speed.

378
00:12:49,405 --> 00:12:52,988
Equals average value,
sorry, absolute value

379
00:12:54,407 --> 00:12:56,069
of the instantaneous philosophy.

380
00:12:56,069 --> 00:12:58,627
So now here I've got
to make an assumption.

381
00:12:58,627 --> 00:13:00,456
This is gonna get a little subtle.

382
00:13:00,456 --> 00:13:04,623
If all we're given is a
horizontal position graph,

383
00:13:05,743 --> 00:13:07,978
we don't really know about
the vertical position.

384
00:13:07,978 --> 00:13:10,503
This turtle could've gone back and forth,

385
00:13:10,503 --> 00:13:12,736
or the turtle could've
been like flying upward,

386
00:13:12,736 --> 00:13:15,579
as she went back and forth.

387
00:13:15,579 --> 00:13:18,264
And if the horizontal location
was the same the whole way,

388
00:13:18,264 --> 00:13:19,527
this would've looked exactly the same

389
00:13:19,527 --> 00:13:20,708
regardless of whether the turtle

390
00:13:20,708 --> 00:13:22,982
had any vertical motion at all.

391
00:13:22,982 --> 00:13:23,815
So we gotta be careful,

392
00:13:23,815 --> 00:13:26,565
'cause the speed is the
magnitude of the total velocity.

393
00:13:26,565 --> 00:13:29,369
This is the just the
velocity in the x direction.

394
00:13:29,369 --> 00:13:30,753
So we're gonna make an assumption.

395
00:13:30,753 --> 00:13:34,043
I'm gonna assume this turtle
was just moving horizontally.

396
00:13:34,043 --> 00:13:35,585
Instead of having the vertical motion.

397
00:13:35,585 --> 00:13:37,136
She's not ready for that yet.

398
00:13:37,136 --> 00:13:40,226
Alright so how do you get this?

399
00:13:40,226 --> 00:13:42,378
Speed is just the absolute value.

400
00:13:42,378 --> 00:13:45,384
The magnitude of the
instantaneous velocity.

401
00:13:45,384 --> 00:13:47,216
And if this is the only
component of velocity,

402
00:13:47,216 --> 00:13:50,180
then I can just figure
this out pretty easy

403
00:13:50,180 --> 00:13:53,913
by saying that, oh I gotta give you time,

404
00:13:53,913 --> 00:13:56,758
makes no sense to say instantaneous speed.

405
00:13:56,758 --> 00:13:58,961
I gotta say instantaneous
speed at a given moment.

406
00:13:58,961 --> 00:14:01,803
'Cause the instantaneous
speed here was zero.

407
00:14:01,803 --> 00:14:04,891
The instantaneous speed at this point

408
00:14:04,891 --> 00:14:06,034
would've been what?

409
00:14:06,034 --> 00:14:07,831
Well it would've been the
absolute value of this.

410
00:14:07,831 --> 00:14:12,014
So it would've been positive
four meters per second.

411
00:14:12,014 --> 00:14:13,565
That would've been the instantaneous speed

412
00:14:13,565 --> 00:14:15,898
at three seconds or any time

413
00:14:17,388 --> 00:14:20,305
between two to four seconds really.

414
00:14:21,717 --> 00:14:22,550
That was a lot.

415
00:14:22,550 --> 00:14:23,976
I told you there was a lot in there.

416
00:14:23,976 --> 00:14:26,132
So recapping really quick.

417
00:14:26,132 --> 00:14:29,036
The value of the horizontal
position versus time graph.

418
00:14:29,036 --> 00:14:32,171
Gives you the horizontal
position, surprise surprise.

419
00:14:32,171 --> 00:14:35,060
The slope of a horizontal
position versus time graph

420
00:14:35,060 --> 00:14:37,265
gives you the velocity in the x direction.

421
00:14:37,265 --> 00:14:40,883
The average slope gives
you the average velocity.

422
00:14:40,883 --> 00:14:44,139
The instantaneous slope give
you the instantaneous velocity

423
00:14:44,139 --> 00:14:46,172
and if it's a straight
line with no curvature

424
00:14:46,172 --> 00:14:48,738
these are gonna be the
same on any given line.

425
00:14:48,738 --> 00:14:49,835
They work the same here.

426
00:14:49,835 --> 00:14:51,019
You're like what hold on,

427
00:14:51,019 --> 00:14:51,875
these aren't the same?

428
00:14:51,875 --> 00:14:52,708
We'll that's because I averaged

429
00:14:52,708 --> 00:14:54,493
over this whole thing right here.

430
00:14:54,493 --> 00:14:57,955
I took the average velocity
over the whole time

431
00:14:57,955 --> 00:14:59,542
this slope was changing.

432
00:14:59,542 --> 00:15:01,617
So what I really got was
the average of all of these

433
00:15:01,617 --> 00:15:03,202
and that's why these weren't equal.

434
00:15:03,202 --> 00:15:06,737
But if I can restrain myself
to just the average value

435
00:15:06,737 --> 00:15:09,258
along a line that
doesn't change its slope,

436
00:15:09,258 --> 00:15:12,433
that will equal the
instantaneous slope at any point.

437
00:15:12,433 --> 00:15:13,730
And the instantaneous speed

438
00:15:13,730 --> 00:15:16,937
is the magnitude of the
instantaneous velocity,

439
00:15:16,937 --> 00:00:00,000
assuming you only have
motion in one direction.