WEBVTT

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We've now described what we call
the average velocity for a time

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interval between when the
runner started at time t

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to a later time
at t plus delta t.

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And we described
that as the component

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of the displacement vector
divided by the time interval

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times a unit vector, i hat.

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Now what we'd like to ask
is a separate question.

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So our question
now is, what do we

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mean by the velocity at
some specific time, T1?

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Now in order to understand
that, let's just

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make a plot of the
position function.

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So remember we called the
component of the position

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function x of t.

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So we're going to plot the
component of the position

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function with respect to time.

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Now let's just say
that the runner started

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at the origin of time equal 0.

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So I can make some type of
arbitrary plot of that position

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function.

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And let's indicate in
particular, the time T1.

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So what this
represents is x of T1.

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And so first I'd like to
consider the interval T1

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and T1 plus some
later time, delta T.

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So let's make this
T1 plus delta t.

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This is the time
delta t and up here

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we have our position
function at T1 plus delta T.

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Then for this time interval,
the average velocity,

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so for this particular
time interval,

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the average represents
delta x over delta t.

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So it's just rise over run.

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It's just the slope
of this straight line.

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So for this particular
interval, the average

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is the slope of the line
shown here on the figure.

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Now this is just
an average velocity

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and now what we would
like to do is shrink down

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our interval delta T. So now
let's make another case where

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we shrink delta t and
let's again calculate

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the average velocity.

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So for instance, suppose
we have a smaller delta t

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and we draw that line.

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Then our average velocity
represents that slope.

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And again, we keep
on taking a limit.

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So now we have another slope so
we have one slope, two slopes,

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and now we shrink
again to a new delta t

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and you can see that
the slope is changing.

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And if we consider
the limit as delta t

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goes to 0 of this
sequence of slopes,

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then what are we getting,
you can see graphically,

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that eventually we will
get to a line which

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is the slope of the
tangent line at time T1.

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And so in this
particular case, what

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we mean by the instantaneous
velocity, v at time T1,

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is the limit as delta
x goes to 0 i hat here

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where we're taking,
this is the limit,

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delta t goes to 0 x at
T1 plus delta t minus x1

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of t divided by delta t and the
whole thing is a vector, i hat.

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So what a limit is, is
a sequence of numbers.

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So we take a fixed delta
t, we calculate the slope.

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We take a smaller delta
t, calculate the slope.

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And each time we do that,
the slopes represent

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a sequence of numbers and
the limit of that sequence

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you can see graphically,
is the slope

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of the tangent line at time t 1.

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And so what we say is, v of T1
is the instantaneous velocity

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at time t equals t1.

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And that's how we describe
instantaneous velocity

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at some specific time.

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If we were now being a
little bit more general,

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we could just say
that v at any time t

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is the limit delta
t goes to 0 delta

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x over delta t ball
direction i hat,

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and the only thing here is
we're no longer considering

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T1 but an arbitrary time t.

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This quantity, the limit, is
awkward to write every time.

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It has a name.

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And that's precisely what
we call the derivative

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of the position function.

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So our instantaneous velocity
is the time derivative

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of the position function
at any instant in time.