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We've now described what we
call the instantaneous velocity

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at some time t.

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And we talked about it
as a limit as delta t

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goes to 0 of the displacement
over the time, which

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we wrote as the derivative
of the position function

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in terms of i-hat.

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Now this derivative, we're
going to use a notation.

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We'll just write that as
v of t i-hat, where v of t

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is the component of the
instantaneous velocity at time

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t.

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Now remember, this
is just a symbol.

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But we describe this
as the derivative

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of the position
function, which I'll

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indicate as a function of time.

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And that's what we
mean by the component

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of the instantaneous velocity.

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Now what we would like
to do is ask ourselves,

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how does the velocity
change in time,

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if the runner is
increasing their velocity?

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Well, in general, we'll do
exactly what we did before.

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What we'd like to
introduce first

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is the concept of the
change in the velocity

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delta v, which will describe
as the velocity at time

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t plus delta t minus
the velocity at time t.

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So what we have in here is
the change-- the component

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of the change of the velocity.

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And this quantity
is how the velocity

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changed in into time interval
between t and t plus delta t.

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Remember, this is
very specifically

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for this time interval.

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Now what we would like to do is
take the same limiting process.

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Let's take the limit
as delta t goes

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to 0 of this change
in velocity over time.

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So we have the limit
as delta t goes to 0

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of delta v, delta t, i-hat.

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And this quantity
is what we call

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the instantaneous acceleration.

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Now as before,
what we're doing is

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we're plotting the
component of the velocity

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as a function of time.

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Let's just say, again, that
we have some unusual function.

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I'll just draw it like this.

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And here's the
picture at time t.

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Here is the picture at
time t plus delta t.

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This change, delta v over
delta t, this is v at t.

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Up here, this is what we mean
by-- let's just-- remember,

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we're plotting the velocity
as a function of time v of t

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plus delta t and
this quantity here,

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which is delta v over delta t.

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Again, we can even call
as we said before, we can

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call this average acceleration.

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But what we're interested
in is that as we

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take the limit as delta t goes
to 0, then the slope changes.

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You can see.

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And we continue this
limiting process,

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until we shrink
delta t down to 0.

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And that what we
have here is this

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is the slope of
the tangent line.

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And that's what we call the
instantaneous acceleration.

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So if we want to
use a notation, we

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don't want to keep on writing
limit delta t goes to 0.

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So what we can write is a of t.

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It has a component a of t i-hat
and that component of a of t.

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This is precisely the derivative
of the velocity function

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as a function of time.

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And so now we've described the
position vector, the velocity

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vector, and the acceleration
vector associated with motion.