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Let's consider a
very simple example

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of a runner in which our
position function x of t

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is given as a quadratic
function in time.

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It will be a constant
b times t squared.

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Here, b is a constant.

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Now it's always
important in SI units

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to consider what the
units of this constant is.

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Because a position function
is measured in meters

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and time is measured in
seconds, b is a constant

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and it has units of
meters per second squared.

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And there's an
example of a runner--

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and let's make a plot of
that position function.

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So we're going to plot x
of t as a function of time.

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b here-- let's make b
a positive constant.

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And so our function looks
something like that.

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Now, the velocity-- component of
the velocity-- remember v of t

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is given by dx dt
of this function.

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And the derivative of
a polynomial t squared

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is very simple.

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That's just simply 2b times t.

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Again, let's look at our units.

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Because b has the units of
meters per second squared,

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and when multiplying
that by second,

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we have the units of velocity in
SI units as meters per second.

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Now let's plot that function.

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Notice this is a
linear function.

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And so if I plotted
underneath here, the velocity

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as a function of time, it
starts off with a zero slope.

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Remember we're looking-- our
velocity at any given time

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corresponds to the slope of a
tangent line to the position

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function.

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And you can see that
slope is increasing.

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Now you wouldn't know
it from this graph,

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but if you did plot t squared,
it's increasing linearly.

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And so our velocity
function-- the initial slope

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is 0 at t equal 0, and it's
increasing linearly in time.

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So we'll just draw that
as some linear function.

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And the slope here
of this function

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will be now the acceleration.

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So a of t is the derivative of
the component of the velocity

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function as a function of time.

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And this derivative
is quite easy.

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It's just simply 2 b.

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Now notice those have
the units of meters

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per second squared, which
are units for acceleration.

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When we, again,
look at the slope,

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notice that at
every single point,

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the slope of the velocity as a
function of time is a constant.

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The slope here is
just equal to 2b.

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And so now if we plotted
our acceleration function,

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we have this point 2 b,
and every single point

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has the same value
of acceleration.

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So here the acceleration
is an example

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of constant acceleration.

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And this is our simplest case.

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Notice we started with
the position function.

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We differentiate to get the
component of the velocity

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and to get the component
of the acceleration.

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So this is a very simple
model for a runner

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whose increasing speed
linearly, accelerating

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at a constant rate.