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I'd like to now show you two
mathematical facts about how we

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integrate quantities of motion.

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Suppose we have an object, let's
call this the i hat direction,

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and it's moving, and it has
an x component of velocity.

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And suppose it starts
at some initial position

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and goes at some final position.

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And in the initial
position, it was at time ti,

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and the final position
was at t final.

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Now, we have two very
important integrals

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that we want to look at
which govern how we describe,

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how we integrate acceleration
with respect to time,

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and also how we'll
integrate acceleration

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with respect to space.

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Now, recall that our
acceleration, which

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may be a function of
time, is the derivative

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of the x component of
velocity with respect to time.

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So, let's first look
at a simple integration

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of acceleration with respect
to time and see what we get.

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So, if we integrate
acceleration,

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now I'm going to introduce an
integration variable, dt prime.

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And our integration variable
goes from our initial time

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to some final time.

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And if we use our
fact that acceleration

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is the derivative
of the velocity,

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then we can write this is
dvx, dt prime, dt prime again.

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After a while I'll drop
the dummy variables

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and the endpoints
of the integral.

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And this simply
becomes the integral

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from t prime, t initial.

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t prime equals t
final of dvx prime.

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Now notice we've done
a change of variables

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in the integration.

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So instead of now talking
about the endpoints

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of the integral from
t prime, t initial

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to t final, now what we're doing
is we changed our integration

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variable.

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And so what we have is, we
have the velocity integration

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variable is going from
some initial value.

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And that integration variable
is going to some final value.

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So again, our initial
conditions may

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have some initial velocity
and some final velocity--

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three different
ways of describing

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the initial and final states.

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This integral is a very
straightforward integral,

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the x final minus
vx initial, which

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is the change in the x
component of the velocity.

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And that's our classic
result that we've

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done that the integration
of acceleration

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with respect to time is
a change in velocity.

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Now, let's see what
happens as a comparison

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when we integrate our
acceleration not with respect

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to time, but suppose that
it's a function of space.

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And so now we're
integrating again.

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We have a dummy variable.

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We have to be
careful, because this

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is the x component
of acceleration,

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but x prime is our
integration variable,

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and that's going from
some initial position

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to some final position.

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Now, we can write this as
again, make the substitution,

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dx dt, dx prime, going from
the initial to the final.

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But now notice we're going to
rewrite this integrand as dvx

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times dx prime, dt prime, dt.

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And when we do that, we have
this result that dx prime,

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dt prime is precisely
what we mean by vx.

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Now, I'll introduce some
dummy variables there as well.

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So what our integrand becomes
is dvx prime, vx prime.

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That's not a function,
it's a product

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of a differential
times vx prime.

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And now our new
integration variable

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is going from some initial
value to some final value.

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Once again, this is a
straightforward integral.

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For those who haven't
seen integrals like this,

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it's just something like
x dx is x squared over 2,

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but our integration
variable is vx prime.

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So what we get is
1/2 vx final squared

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minus vx initial squared.

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And so what we see here is two
fundamentally different facts

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that if you integrate
acceleration

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with respect to time, you
get the change in velocity.

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But if you integrate
acceleration

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with respect to space, you
get 1/2 times the change

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not of velocity, but
of the x component

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of the velocity squared.

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And both of these facts
are central to how

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we'll analyze the
concept of work

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and how we applied
Newton's second law.