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We are now going to derive
the work-energy principle.

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We're going to do so by
using Newton's second law,

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but we're not going to consider
it with respect to time

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as what we've previously
done, but we're now

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going to consider it
with respect to space.

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So F equals ma, and
we're going to take

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the x component of that.

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And now, we're going
to integrate this.

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So Fx dx prime, and it
runs from x prime equals

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x initial to some final value.

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And then over here, we're going
to pull the m out. m integral,

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and it also goes from some
initial to some final value.

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And we're going to have ax dx.

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We have previously shown
that this integral here

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is one-half v final squared
minus v initial squared.

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And if we're going
to plug this in here,

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we're going to get
that this expression is

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one-half mv final squared
minus v initial squared.

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And you should
recognize this term.

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This is the change
in kinetic energy.

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And what we have here,
on the other side,

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this integral fx
dx is defined to be

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the work-- work done
on the system, which

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we're going to call w.

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And that work done
on the system,

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it's change in kinetic energy.

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So this is the
work-energy principle.