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For a rocket and
an external force,

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we had the rocket
equation, which

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we wrote as mass of the
rocket-- now remember,

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this is a function of
time-- times the derivative

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of the rocket dr dt.

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And we also had this
second term that

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came from the ejecting fuel.

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Now from our mass
conservation equation,

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we can rewrite this equation
as a differential relationship

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that the change of mass
of the fuel in time

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is equal to minus the
change of the rocket.

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It doesn't matter which side
we put the minus sign on.

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So now I want to interpret this
equation in a different way,

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and it will come back to
what we mean by our system.

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Let's first off bring this
other term over to this side.

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So we have plus dMr dt u.

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And that's equal to Mr dVr dt.

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Now separately, let's make this
substitution again and go back

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to our fuel term.

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So that's minus dMf
dt u equals Mr dVr dt.

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Now notice that over
here we have mass times

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the acceleration of the rocket.

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So if we just rethought our
system as simply the rocket--

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Mr-- then we have two
forces acting on the rocket.

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The external force might
be the gravitational field,

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but we have a new
term here which we're

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going to refer to as thrust.

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And so our thrust can be
thought of as an external force

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simply on the
rocket as a system.

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And that's equal to
minus the f dt u.

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Now again, if I chose
j-hat up-- let's

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just look at this
in components--

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then our thrust
as a vector, we'll

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write it as a y-component,
is equal to minus dMfuel dt.

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Now what is that relative speed?

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Well, the fuel is being
ejected backwards,

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so that's minus mu j-hat.

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The external force, by the
way, would be minus mg j-hat

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if it's near the
surface of the Earth.

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So we have another
minus u j-hat.

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And we see that we
have a positive thrust

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force in the vertical
direction that

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is giving us an additional force
other than the gravitational

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field.

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And in Cartesian-- in
our unit vectors here,

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we have minus mg plus dMfuel
dt times u is equal to Mr dV

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ydt of the rocket.

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And so this is the rocket
equation in components.