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We'd like to analyze a
physics problem called

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the rocket problem, in which
we have a continuous mass

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transfer.

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Let's see what we mean by that.

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Let's consider a
rocket launching.

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So here's a classic example of
a rocket that's launching off

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in a gravitational field.

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As a rocket starts at rest,
it starts to burn fuel,

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and ejects the fuel backwards.

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And as the ejected
fuel backwards

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is producing a thrust on the
rocket-- and you can see,

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as the rocket starts
to rise-- it's

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starting to accelerate upwards
as fuel is ejected backwards.

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Now, this process goes on
for some time interval.

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And a lot of fuel
is ejected backwards

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during that time interval.

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In this particular case, we
could analyze the situation

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from the ground
frame of the rocket--

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from the ground
frame of the earth--

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and the fuel is
ejected backwards

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relative to the rocket,
rocket moves up forward.

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So how do we analyze
this situation?

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If we just looked at the rocket
when the rocket initially

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launched, the mass of the rocket
is the dry mass of the rocket

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plus all the fuel in there.

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It's at rest.

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In a state much later on,
the mass of the rocket

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has decreased and the
speed has increased.

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But we can't just compare this
initial to this final state,

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because throughout
this process, there

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was a continuous mass transfer.

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Fuel was ejected backwards.

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So how do we analyze these types
of continuous mass transfer

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problems?

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And the key is, again,
identifying states.

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So, for a continuous
mass transfer problem,

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we always have to choose
a reference frame.

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So here, in our example,
we were choosing

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the ground frame of the earth.

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And then, we want to
identify two states.

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We don't want to identify the
initial and the final states.

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We want to pick two arbitrary
states that are separated

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by a small time interval.

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So we can take a
state at time t.

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And then our second state occurs
at a small time afterwards,

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time t plus delta t.

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And what we want to do is now
represent the state changes

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by momentum diagrams.

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And that will lead us to a
differential equation, which

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is called the rocket equation.

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And that's how we'll analyze
all continuous mass transfer

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problems.

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In applying the
momentum principle,

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our first thing is just to
choose a reference frame.

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So in our example,
we're going to choose

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the ground frame of the
earth as a reference frame.

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Now, the next step is the
key to these processes.

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Instead of choosing some initial
state where the rocket is

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at rest and some final
state much later on,

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we want to choose
two states that

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are separated by a small
time interval, delta t.

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So our first state, at time t.

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And our second state, we
choose at time t plus delta t.

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Now, the next step
is that we have

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to identify what is our system.

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And so in this
case, our system is

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going to be the mass of
the rocket at time t.

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That mass is the dry
mass of the rocket

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plus all the fuel that's
still in the rocket at time t.

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And this mass will stay the
same between times t and t

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plus delta t, even
though some of the mass

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has been ejected downward.

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So our next step is to look
at that mass conservation

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of the system.