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Let's look at this
rocket sled here.
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It's gone on in the snow,
but it wants to stop.
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There are two little
devices mounted on the sled,
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and they can eject gas.
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And so the forward one
is used to eject gas
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to make the sled stop.
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We want to derive a relation
for the differential
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between the speed of the
sled and the differential
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of the mass of the rocket sled.
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But before we do that
with a rocket equation,
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we need to actually
consider what
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else we know about this system.
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Well, we know that the
dry mass has a mass of m0.
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The fuel mass is
given also as m0.
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And we know that, at time of t
equals 0, the speed of the sled
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is v0.
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We also know that, at a
later time, t plus delta t,
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we have the sled here whose
mass is now m of t plus delta t.
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And we have this low mass
parcel that has been ejected,
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so the gas.
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And that has the mass of delta
mf, so the mass of the fuel.
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We furthermore know that,
relative to the sled,
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this little gas parcel
is moving with a speed u.
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The rocket equation says that
my force-- my external force--
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has two terms.
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We have the mass of the
rocket times the acceleration
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of the sled minus
the differential here
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of the mass times the speed u.
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And these are
actually all vectors.
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And so this describes this
little gas parcel here,
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and this one describes
rocket and we
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know that v equals v i hat.
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i hat is going in the right--
in the direction of motion,
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and u equals u i hat.
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We also know that no external
forces apply to the sled,
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so that's actually 0.
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We can then turn this equation.
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We can apply all of this.
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We will get 0 equals mr
dvr/dt minus dmr/dtu.
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We can bring this one
on the other side,
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and we end up with the
relationship that we wanted
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to obtain-- namely,
an equation that
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contains the differentials
of the speed of the rocket,
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that sled, and the
differential of the mass.