WEBVTT

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Let's consider a
gravitational slingshot.

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What is that?

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Well, once in a while, people
like to send spacecrafts out

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into the solar
system, particularly

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the outer solar system, to
explore what's going on there.

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And because we can't on
Earth give enough speed

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to these little spacecrafts, we
need the big planets around us

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to help us a little.

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And so we can consider a large
planet like Jupiter or Saturn.

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Here we have Saturn.

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And if we have a
little spacecraft

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and we make it fly by close,
then what actually happens

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is it will, due to the
gravitational attraction

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of Saturn, acquire
a kick in velocity.

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And that is a
gravitational slingshot.

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So let's look at that.

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So our little spacecraft comes
in with an initial velocity.

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And Saturn, of course,
also has a velocity.

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And once it has passed,
our little spacecraft

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will have a final velocity.

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And in order to calculate what
this final velocity is going

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to be, what the increase
in speed is going to be,

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we need the concept
of relative velocity.

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And for that, we need to first
consider some initial state.

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So we have the relative
velocity initially.

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And that is the difference
between those two velocities,

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between the
spacecraft and Saturn.

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And what becomes
very important here

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is that we look at
the coordinate system.

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And keep that in mind.

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Otherwise, we're going
to get a few sign errors.

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So the initial velocity
of the spacecraft

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goes in the i hat direction.

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And the velocity--
the relative velocity

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is, of course,
the difference, so

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minus the velocity of Saturn.

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But that one goes in the
minus i hat direction.

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And then we have
the final state.

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So V final relative.

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And here we have
the final velocity

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of the spacecraft now
going in the minus i hat

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direction minus the
velocity of Saturn

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that also goes in the
minus i hat direction.

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Now, there is one thing
that we need to consider,

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which is this
velocity of Saturn.

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This one actually
is, of course, here,

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the initial velocity of Saturn,
and this one is the final one.

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But because the
mass of Saturn is

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much larger than the
mass of the spacecraft,

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we can actually set the
initial velocity of Saturn

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to the final velocity of Saturn.

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So we can turn this-- we can
take this away here again,

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and just consider one
velocity of Saturn.

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OK, good.

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There's one more thing that we
need in order to solve this,

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because we need to know how
the relative velocities are

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actually related.

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And the energy momentum
law helps us there,

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because that gives us that
the initial relative velocity

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equals minus the final
relative velocity.

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OK, so we can plug that in now.

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What do we have here for
the initial velocity?

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We have Vi minus two i's gives
us a plus the Saturn velocity.

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And that equals--
minus minus gives us a

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plus-- the final velocity
of the spacecraft.

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And then we have
three minuses here,

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so that gives us a minus
the Saturn velocity.

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And we know from the problem
that the initial velocity here

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of the spacecraft
was actually given

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at three Saturn velocities.

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So we can tally this up now.

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We have three-- we have
three plus one is four.

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And we'll put this
over on the other side.

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That gives us five.

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Five Saturn velocities
equals our final velocity.

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So that's quite a good
gain, I would say.

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And it nicely illustrates
why big planets like Jupiter

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and Saturn are really, really
helpful for the exploration

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of the solar system.

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And that is actually how the
New Horizons mission made it out

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to Pluto, all the way out there.