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Now that we know the
magnitude of the acceleration

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due to gravity at 400 kilometers
above the surface of the earth,

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where the space
station might hang out,

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what I want to do in this video
is think about how fast does

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the space station need to be
moving in order to keep missing

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the earth as it's falling, or
another way to think about it,

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in order to stay
in orbit, in order

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to maintain its circular
motion around the earth.

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So we know from our
studies of circular motion

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so far, what's keeping it going
in circular motion, assuming

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that it has a constant
speed, is some type

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of centripetal acceleration.

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And that centripetal
acceleration

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is the acceleration
due to gravity.

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And we figured out what
it was at 400 kilometers.

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And so we know that that
centripetal acceleration-- let

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me write it here
in pink-- we know

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that the magnitude of that
centripetal acceleration

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has to be equal to the speed or
the magnitude of the velocity

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squared divided by
divided r, where

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r is the radius of
the circular path.

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So in this case,
it'd be the radius

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of the orbit, which
would be the radius

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of the Earth, plus the altitude.

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So that, we already figured
out in the last video,

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is 6,771 kilometers.

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Let's just solve this
for v, and then we

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can put in the numbers
in our calculators.

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So you multiply both sides by
r, and you flip the two sides.

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You get v squared is
equal to the magnitude

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of our acceleration
times the radius.

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The magnitude of our
velocity or speed

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is equal to the square root
of our acceleration times--

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or the magnitude of
our acceleration--

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times the radius.

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And so let's get
our calculator out,

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and you can verify that
the units work out.

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This is meters per second
squared times meters, which

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gives us meters squared
per second squared.

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Take the square root
of that, and you

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get meters per second, which
is the appropriate units.

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But let's get our calculator
out and actually calculate this.

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Let me see.

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My calculator's sitting
on my other screen.

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There you go.

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And then we want to calculate
the principal square root

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of the acceleration due to
gravity at this altitude,

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the magnitude of
the acceleration

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due to gravity at that
altitude is 8.69 meters

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per second squared times the
radius of our circular path.

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That's going to be the
radius of Earth, which

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is 6,371 kilometers plus the
400 kilometers of altitude

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that we have in this scenario.

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So that gives us 6.-- and we
did this in the last video--

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6.771 times 10 to
the sixth meters.

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And it's important that
everything here is in meters.

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Our acceleration is in
meters per second squared.

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This right over
here is in meters.

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So the units don't
do anything strange.

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And then we get a drum
roll for how fast--

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and this is going to be
in meters per second.

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We already thought about
how the units will work out.

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We get 7,670-- I could say 71.

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But actually I'm
just going to stick

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to three significant digits.

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7,670 meters per second.

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So let me write that down.

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The necessary velocity
to stay in orbit

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is 7,670 meters per second.

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So let's just conceptualize
that for a second.

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Every second, it's
going over 7,000 meters.

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Or every second, it's
going over 7 kilometers.

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Every second.

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It's going at this super-- if
we assume that's the direction

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it's traveling, it's going at
this super-, superfast speed.

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And if we want to translate
that into kilometers per hour,

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you just take 7,670
meters per second.

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If you want to know
how many meters it's

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going to do in an
hour, you just say,

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well, there's 3,600
seconds per hour.

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And so if you
multiply that, that's

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how many meters it
will travel in an hour.

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But if you want
that in kilometers,

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you just divide by 1,000.

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You have 1 kilometer
for every 1,000 meters.

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Meters will cancel out.

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Seconds will cancel out.

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And you are left with
kilometers per hour.

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So let's do that.

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So that was our previous answer.

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We multiply by 3,600 and
then divide by 1,000.

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So we really could have
just multiplied by 3.6.

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And then we get 27,000, roughly
27,600 kilometers per hour.

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So this is really an
unfathomable speed.

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And you might be wondering, how
does such a big thing maintain

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that type of speed, because even
a jet plane, which is nowhere

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near this fast, has to have
these huge engines to maintain

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its speed.

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How does this thing maintain it?

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And the difference between
this and a jet plane

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is that a jet or a car or if
I throw a ball or anything

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like that-- but a jet plane.

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Let's focus on a jet
plane so we don't

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have to worry
about other things.

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A jet plane has to
travel through the air.

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It has to travel
through the air.

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And actually, it uses the air as
kind of its form of propulsion.

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It sucks in the air, and then it
spits out the air really fast.

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But it has all of
this air resistance.

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So if the engines were
to just shut down,

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all the air would
bump into the plane

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and provide essentially
friction to slow down the plane.

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What the space station or the
space shuttle or something

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in space has going for
itself is that it's

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traveling in an almost
complete vacuum.

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Not 100% complete vacuum,
but almost complete vacuum.

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So it has pretty much
no air resistance,

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negligible air resistance
to have to deal with.

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So we know from Newton's
laws, an object in motion

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tends to stay in motion.

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So once this thing
gets going, it

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doesn't have air
to slow it down.

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It'll keep staying that speed.

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In fact, if it did
not have gravity,

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which is causing this
centripetal acceleration

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right over here, it would
just go in a straight path.

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It would go in a straight
path forever and ever.

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And that brings up
an interesting point.

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Because if you are
in orbit like this,

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traveling at this very,
very, very fast speed,

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you have to make sure that
you don't vary from this speed

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too much.

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If you slow down, you will
slowly spiral into the Earth.

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And if you speed up a
lot beyond that speed,

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you will slowly spiral
away from the Earth.

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Because then the centripetal
acceleration due to gravity

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won't be enough to keep you
in a perfect circular path.

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So you really have to stay
pretty close to that speed

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right there.