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- [Voiceover] Let's say I'm
person A here in my ship

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traveling through the universe
at a constant velocity,

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so that is person A right over there.

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Let me write it a little bit bigger.

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Person A.

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And let's say that I
have a friend, Person B,

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and they are in another ship.

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And in my frame of reference,
so this is Person B,

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in my frame of reference,
they are traveling,

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so their velocity vector looks like this,

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they are traveling at 8/10
of the speed of light,

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0.8 c.

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And so once again this is all given

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in A's frame of reference
and my frame of reference.

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Now the question I have for you is

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surely there must be some third party,

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let's call them C for convenience.

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Surely there must be some third party

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or some third party frame of reference

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that we could imagine it
might be someone in some ship,

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where their velocity in
my frame of reference

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is in between being
stationary and traveling

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0.8 times the velocity of light.

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And even more, from
their frame of reference,

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A and B should be leaving
or going outward from them

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at the same velocity.

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So what am I talking about?

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So, this is...

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we're gonna call,

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I'll just call this
the frame of reference.

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So this first row is
A's frame of reference

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that we just described.

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And C is going to be going
with some velocity, V,

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away from A.

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So some velocity, V, we
haven't figured it out yet.

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Now let's think about
C's frame of reference.

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So frame of reference C, so
C and C's frame of reference

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is just going to be stationary,

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and we want to figure
out a V so that A and B

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are moving away from C
at the same velocity.

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So in this frame of
reference, A will look like,

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A will look like it's moving to the left

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with a velocity of negative,

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here it's magnitude is the same,

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but just in the other direction.

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And B will also be moving
away with velocity V.

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So B, right over here, is
going to be moving away

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with velocity V.

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So this is a really interesting question.

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Can we figure out what V is going to be?

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If we were dealing with
the Galilean world,

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you might say well V is
just going to be half way

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in between these two things.

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If we were just on the
highway in the Galilean

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or Newtonian world, and B
is going 80 miles per hour,

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and A is stationary, well
then if C goes half way,

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if C is going 40 miles per hour,
then from C's point of view

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it looks like A is going
backwards at 40 miles per hour,

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and it'll look like B is going
forward at 40 miles per hour.

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But we know by now that we don't deal,

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we aren't living in a
Newtonian or Galilean universe,

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we're living in one defined
by special relativity.

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So I encourage you to pause the video

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and figure out what this in
between frame of reference,

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what it's velocity needs
to be relative to A.

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And I'll give you a hint,
it's going to involve

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the Einstein Velocity Addition Formula.

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So let's work through this together.

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So I'm just gonna write down

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the Einstein Velocity Addition Formula.

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So it tells us that the change

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in x prime

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with respect to t prime is equal to

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u minus v

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over one minus uv

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over c squared.

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Now let's think about
how we might apply it.

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And the trick here is
to really think about it

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from C's frame of reference,

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it's to think about it from
C's frame of reference.

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So if you think about it
from C's frame of reference,

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you could say, you could
say that v right over there,

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let me put this in a different color,

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you could say that v is the velocity

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that A is moving away from C at,

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so velocity A

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moving...

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from C.

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And then you could say that u

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is the velocity that B is moving from C.

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Remember, we're dealing
in C's frame of reference.

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So velocity...

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velocity that B moving...

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from...

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from C, actually let me
make everything upper case.

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This should be upper case A,

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and this should be upper case C.

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And in that case, what is delta
x prime over delta t prime?

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Well, that would be the
velocity that B is moving

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away from A in A's frame of reference.

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I know that this can get
a little bit confusing,

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but I really want you to pause it,

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watch it in slow mo, really
think about what we're doing.

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I'm kind of starting now,
I know I started this video

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in A's frame of reference,
and this is really

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the trick of the problem
is I'm now shifting

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over to C's frame of reference.

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I'm like, okay, V is the
velocity A is moving away from C,

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B is the velocity, or u is the
velocity B is moving from C,

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and in that case we can view delta x prime

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over delta t prime as the
velocity that B is moving

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away from A in A's frame of reference.

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So B moving...

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moving from A

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in A's frame of reference.

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So in A's reference.

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And both of these are in C's reference.

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So let me write that down.

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In C's reference,

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in C's

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in C's reference.

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I really want you to think about this,

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this is a little confusing,
but hopefully this helps you

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appreciate how this Einstein
Addition, Velocity Addition

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can be valuable.

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Well now, we can substitute what we know.

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We know the velocity B
is moving away from A,

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it is 0.8 c.

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So we can write, we can write

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0.8 c

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is going to be equal to,

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is going to be equal to u,
the velocity B is moving

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from C in C's frame of reference.

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Well we say that this
is just going to be B,

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that's going to be a positive v.

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So that right over there is
going to be a positive v.

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Let me do that in,

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let me do that in the same color.

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It's going to be a positive v.

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And then from that, we are
going to subtract the velocity A

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is moving from C in
C's frame of reference.

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So A is moving from C with
the velocity negative v.

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I know it's kind of
confusing to replace a v

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with a negative v, but this
is a generalized formula,

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while this is the actually value

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that we're using in this case.

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So minus negative V all of that over

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all of that over one minus,

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all of that over one
minus the velocity of u

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times the velocity of v, well,

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or that's just going to
be, that's going to be v

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times negative v or negative v squared.

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So I'll just write that
as negative v squared

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over c squared.

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And just like that we have set it up

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so we can solve for v.

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And the key realization
is is that we said,

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"Okay, there must be some spaceship C

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"that defines a frame of reference

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"where A and B are moving away from it

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"with velocities of equal magnitude."

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And so we use that information

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to go into C's frame of reference

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and use the Einstein
Velocity Addition Formula.

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But instead of knowing what,

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instead of knowing what these are,

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and then solving for this,

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we know what this is, and we're assuming

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that these two have the same magnitude,

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and we're able to solve for v.

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And so let's do that right now.

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In fact, I will do that in the next video,

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so that we will have enough time.

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And I encourage you to solve for v

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on your own before you
watch the next video.