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## Einstein velocity addition

# Finding an in-between frame of reference

## Video transcript

- [Voiceover] Let's say I'm
person A here in my ship traveling through the universe
at a constant velocity, so that is person A right over there. Let me write it a little bit bigger. Person A. And let's say that I
have a friend, Person B, and they are in another ship. And in my frame of reference,
so this is Person B, in my frame of reference,
they are traveling, so their velocity vector looks like this, they are traveling at 8/10
of the speed of light, 0.8 c. And so once again this is all given in A's frame of reference
and my frame of reference. Now the question I have for you is surely there must be some third party, let's call them C for convenience. Surely there must be some third party or some third party frame of reference that we could imagine it
might be someone in some ship, where their velocity in
my frame of reference is in between being
stationary and traveling 0.8 times the velocity of light. And even more, from
their frame of reference, A and B should be leaving
or going outward from them at the same velocity. So what am I talking about? So, this is... we're gonna call, I'll just call this
the frame of reference. So this first row is
A's frame of reference that we just described. And C is going to be going
with some velocity, V, away from A. So some velocity, V, we
haven't figured it out yet. Now let's think about
C's frame of reference. So frame of reference C, so
C and C's frame of reference is just going to be stationary, and we want to figure
out a V so that A and B are moving away from C
at the same velocity. So in this frame of
reference, A will look like, A will look like it's moving to the left with a velocity of negative, here it's magnitude is the same, but just in the other direction. And B will also be moving
away with velocity V. So B, right over here, is
going to be moving away with velocity V. So this is a really interesting question. Can we figure out what V is going to be? If we were dealing with
the Galilean world, you might say well V is
just going to be half way in between these two things. If we were just on the
highway in the Galilean or Newtonian world, and B
is going 80 miles per hour, and A is stationary, well
then if C goes half way, if C is going 40 miles per hour,
then from C's point of view it looks like A is going
backwards at 40 miles per hour, and it'll look like B is going
forward at 40 miles per hour. But we know by now that we don't deal, we aren't living in a
Newtonian or Galilean universe, we're living in one defined
by special relativity. So I encourage you to pause the video and figure out what this in
between frame of reference, what it's velocity needs
to be relative to A. And I'll give you a hint,
it's going to involve the Einstein Velocity Addition Formula. So let's work through this together. So I'm just gonna write down the Einstein Velocity Addition Formula. So it tells us that the change in x prime with respect to t prime is equal to u minus v over one minus uv over c squared. Now let's think about
how we might apply it. And the trick here is
to really think about it from C's frame of reference, it's to think about it from
C's frame of reference. So if you think about it
from C's frame of reference, you could say, you could
say that v right over there, let me put this in a different color, you could say that v is the velocity that A is moving away from C at, so velocity A moving... from C. And then you could say that u is the velocity that B is moving from C. Remember, we're dealing
in C's frame of reference. So velocity... velocity that B moving... from... from C, actually let me
make everything upper case. This should be upper case A, and this should be upper case C. And in that case, what is delta
x prime over delta t prime? Well, that would be the
velocity that B is moving away from A in A's frame of reference. I know that this can get
a little bit confusing, but I really want you to pause it, watch it in slow mo, really
think about what we're doing. I'm kind of starting now,
I know I started this video in A's frame of reference,
and this is really the trick of the problem
is I'm now shifting over to C's frame of reference. I'm like, okay, V is the
velocity A is moving away from C, B is the velocity, or u is the
velocity B is moving from C, and in that case we can view delta x prime over delta t prime as the
velocity that B is moving away from A in A's frame of reference. So B moving... moving from A in A's frame of reference. So in A's reference. And both of these are in C's reference. So let me write that down. In C's reference, in C's in C's reference. I really want you to think about this, this is a little confusing,
but hopefully this helps you appreciate how this Einstein
Addition, Velocity Addition can be valuable. Well now, we can substitute what we know. We know the velocity B
is moving away from A, it is 0.8 c. So we can write, we can write 0.8 c is going to be equal to, is going to be equal to u,
the velocity B is moving from C in C's frame of reference. Well we say that this
is just going to be B, that's going to be a positive v. So that right over there is
going to be a positive v. Let me do that in, let me do that in the same color. It's going to be a positive v. And then from that, we are
going to subtract the velocity A is moving from C in
C's frame of reference. So A is moving from C with
the velocity negative v. I know it's kind of
confusing to replace a v with a negative v, but this
is a generalized formula, while this is the actually value that we're using in this case. So minus negative V all of that over all of that over one minus, all of that over one
minus the velocity of u times the velocity of v, well, or that's just going to
be, that's going to be v times negative v or negative v squared. So I'll just write that
as negative v squared over c squared. And just like that we have set it up so we can solve for v. And the key realization
is is that we said, "Okay, there must be some spaceship C "that defines a frame of reference "where A and B are moving away from it "with velocities of equal magnitude." And so we use that information to go into C's frame of reference and use the Einstein
Velocity Addition Formula. But instead of knowing what, instead of knowing what these are, and then solving for this, we know what this is, and we're assuming that these two have the same magnitude, and we're able to solve for v. And so let's do that right now. In fact, I will do that in the next video, so that we will have enough time. And I encourage you to solve for v on your own before you
watch the next video.