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## Physics library

### Course: Physics library > Unit 16

Lesson 4: Einstein velocity addition# Finding an in-between frame of reference

Let's use Einstein velocity addition to find a frame of reference where A and B are traveling in opposite directions at the same speed.

## Want to join the conversation?

- At about 5: 20, why is Δx'/Δt' equal to B's velocity in A's frame of reference?(8 votes)
- 2 yrs late lol but been thinking about this today and this is how I see it: Δx'/Δt' represents u' when in C's frame which is exactly the same thing as u in A's frame. u in A's frame is 0.8c so when taking C as stationary, u' is whatever u was in A's frame i.e. 0.8c. Just like how by the end of the calculation in the next lesson, u in C's frame was what u' wouldve been in A's frame as A will see C flying away at 0.5c.

Hope I've explained that correctly and well enough to understand, if not please feel free to correct me.(1 vote)

- At3:58, why does A's velocity have to be V in the formula? I know that if we place it as U, we would get -2V on the numerator vs. getting 2V, so it is different. But how do you choose which velocity goes where properly?(2 votes)
- I believe, if I understand you correctly, that you are asking if there would be any change, or if this proof would be made different if the axis used to define the motion of the objects were changed. To that, the answer is actually no, for any inertial reference frame, and any set of coordinates in which you define the motion of the bodies handled in this video, you will consistently get the same result for the dilation that occurs between the two.(1 vote)

- Can anyone summarize for me what frame of reference is? I'm asking this because I'm preparing for an assessment but couldn't see the summary anywhere.(1 vote)
- A frame of reference is essentially a viewpoint.(3 votes)

- Is there a way to calculate relative velocity vectors in two or three dimensions?(1 vote)
- Sure, it just requires the use of a 2 or 3 dimensional velocity vector where you define the the velocity in each axes as the difference in velocity in that direction.(2 votes)

- At2:19why the velocities of both A and C have same magnitude?

Shouldn't we use Einstein's velocity addition formula to determine velocity of B in C's frame?(1 vote)- He's finding a velocity V in reference frame A that will lead C to see A and B moving away with the same magnitude.(2 votes)

- What do you do if one or more of the subjects are accelerating? I know special relativity doesn't apply in this case, correct?(1 vote)
- This version of SR doesn't apply. You could use General Relativity. There's also a sort of "enhanced" version of special relativity that can deal with accelerations.(1 vote)

- From C frame of reference, why do both A and B have the same speed?(1 vote)
- I don't get why B's velocity in C's frame of reference is +v? Shouldn't it be (v-0.8c)? Why are the magnitudes of velocity of A and B equal in C's frame of reference?(1 vote)
- At5:24, the Right Side of the equation gives the relative velocity of A and B in C's frame of reference. The left side of the equation gives the relative velocity of A and B in A's frame of reference. Are they equal as used in that equation?(1 vote)
- They have to be equal, because velocity is relative. There's no "correct" answer about who's moving toward whom, so the answer can't depend on which perspective you take.(1 vote)

- Could delta_X_(prime) / delta_t_(prime) have been A moving from B in B's Reference? Essentially, could we have interchanged between those two reference frames A and B?(0 votes)

## 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.