Finding an in-between frame of reference

- [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.