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Current time:0:00Total duration:8:12

Finding an in-between frame of reference

Video transcript

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 zoo person a and let's say that I have a friend person B and they are on 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 eight tenths of the speed of light 0.8 C and so once again this is all given an ace frame of reference in my frame of reference now the question I have for you is surely there must be some third party let's call them let's call them C for convenience surely there must be some third party or some third party frame of reference and 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 going to just call them alright the frame of reference so this first row is a frame of reference that we just described and C is going to be going with some velocity V away 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 a 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 with a velocity of negative V or its magnitude is the same but just in the other direction and B will also be moving away with velocity V so be 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 BV is just going to be halfway in between these two things if we were just on the highway in a Galilean or Newtonian world and B is going 80 miles per hour and a is stationary well then a C goes halfway if C is going 40 miles per hour then from cease 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 where we're living in one defined by special relativity so I encourage you to pause the video and figure out what this 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 going to write down the Einstein velocity addition formula so it tells us that the change in X rx chicks your change in X Prime with respect to T prime is equal to u minus V over 1 minus u V over C squared now let's think about how we might apply it and the trick here is to really think about it from from C's frame of reference is 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 just 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 or dealing in C's frame of reference so velocity velocity that be moving from from C and actually let me make everything uppercase this should be uppercase a and this should be uppercase e 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 1/8 frame of reference I know this can get a little confusing but I really want you to pause it watch it in slow-mo really think about what we're doing from this I'm kind of starting now I know I started this video in ace 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 Hugh 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 and A's frame of reference so be moving moving from a an ace frame of reference so in a reference and both of these are in in C's reference so let me write that down in C's reference in C's in CS and C's reference really want you to think about this this is a little confusing but hopefully this helps you appreciate how this I sine addition of a velocity addition can be valuable well now we can we can put it 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 and C's frame of reference well we say that's just going to be V 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 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 and C's frame of reference so a is moving from C with the velocity negative V and it was kind of substance it's confusing to replace a V with a negative V but this is the generalized formula well this is the actual value that we're using in this case so minus negative V all of that over all of that over 1 minus that over 1 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 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 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 have enough time and I encourage you to solve for V on your own before you watch the next video