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Current time:0:00Total duration:7:36

Einstein velocity addition formula derivation

Video transcript

let's say this is me and I am floating in space my coordinate system my frame of reference we've seen it before we'll call it the s frame of reference any space in any point in space-time we give it an X and y coordinates and let's say that we have my friend and we've we've involved her in many other videos before she is traveling if with a relative velocity to me of V so V right over there her frame of reference we call it the S prime frame of reference and you can denote any event in space-time by the coordinates X Prime and X Prime and see T Prime now you could also do it as T prime if you like what we've been doing in a CT prime so that we have similar units and let's say that we have a third character now this is going to get interesting let's say this third character is traveling with a velocity u in my frame of reference so U is equal to change in x over change in time so if we know all of this information let's see if we can come up with can formulate away if we know what the change in X and change in time is and we know what V is if we can figure out what is this velocity going to be in the S prime frame of reference and this purple friend's frame of reference so what we want to figure out is what is let me do it in that purple color we want to figure out what is going to be the change in X prime over change in T prime if we figure this out then we know what will this velocity look like in the S prime frame of reference well let's just go back to the Lorentz transformations let's first think about what our let's first think about what change in X prime is going to be well our change in X prime is going to be the Lorentz factor times our change in X change this in white times our change in X minus beta times change in CT and let me close those parentheses and then we want to divide that by our change in time we want to divide it by our change in time so change in time or change in T Prime I should say well let me just go back to the Lorentz transformation let me write it the way that I'm used to writing it I'm used to writing it as I'm used to writing it as change in CT prime which is the same thing as C times the change in T prime is equal to is equal to the Lorentz factor times times this is going to be C change in CT or I could write it as C change in T minus beta minus beta times change in x times change in X and once again I could have written this as change in CT or I could write this as C times change in T because the C isn't changing so we know that already so if we wanted to solve for just change in T prime we could just divide both sides by C so let's do that if we divide all of this if we divide all of this by C what do we get and this is a form we've seen this in other videos that you might recognize and you might see this in some textbooks our change in T prime is going to be equal to gamma times well C times delta T divided by C is just going to be delta T delta T and then if you take so minus beta divided by C well let's just remind ourselves that beta is equal to V over C so beta divided by C is V over C squared so I could write this as minus V over C squared change in X change in X so change in T Prime let's write that now that's going to be equal to you have your gamma your gamma times change in time in my coordinate system not the t-prime just change in T change in T minus V over C squared change in X so immediately there's at least one simplification we can do we can divide both the numerator and the denominator by gamma keep wanting to change so we can divide the numerator and denominator by gamma that simplifies it a little bit and so we can write this as our change in X prime over change in T prime is equal to in the numerator we have change in X and actually let me just write out what beta is so beta actually be write it over well I'll do it over here minus V over C that's what beta is and actually let me take the C out if change in CT is the same thing as C times change in T so times C delta T well V divided by C times C is just going to be V so let me write it that way so this part these cancel out this is the same thing as change in X minus V change in T so let me write it just that way simplify so minus V change in T that's our numerator and then our denominator our denominator is change in t minus V over C squared change in X so it looks like we're getting close here but we we don't have the change in X change in T we have how much change in how much of a change in X we have forgiven change in T so what we could do is we can divide the numerator and the denominator by delta T so that we could multiply the numerator by 1 over delta T and the denominator by 1 over delta T which is equivalent to dividing the numerator in the denominator each by delta T and why am i doing that well if I do that this first term right over here is going to be Delta x over delta T which we know and then we have minus V minus V because delta T divided by delta T is just going to be 1 and then our denominator delta t divided by delta t is just 1 and then we're going to have minus V minus V over C squared and then times Delta X divided by delta T Delta X / delta T so this is cool we've just been able to figure out what our velocity in the primed coordinate system and the s prime frame of reference is in terms of the relative velocity between the s prime frame of reference and my frame of reference Delta X delta T the velocity in my frame of reference and well we know all of this other stuff - this is just Delta X delta T and this is V or we could even write it like this we said Delta x over delta T is going to be equal to u so it's equal to u minus V over 1 minus Delta X over Delta T's just to you so this is just u here so it's U times V U times V or V times u all of that over C squared so this is a really really really useful derivation which we're going to apply some numbers to in the in the next video so we can appreciate kind of how fun this is on some level