A2.3 Velocity Transformation
The velocity transformation allows you to work out what the velocity measured in one reference frame will be in a different frames. Hence it allows us to combine velocities measured in different reference frames.
A Gedankenexperiment: suppose that Nefertiti’s friend Lucrezia Borgia is riding her bike, in the same direction as Cormorant is flying, thrilled with the excitement of pursuing a seabird:
Here, Nefertiti is stood still and all three protagonists are colinear. wN and uN are the individual velocities of Lucrezia and Cormorant in Nefertiti's frame of reference. Now let's consider the relative velocity of Lucrezia and Cormorant. First, in Nefertiti's frame of reference:
Cormorant, however, wants to know how quickly Lucrezia is approaching him in his own frame of reference. This means that we're going to have relativistic issues to deal with, because we only know the relevant velocities in Nefertiti's frame.
To do this, we can use the Lorentz transformations that we derived in the last section. These will allow us to find expressions for Lucrezia's position (xC) and time (tC) in Cormorant's frame of reference, in terms of the corresponding values in Nefertiti's reference frame. Then we can use a bit of calculus to find Lucrezia's velocity, dxC/dtC.
You should end up with this expression:
This velocity addition formula is based on the specific case of reference frames centred on Nefertiti (whose frame is the rest frame (0) because the original length and time measurements are made in this frame)and Cormorant whose reference frame is moving (1), with u being the relative velocity of Cormorant’s frame relative to Nefertiti’s. We can generalise this for any two inertial reference frames:
where u is the velocity of frame 1 relative to frame 0.
If we work through a similar derivation to find an expression for w0 in terms of w1, we get:
Now that we can convert velocity measured in one reference frame to another frame, we can deal with a couple of well known paradoxes that are, perhaps, not really paradoxes.
