How Do You Solve for B in the Equation y = x(1 - B)?

[ck99]

Hi folks, I am working through a section on the classical limit of the relativistic doppler shift, but I am stuck on some algebra, I think it should be really basic but I just can't get it to work! My notes go from

w = w'(1 - B)

to

(w' - w)/w = B

The context doesn't matter here really, I am just trying to work out the algebraic steps to go from

y = x(1 - B)

to

( x - y) / y = B

I have spent over an hour going around in circles and it seems impossible, have I copied something down wrong or can it be done?

 

[FalseVaccum89]

I'm almost certain your derivation is completely messed up. It would help if you posted an example of your logical progression from the first to the second equation.

For instance, the primed W is a multiplied variable on the RHS, but a subtracted one on the LHS. Then you've got the constant "1" that magically disappears. . . :what:

 

[Norwegian]

My notes go from

w = w'(1 - B)
to
(w' - w)/w = B

This becomes correct if you replace the last w with w'

 

[ck99]

That's what I did first of all, but the classical doppler shift equation is Δw/w = B and not Δw/w' = B.

The starting point comes from the Lorentz transform between frames of a photon emitted parallel to the particles motion, in the non-relativistic limit (ie using a Mclaurin series to get the (1-B) factor where B = v/c).