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Taylor approximation of the Doppler Law for slow-moving emitters

  1. Apr 13, 2012 #1
    The relationship linking the emitted frequency Fe and the received frequency Fr is the Doppler Law:

    [tex]F_r = \sqrt \frac{1-\frac{v}{c}}{1-\frac{v}{c}} F_e[/tex]

    The Taylor series for the function [tex]\sqrt\frac{1+x}{1-x}[/tex] near x = 0 is [tex]1+x+\frac{x^2}{2}+\frac{x^3}{3}+...[/tex]

    On Earth, most objects travel with v much smaller than c. That is, the ratio v/c is very small. Use this fact to obtain the approximation to the Doppler Law for most objects on Earth:

    [tex]F_r ≈ (1-\frac{v}{c})F_e[/tex]

    I'm not really sure where to go with this. I know it will involve using the Taylor series 1 + x + x2/2 + ... probably only using the 1 + x part, but I'm not sure what to do from here.
     
  2. jcsd
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