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How can I show that z = v/c for small velocities

  1. Jan 21, 2007 #1
    How can I show that

    [tex]1+z=\sqrt{\frac{1+v/c}{1-v/c}}[/tex]

    becomes [tex]z \simeq v/c[/tex] for small velocities? Please give me a hint.
     
  2. jcsd
  3. Jan 21, 2007 #2

    cristo

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    Try writing the right hand side as [tex](1+v/c)^{1/2}(1-v/c)^{-1/2}[/tex]. Can you expand this?
     
  4. Jan 21, 2007 #3
    So

    [tex]1+z=\frac{\sqrt{1+\beta}}{\sqrt{1-\beta}}\simeq \left( 1+\frac{1}{2}\beta-\frac{1}{8}\beta^2+... \right) \left( 1+\frac{1}{2}\beta+\frac{3}{8}\beta^2+... \right) = 1+\beta+\frac{5}{8}\beta^2+...=1+\frac{v}{c}[/tex]

    Correct?
     
  5. Jan 21, 2007 #4

    cristo

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    Yup, that's correct.
     
  6. Jan 21, 2007 #5
    cristo scroll up, lol
     
  7. Jan 21, 2007 #6

    Gib Z

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    I would think there was an easier way...v is approaching zero, sub v=0 into the right hand side and its easy to see z is also approaching zero...

    PS: Cristo has 666 posts...ooh
     
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