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Solve relativistic velocity in terms of momentum (vector equation)

  1. Dec 3, 2012 #1
    Given the formula [itex]\vec{p}=\frac{m_0}{\sqrt{1-\frac{|\vec{v}|^2}{c^2}}}\vec{v}[/itex], I'd like to make [itex]\vec{v}[/itex] the subject, so I can do a numerical approximation for some relativistic motion problem. I want to treat it as a vector equation, but since it is non-linear, the only way I can think of is to split it into 3 equations with [itex]|\vec{v}|^2=v_x^2+v_y^2+v_z^2[/itex]. This is however very complicated, though Mathematica gave me the answer analogous to the equation with only magnitudes of the vectors. Is there a simple way I can solve such kind of vector equations?

    Edited 3 Dec:
    Solution: [itex]\vec{v}=\frac{\vec{p}}{\sqrt{m_0^2+\frac{|\vec{p}|^2}{c^2}}}[/itex]
    Last edited: Dec 3, 2012
  2. jcsd
  3. Dec 5, 2012 #2

    Meir Achuz

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    1. Square the equation to get p^2=f(v^2).
    2. Solve this for v^2=g(p^2).
    3. Put this nto the square root to replace the v^2 by p^2.
    4. Rewrite the original equation with the new square root.
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