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Math Help - Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

  1. #1
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    Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

    * Could you help to integrate
    dE = (m+dm)*v*dv + (c^2)*dm


    I would like to compare the result with relativistic kinetic energy
    which as you know is calculated by equation


    E = mc^2( 1/(1-v^2/c^2)^(1/2) -1 )
    Kinetic energy - Wikipedia, the free encyclopedia


    Thank you very much for any possible help.
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  2. #2
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    Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

    I am sorry, likely my initial question was wrong formulated.
    Maybe this question would be more correct and easier:
    How to prove
    dm = m*v*dv / (c^2 - v^2)
    is equivalent to
    m = m0 / (1-(v^2 / c^2))^(1/2)
    ?

    It is easy to write a program to calculate
    dm = m*v*dv / (c^2 - v^2)
    numerically by expression
    m += m*v*dv / (c^2 - v^2)
    inside a loop.
    But I need analytical derivation.

    Thank you
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  3. #3
    A Plied Mathematician
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    Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

    It's a fairly straight-forward differential equation. I'm not sure I agree with your results. Here's WolframAlpha's solution (this DE is a first-order separable ODE, with decently straight-forward integrals).
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  4. #4
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    Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

    Quote Originally Posted by Ackbeet View Post
    It's a fairly straight-forward differential equation. I'm not sure I agree with your results. Here's WolframAlpha's solution (this DE is a first-order separable ODE, with decently straight-forward integrals).
    Thank you very much. It looks it works well.
    Now I can write better equation for kinetic energy:


    dE = ( m*c^2*v / (c^2 v^2) ) dv = ( (m0/(1-v^2/c^2))*c^2*v / (c^2 v^2) ) dv


    I think this is equal to relativistic kinetic energy
    E = m0*c^2( 1/(1-v^2/c^2)^(1/2) -1 )
    I was mentioned by my first post.


    By using WolframAlpha I have got :
    E = m0*c^2 / (1-v^2 / c^2)^(1/2) + constant.


    http://www.wolframalpha.com/input/?i=int+m_0*c^2*v%2F%28%281-v^2%2Fc^2%29^%281%2F2%29*%28c^2+-+v^2%29%29+dv


    Maybe you have some hint how to prove this constant equals to
    -m0*c^2 ?

    ----------------------------------------------------------------------------
    Thank you, I have solved it by myself. You may close the tread.
    Last edited by NewR; October 4th 2011 at 04:16 AM.
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  5. #5
    A Plied Mathematician
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    Re: Could you help to integrate dE = (m+dm)*v*dv + c^2*dm

    Glad you got your solution.

    We don't generally close threads around here unless there's something wrong with them. I have marked your thread as "SOLVED", though.

    Cheers.
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