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Homework Help: Derive p^2/2m from relativistic equations

  1. May 7, 2016 #1
    Given the relativistic equation for energy E2 = (pc)2 + (mc2)2
    I want to find the non-relativistic approximation for kinetic energy in non-relativistic terms,
    Knr = p2/2m

    I start off with subtracting the rest energy
    from the above equation.

    So K = E - E0

    and assume that c is very large.
    I've messed around for hours on the algebra and I need help.
    I want to show that K ≈ Knr

    I am doing this using a linear approximation. I've written the energy as E = E0 √1+x

    And using the function f(x)=√1+x about x = 0

    I've derived the linearization as L(x) = 1 + x/2

    But I am struggling with relating to the equations above to show that K ≈ Knr
    Last edited: May 7, 2016
  2. jcsd
  3. May 7, 2016 #2


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    You have the right approach. Can you tell us how x is related to p when you write E = E0 √(1+x) ?
  4. May 7, 2016 #3
    That's where I get stuck, relating the two equations

    1. E = E0√(1+x) and
    2. E = √( (pc)2 + (mc2)2)

    I've arranged the relativistic equations as such
    3. K = E - E0
    3. K = √( (pc)2 + (mc2)2) - mc2

    I don't know what x represents in equation 1 and so I don't know where I go next.

    I tried a lot of algebraic manipulation of equation 2. including substituting mv for p
    In an effort to find a structural relationship, but I'm lost. I would appreciate help relating x
    to p.
    This question comes from a numerical methods section of a calculus course. I completely understand the numerical methods, but I'm struggling with the physics and how to apply a linear approximation to relate the two equations. I don't understand which part of equation 3. I am substituting with Eq. 1 and if any variables should be eliminated or rearranged.
  5. May 7, 2016 #4


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    Can you rewrite equation 2 using E0 instead of mc2? That might help you see how to identify x in equation 1.
  6. May 7, 2016 #5

    mc2 = √(E2 - (pc)2)

    sub into eq 1?

    E = √(E2 - (pc)2)⋅√(1 + x)

    I still dont see it. Please help
  7. May 7, 2016 #6


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    Your equation 2 can be written as ##E = \sqrt{(pc)^2 + E_0^2}##. Is there any way to "pull" the ##E_0## outside the square root so that it looks more like equation 1?
  8. May 7, 2016 #7

    E = E 0 √((pc)2/E02 +1)

    So x represents (pc)2/E02

    This problem has taken up my entire night and it's only worth 0.2% of my final mark haha. The problem was so interesting that I could not put it down. I will tackle the rest of the problem in the morning, might bug you again for a hint if I get stuck. Thanks so much for your help, you're awesome ! :approve:
  9. May 7, 2016 #8


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    Yes, good.
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