1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
    E0=mc2
    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

    TSny

    User Avatar
    Homework Helper
    Gold Member

    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

    TSny

    User Avatar
    Homework Helper
    Gold Member

    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
    Yes,

    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

    TSny

    User Avatar
    Homework Helper
    Gold Member

    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
    Yes!

    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

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Yes, good.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted