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Special Relativity derivations ...

  1. Sep 1, 2015 #1
    The problem statement, all variables and given/known data
    Using the special relativity formulae
    p = mv / [1 - (v/c)2]
    E2 = p2c2 + m2c4
    derive linear relations between:
    (i) momentum and mass;
    (ii) energy and mass;
    (iii) energy and momentum,
    which involve only c, c2, β = v/c, and γ (= 1/sqrt(1 - β2))

    The attempt at a solution

    I am pretty sure the answer to (i) is p = γmv = γmβc, although I am unsure if this counts as a linear relation.
    I suppose for (ii) I should be aiming for E = mc2, and for (iii) maybe I should be trying to get to E = pc (although I think this only applies to massless particles), but I haven't had much luck thus far.
    I know these aren't really that hard, but for some reason my brain is just drawing blanks with these.

    Many thanks for help and patience.
     
  2. jcsd
  3. Sep 1, 2015 #2

    robphy

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    I think that here "linear" means that the two variables being related are each of first-power (no squares or square roots).
    So, (i) is fine.
    It seems that you have three quantities: p,m,E (mutually related by the above).... and you wish to find the relations between pairs chosen from those three.
    So, what would you get for (ii)?
     
  4. Sep 2, 2015 #3
    Sorry, a bit confused. I am pretty sure for (ii) I am meant to get E = mc2. My main problem is getting there from the formulae the question provided. Does that make sense? ...
     
  5. Sep 2, 2015 #4

    robphy

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    From your given relations,
    E is the relativistic energy, p is the relativistic momentum, and m is the [invariant] rest mass.
    With these symbols, ##E\neq mc^2## in general.
    You can see this immediately by plugging in "what you think your E should be" into "E2 = p2c2 + m2c4".
    When (that is, Under what conditions) will E="what you think your E should be" be true?
     
    Last edited: Sep 2, 2015
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