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Centre of mass (in relativity).

  1. Aug 4, 2007 #1
    consider a head on elastic collision of a bullet of rest mass M with a stationary target of rest mass m.
    prove that the post-collision gamma factor of the bullet cannot exceed (m^2+M^2)/(2mM)

    im given as a hint to prove that if P,P' are the pre and post 4 momentum of the bullet and Q,Q' are of the target, then in the CM frame (P'-Q)^2>=0
    well i understand how from the hint i get the answer, but how to arrive at this, i mean the momentum is conserved, i.e: P'+Q'=P+Q
    so by interchanging we get:
    P'-Q=P-Q'
    now in the cm, we must have: (P'-Q)^2-(P-Q')^2=(M+m)^2c^2
    but i don't quite see why this is correct, i mean
    P=(E/c,p)
    Q=(mc,0)
    P'=(E'/c,p')
    Q'=(E''/c,p'')
    where in the cm we have: p'=-p''
    so (cause the 3 momentum there is zero) (P'-Q)^2=(E'/c)^2+(mc)^2-2*E'*m
    (P-Q')^2=(E/c)^2+(E''/c)^2-2EE''/c^2
    here's where im stuck, can anyone help?
     
  2. jcsd
  3. Aug 4, 2007 #2

    Gokul43201

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    Are you asking why the square of a real number must be non-negative? :confused: Maybe I'm misunderstanding...
     
  4. Aug 4, 2007 #3
    your'e correct, i thought about it myself, and there's not a lot to prove here, but then why is the hint to use the cm system to prove that (P'-Q)^2>=0?
     
  5. Aug 4, 2007 #4

    CompuChip

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    P and P' are four-vectors, so I assume that (P - P')^2 is actually
    [itex](P - P')_0^2 - (P - P')_1^2 - (P - P')_2^2 - (P - P')_3^2[/tex]
    where the [itex]P_i[/itex] are real numbers, in which case it is not trivial at all that the "square" is positive.
     
  6. Aug 4, 2007 #5
    so how would you go around proving it?
    my exam is tomorrow... (-:
     
  7. Aug 4, 2007 #6

    Gokul43201

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    Next time, I'll actually read the question properly before I shoot my mouth... err, keyboard off!!
     
  8. Aug 5, 2007 #7

    CompuChip

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    I'm a bit confused by all the primes, I think you still need that
    (E/c)^2+(E''/c)^2-2EE''/c^2 >= 0.
    Can't you use some estimate like E > E'', which would give
    (E/c)^2+(E''/c)^2-2EE''/c^2 >= (E/c)^2+(E/c)^2-2E^2/c^2 = 0
    where the estimate is justified by a physical argument?
     
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