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Calculate the velocity of the centre of mass at threshold

  1. Jan 1, 2013 #1
    1. The problem statement, all variables and given/known data

    A high-energy photon γ1 of energy 261 GeV collides with an infra-red photon γ2 of energy 1eV to produce an electron-positron pair via the reaction

    γ1 + γ2 → e- + e+


    3. The attempt at a solution

    Conservation of momentum:

    p1 - p2 = (0.5)pe

    But since p1 is 1011 times more than p2,

    pe ≈ (0.5)(p1) = 6.96 x 10-17 kg m s-1

    But when i try to calculate the velocity of the positron/electron using the momentum i get v = c...

    Comparing orders of magnitude

    pe ≈ 10-17

    m(γv) = (10-23) (γ)

    This implies that γ ≈ 106....

    Am I doing something wrong here?
     
  2. jcsd
  3. Jan 1, 2013 #2

    mfb

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    First, working in SI units is a bit impractical here - you get large powers of 10 everywhere.
    Second: How can you know that the reaction is at threshold? Otherwise p1 - p2 = (0.5)pe is wrong.

    You should get something really close to, but below c.

    The order of magnitude is correct.
     
  4. Jan 1, 2013 #3
    I tried to calculate it but my calculator simply gives c...
     
  5. Jan 1, 2013 #4

    mfb

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    ##\frac{v-c}{c} \approx 5 \cdot 10^{-13}##, you would need ~13 digits to see a difference.
     
  6. Jan 1, 2013 #5
    Is this question a trick question?
     
  7. Jan 1, 2013 #6

    mfb

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    No, why?
    What is the problem statement, by the way? What are you supposed to calculate?
     
  8. Jan 2, 2013 #7
    It's a 7 marks question, where i'm supposed to find the velocity of the positron/electron pair at threshold energy.
     
  9. Jan 2, 2013 #8

    mfb

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    Well, you can give the relative deviation from c as result (similar to my post), as the value in m/s is not really interesting.
     
  10. Jan 2, 2013 #9
    I see, so this is actually an approximation question?
     
  11. Jan 2, 2013 #10

    mfb

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    I don't understand your question. You can give an exact result, where is the problem?
     
  12. Jan 3, 2013 #11
    The problem is that my calculator's only able to calculate up to 8 decimal places.. so it can only give 0.99999999c anything beyond that it automatically registers c.
     
  13. Jan 3, 2013 #12

    mfb

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    You can calculate (v-c)/c with a precision of 8 decimal places then.
     
  14. Jan 4, 2013 #13
    Here's what I got; 12 decimal places. When i substituted it into my calculator it gave me a value for gamma as 7*10^5

    1zv6b5w.png
     
  15. Jan 4, 2013 #14

    mfb

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    ##\beta = \sqrt{1-10^{-12}} \approx 1-\frac{1}{2}10^{-12}##
    ##1-\beta \approx 5 \cdot 10^{-13}##
    This allows to determine (1-β) with the same relative precision as γ.
     
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