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Length contraction numerical problem

  1. Jul 11, 2017 #1
    1. The problem statement, all variables and given/known data
    upload_2017-7-11_10-41-45.png

    2. Relevant equations
    1) k.e. = (1/2)mv2, non- relatvistic

    2)K.E. = m0 c2 (ϒ -1), relativistic

    3) L=L0/ϒ , Length contraction

    3. The attempt at a solution
    (A) Taking 1019 eV to be the kinetic energy of the proton,
    Non- relativistic calculation
    (1/2)mv2 = 1019 eV = 1.6

    v = √[ (3.2/1.67)* 1027 ] = 4.3 * 1013 m/s

    How is kinetic energy defined relativistically?

    K.E. = m0 c2 (ϒ -1)

    We can't take 1019 eV as total energy of the proton as if we take so then we won't be able to calculate speed of the proton.

    So, for calculating speed of the proton, I have to take 1019 eV as the kinetic energy of proton.

    So,
    ϒ2 = [ {1019 eV/ m0 c2}+1]2

    m0 c2 = 939 MeV

    Solving it gives,
    1 - v2/c2 = [939 * 10-13]2

    v ≈ c

    Hence, w.r.t. galaxy frame the proton will take 105 years to cross the galaxy.



    B)

    W.r.t. the proton frame,
    the length of the galaxy is shortened to 105 [939 * 10-13] light years.


    Hence, the time taken = 105 [939 * 10-13] years


    Is this correct?
     
  2. jcsd
  3. Jul 11, 2017 #2
    Hi Pushoam:

    Your work looks OK to me. However, I suggest converting the (B) answer to seconds.

    Regards,
    Buzz
     
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