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Proof regarding composition of velocities

  1. Jun 30, 2010 #1
    1. The problem statement, all variables and given/known data

    This is a problem from D'Inverno's "Introducing Einstein's Relativity".

    If vAB is the velocity of B with respect to A, vBC is the velocity of C with respect to B, and vAC is the velocity of C with respect to A (all velocities are in relativistic units, that is, c=1), prove that if 0<vAB<1 and 0<vBC<1, then vAC<1.


    2. Relevant equations

    The problem should be resolvable with just the equation

    vAC=(vAB+vBC)/(1+vABvBC).



    3. The attempt at a solution

    I understand that there are other ways to prove this, but I want to know this particular approach. I suspect it boils down to showing that the numerator is less than the denominator. I have tried reducing the denominator and/or increasing the numerator to find a greater expression that is less than one, but so far nothing has worked. I know this is really just a mathematical proof and has little to do with conceptual relativity, but I'd still like the solution.
     
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  3. Jul 1, 2010 #2

    Dick

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    You want to show (a+b)/(1+ab)<1 if 0<a<1 and 0<b<1, right? Write it as a*(1-b)+1*b<1. a*(1-b)+1*b is a linear function of b as b goes from 0 to 1, right also? It's a weighted average of a and 1. So it must hit it's max and min at b=0 or b=1.
     
    Last edited: Jul 1, 2010
  4. Jul 3, 2010 #3
    I'm sorry, I don't really understand how those two (the original expression and the linear function) are equivalent. Could you explain a little more deeply?
     
  5. Jul 3, 2010 #4
    Since a<1, b<1, we can rewrite them as: a=1-x, b=1-y, where x>0, y>0. This will make it a lot easier :smile:
     
  6. Jul 3, 2010 #5
    Ok, that works. Thanks a lot!
     
  7. Jul 3, 2010 #6

    Dick

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    (a+b)/(1+ab)<1. Multiply both sides by (1+ab). (a+b)<1+ab. Subtract ab from both sides. a+b-ab<1. Collect terms and factor. (a-ab)+b<1, a*(1-b)+1*b<1.
     
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