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Relativistic version of newton's second law with parallel force

  1. Oct 5, 2009 #1
    1. The problem statement, all variables and given/known data
    "Given F=dp/dt. If the force is parallel to velocity show that F=γ3ma."


    2. Relevant equations
    F=dp/dt and p=γmv


    3. The attempt at a solution
    Since the force is the first derivative of the momentum with respect to time, and γ and v both vary with time since there is a net force causing both quantities to change, take the derivative of p using multiplication rule: dp/dt = m[γ'v+γa] γ'=γ3*v/c2 so this would change dp/dt to dp/dt = m[γ3(v/c)2+γa] After this I can't think of anywhere to go that could simplify this to the desired equation F=γ3ma.
     
  2. jcsd
  3. Oct 5, 2009 #2
    are you sure about [tex]\gamma[/tex] and its derivative? there must be something wrong about it
     
  4. Oct 5, 2009 #3
    Yeah, I did the derivative wrong, I forgot to do chain rule for the v2 so the derivative of γ would be γ3*va/c2

    So then dp/dt = γma[γ2(v/c)2+1]. Assuming the math I did was right the only way I could get what I'm looking for is if (v/c)2 is equal to 1-1/γ2 so when you distribute you'd get γ2-1+1, then it would give dp/dt = F = γ3ma what I'm looking for.

    I'll have to mess around with this.

    [edit]
    Coincidentally (v/c)2 does equal what it needs to equal to simplify to what the equation needs to be.

    Thank you a lot Pull :)
    [/edit]
     
    Last edited: Oct 5, 2009
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