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D/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t) [this has been posted before.I need more help]

  1. Sep 27, 2012 #1
    1. The problem statement, all variables and given/known data

    If vector r(t) is not 0, show that d/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t).

    2. Relevant equations

    The hint given was that |r(t)|^2 = r(t) dot r(t)

    3. The attempt at a solution

    the post I saw about this question said that you should take the derivatives of both sides of the hint equation using chain rule, so I got:

    2|r(t)|(1)=r'(t) dot r(t) + r(t) dot r'(t)
    which is equal to:
    |r(t)|= r'(t) dot r(t)

    aaaaand now I don't know what to do
     
    Last edited: Sep 27, 2012
  2. jcsd
  3. Sep 27, 2012 #2
    Re: d/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t) [this has been posted before.I need more h

    This step is incorrect. You have the equivalent of f(g(x)) -- the chain rule says d/dx f(g(x)) = f'(g(x)) * g'(x).
     
  4. Sep 27, 2012 #3
    Re: d/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t) [this has been posted before.I need more h

    Okay, now I'm confused... in this situation, wouldn't f'(g(x)) be equivalent to 2|r(t)| and g'(x) be equivalent to 0 because |r(t)| is a length and the derivative of a number is 0? So then it would be 2|r(t)|(0)=r'(t) dot r(t) + r(t) dot r'(t) --> 0=r'(t) dot r(t) + r(t) dot r'(t)?
    If this is right, then I still don't know where to go from here.
     
  5. Sep 27, 2012 #4
    Re: d/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t) [this has been posted before.I need more h

    No. |r(t)| has t dependence. Accordingly, we can't assume the magnitude is constant.
     
  6. Sep 27, 2012 #5
    Re: d/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t) [this has been posted before.I need more h

    Oh man...I totally get it now...i feel kinda dumb :P
    the derivative would just be:
    2|r(t)|d/dt|r(t)|= r'(t) dot r(t) + r(t) dot r'(t)
    2|r(t)|d/dt|r(t)|= 2(r'(t) dot r(t))
    |r(t)|d/dt|r(t)|= r'(t) dot r(t)
    d/dt|r(t)|=(1/|r(t)|) ( r'(t) dot r(t))

    Thanks for letting me waste your time :)
     
  7. Sep 27, 2012 #6
    Re: d/dt |r(t)| = (1/|r(t)|) r(t)dot r'(t) [this has been posted before.I need more h

    No problem. Glad to have helped. :)
     
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