Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I The two equivalent parallel velocity vectors

  1. Aug 17, 2016 #1
    This is an exercise from the textbook Apostol Vol 1 (page 525, second edition), and I do not know how to prove it:
    Suppose a curve C is described by two equivalent functions X and Y, where Y(t) = X[u(t)].
    Prove that at each point of C the velocity vectors associated with X and Y are parallel, but
    that the corresponding acceleration vectors need not be parallel.
    I would really appreaciate some enlightenment.
     
    Last edited: Aug 17, 2016
  2. jcsd
  3. Aug 17, 2016 #2

    fresh_42

    Staff: Mentor

    How are velocity and acceleration vectors defined at a certain point, i.e. how would you compute them?
     
  4. Aug 17, 2016 #3
    Here's what I tried:
    Let r(t) be the position vector and T(t) such that:
    T(t) = r'(t)/||r'(t)||
    Now, let w(t) be the speed function and v(t) the velocity vector such that:
    v(t) = (w(t)*T(t))
    Deriving:
    v'(t) = w'(t)T(t) + w(t)T'(t) = a(t)
    So acceleration is written by the sum of the two components above. if the velocity of Y(t) = X[u(t)], then we can define Y'(t) = X'(u(t))u'(t) = w(t). Substituting this relation in the acceleration equation above, and we get that their acceleration can be defined as:
    [X''(u(t))u'(t) + X'(u(t))u''(t)]T(t) + X'(u(t))u'(t)T'(t) = a(t)
    Y''(t)T(t) + Y'(t)T'(t) = a(t)
    ... How should I've written?
     
  5. Aug 17, 2016 #4

    fresh_42

    Staff: Mentor

    This appears to be a little over complicated to me. The more since you have to deal with the crucial part anyway.
    The little given by the text, I assume that ##X## and ##Y## are parametrizations of ##C##. That are functions ##X\, ,\, Y : \mathbb{R} \rightarrow C##. Wouldn't velocity simply be the change of position related to the change in time at a certain point in time?
    But this sounds to me like ## \frac{d}{dt} X(t) |_{t=t_0}## and likewise for ##X(u(t))##. The acceleration then is the second derivative.
    Note that in the text "at each point" is given which means the differentials have to be evaluated at this point.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: The two equivalent parallel velocity vectors
Loading...