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Determining the velocity function

  1. Dec 5, 2016 #1
    1. The problem statement, all variables and given/known data
    Given the ## r(t) = ae^{kt}## , ##θ(t)=kt## find the velocity function that is dependent on ##r##.
    ##v(r)=?##
    2. Relevant equations
    3. The attempt at a solution

    My attempt:
    1)##r(t) = ae^{kt}##
    2)##{\dot r(t)} = ake^{kt}##
    From the first equation:
    ##\ln {\frac{r(t)}{a}}=\ln e^{kt}##
    ##\ln {\frac{r(t)}{a}}=kt##
    ##t=\frac{\ln {\frac{r(t)}{a}}}{k}##
    Replacing the ##t## in the second equation i get:
    ##{\dot r}=akr##
    Shouldn't this be the answer? In the answers it says ##{\dot r}=\sqrt2r##?
     
    Last edited: Dec 5, 2016
  2. jcsd
  3. Dec 5, 2016 #2
    Your original equations are incorrect. ##\theta## is not a vectorl where are your unit vetors i these equations?
     
  4. Dec 5, 2016 #3
    Yeah, no vectors, just the parametric equations of motion given. So whats wrong now?
     
  5. Dec 5, 2016 #4
    If you are going to determine the velocity vector, you need to start out by expressing the position vector as ##\vec{r}=r\vec{i}_r(\theta)## and taking into account the fact that ##\vec{i}_r## is a function of ##\theta##, that ##\theta## is a function of time, and that derivative of ##\vec{i}_r## with respect to ##\theta## can be expressed in terms of ##\vec{i}_{\theta}##.
     
  6. Dec 5, 2016 #5
    So the polar coordinate,
    ##\vec r(t)=ae^{kt}\vec e_r##
    ##θ=kt##
    ##\vec v(t)=\dot r\vec e_r + r\dot θ\vec e_θ##
    ##\vec v(t)=ake^{kt}\vec e_r + ae^{kt}k\vec e_θ##
    ##v(r)=\sqrt2r##
    This should be it.
     
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