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Rate of change of the magnitude of the displacement

  1. Jun 18, 2007 #1
    If the displacement was given by [tex] \overrightarrow x (t) [/tex]
    ( i.e vector valued function ) .

    Then the velocity is [tex] \overrightarrow v = \frac {d \overrightarrow x}{dt} [/tex]

    The speed is the magnitude of v .


    But ..

    What is the derivative of the magintude of the displacement [tex] \frac {d \|\overrightarrow x\|}{dt} [/tex] ?




    To Clerify my question more ..

    Suppose on the xy - plane , there are two point started moving from the origin , the first one in the y - axis direction and its displacement at time t is y =t
    the other point moves in the x - axis direction , and its displacement is [tex]x=t^2[/tex]

    If we want to find the rate of change of the distnace between them as a function of time .. there are 2 approaches ..

    The First : :

    We can say that .. the distnace between them is :

    [tex] r = \sqrt { x^2 + y^2 } = \sqrt { t^2 + t^4 } [/tex]

    Thus simply we differentiate r with respect to t ::

    [tex]\frac {dr}{dt} = \frac { 2t^3 + t } { \sqrt { 1 + t^2 }}[/tex]


    The second ::

    Consider .. the vector [tex]\overrightarrow x = t^2 \mathbf i[/tex] and [tex]\overrightarrow y = t \mathbf j[/tex] ..
    Thus , [tex]\overrightarrow r = \overrightarrow x - \overrightarrow y = t^2 \mathbf i - t \mathbf j[/tex]
    The velocity is
    [tex]\frac { d \overrightarrow r } {dt} = 2t \mathbf i - \mathbf j[/tex]

    Thus the rate of change of the distance between them is
    [tex]\left \| \frac { d \overrightarrow r } {dt} \right \| = \sqrt { 4t^2 + 1 }[/tex]

    -------------------------------
    Notice the first one is :

    [tex]\frac {d \| \overrightarrow r \|}{dt} [/tex]

    AND the second is


    [tex]\left \| \frac { d \overrightarrow r } {dt} \right \| [/tex]


    WHICH ONE IS THE RIGHT ANWER ?
     
    Last edited: Jun 18, 2007
  2. jcsd
  3. Jun 18, 2007 #2

    Dick

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    d|r|/dt, in the case where r the distance from a point to a distinguished origin, is called a 'radial velocity' and is a component (in polar coordinates) of the velocity dr/dt.
     
  4. Jun 18, 2007 #3

    Dick

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    If you want to find the 'rate of change of the distance between them' use the first approach. E.g. if a point is circling the origin |dr/dt| is nonzero, yet the distance is fixed.
     
  5. Jun 18, 2007 #4
    And that is what I thought about ..

    Thanks :smile: ,
     
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