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Velocity of a particle GIVEN position vector!

  1. May 18, 2009 #1
    A [article follows the path given by the position vector
    r= (4t , 3 , t^3)

    what's its' speed at t=1???

    no idea how to solve this
    i know velocity is the derivative of the position with respect to time

    do i just solve, 4, 0, 3t^2, then stick it in?
    4 + 3 = 7m/s??
     
  2. jcsd
  3. May 18, 2009 #2
    Velocity is a vector quantity. The particle's velocity at time t is v = (4,0,3t^2). At time t=1, the velocity of the particle is v = (4,0,3). It's speed at time t=1 is [tex]\sqrt{4^{2}+0^{2}+3^{2}}[/tex] = 5.
     
  4. May 18, 2009 #3

    oh so it's just the magnitude of the differentiated vector?

    this was the original equation
    r= (4t , 3 , t^3)

    so i'd differnetiate it,

    (4 , 0 , 3t^2)
    and find the |r| or the magnitude of it?
     
  5. May 18, 2009 #4
    EDIT: You want the magnitude of v, not r.
     
  6. May 18, 2009 #5
    I find it strange to differentiate the position vector

    what happened to the square on the t^3, 3t^2, it became 4,0,3?
    what about the t^2? or is the ^2 part of the t, and we forget the t's?
     
  7. May 18, 2009 #6
    You wanted the speed at time t=1.
     
  8. May 18, 2009 #7
    You're given the particle's postion as a function of time. If you differentiate, then you know the rate and direction at which the particle's position is changing.
     
  9. May 18, 2009 #8
    so how do i make r= (4t , 3 , t^3) become 4,0,3?
    4*1t , 3*0t, 1*1^3?

    i thought differentiating r= (4t , 3 , t^3) 4t -> 4 ,3 -> 0, t^3 -> 3t^2
    then substituting in 1 for t, you get the velocity vector, 4,0,3?
    then i'd just get |(4,0,3)|
     
  10. May 18, 2009 #9

    Mark44

    Staff: Mentor

    v = dr/dt = (4, 0, 3t2)

    speed = |v| = [itex]\sqrt{4^2 + 0^2 + 9t^4 }[/itex]
    The speed at t = 1 is the value of the radical above at t=1.
     
  11. May 18, 2009 #10
    (4,0,3) is a vector with magnitude 5.
     
  12. May 18, 2009 #11
    yeah thanks i know that, but how did you get (4,0,3) from r= (4t , 3 , t^3)?
     
  13. May 18, 2009 #12

    OH RIGHT,
    thanks Random variable, XD XD XD :rofl:

    Just to confirm, i differentiate r= (4t , 3 , t^3)?????
     
  14. May 21, 2009 #13

    Mark44

    Staff: Mentor

    Hasn't this been answered in post 9?
     
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