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Math Help - Proving a particle is traveling at constant speed

  1. #1
    iva
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    Proving a particle is traveling at constant speed

    Hi there,

    I have a particle traveling on a curve defined by (x,y) = r(t) where r(t) = (cos t, sin t, t)

    I know this is a helix with the image being a circle, and I need to prove that the particle travels at a constant speed. I know grad r(t) will give me the tangent vector at a point ie (-sin t, cos t, 1). I thought that since grad gives rate of change, then a magnitude of 0 would mean constant speed. So trying this:

    grad (cos t, sin t, t) = (-sin t, cos t, 1) and || (-sin t, cos t, 1)|| = root(2)..

    meaning its not a constant speed right? But i know i'm wrong because my problem asks me to prove the speed is constant. CAn anyone advise please?

    Thanks!
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  2. #2
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    Isn't \displaystyle \sqrt{2} a constant?
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  3. #3
    iva
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    Ok, but doesn't that represent a rate of change, so shouldn't that value be 0 to indicate that its constant? What kind of value would then be non-constant speed?

    Thanks
    Last edited by iva; March 6th 2011 at 03:54 AM.
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    Quote Originally Posted by iva View Post
    Ok, but doesn't that represent a rate of change, so shouldn't that value be 0 to indicate that its constant? What kind of value would then be non-constant speed?

    Thanks
    If you get a speed of 0 then that means the particle is not moving...

    To have a non-constant speed, then you would have speed as a function of \displaystyle t.
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    Hello, iva!

    A particle travels on a curve defined by: (x,y) = r(t), where: . r(t) \:=\: (\cos t, \sin t, t)

    I know this is a helix with the image being a circle,
    and I need to prove that the particle travels at a constant speed.

    I know. r'(t) gives me the tangent vector at a point, i.e. (\text{-}\sin t, \cos t, 1).

    I thought that since r'(t) gives rate of change,
    then a magnitude of 0 would mean constant speed. . no!

    \,r'(t) is the velocity of the particle.
    . . It provides the direction and the speed of the particle.

    The speed is the magnitude of the velocity: . s \:=\:|r'(t)|
    . . Hence: . s \;=\;\sqrt{(\text{-}\sin t)^2 + (\cos t)^2 + 1^2} \;=\;\sqrt{2}

    At any time \,t, the speed is \sqrt{2} units/second.
    . . The speed is constant.

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