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Math Help - Vectors, path of a particle

  1. #1
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    Angry Vectors, path of a particle

    vector r of a moving particle has the equation with respect to the origin of:

    r(t) = (r[0]/t^2[0])(t^2i - 2tt[0]j + (t^2 + 2tt[0])k) where r[0] and t[0] are constants

    show that the particle goes through the point P(4,-4,8)r[0]. at what time does it do this?

    what I did is i subbed (4,-4,8) into the equation but it didnt make anything clear to me, anyone any advice on how to tackle this problem?

    Anyone know how id go about this, i need to find this part to complete parts b) and c) of which i know how to do which is the frustrating part!

    i got the following:

    r(4,-4,8)= (r[0]/t[0])*(16i/t[0] + 8j + (64/t[0] +16)k)?
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  2. #2
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    let's use a = r_0 and b = t_0 as the given constants to minimize the confusion of the notation.

    if this is r(t) ...

    \displaystyle r(t) = \frac{a}{b^2}\left[t^2 \vec{i} - 2bt \vec{j} + (t^2 + 2bt) \vec{k} \right]

    show that the particle goes through the point P(4,-4,8)r[0].
    \displaystyle \frac{a}{b^2}t^2 = 4a<br />

    t^2 - 4b^2 = 0

    (t - 2b)(t + 2b) = 0


    \displaystyle -\frac{a}{b^2}(2bt) = -4a

    t = 2b


    \displaystyle \frac{a}{b^2}(t^2 + 2bt) = 8a

    t^2 + 2bt - 8b^2 = 0

    (t + 4b)(t - 2b) = 0


    common solution for all three equations is t = 2b
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