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Math Help - 2 tough questions i am stuck on

  1. #1
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    2 tough questions i am stuck on

    Hi there i need help with these two questions before my exam next week, if anyone can help that would be great:

    firstly this one:
    A moving particle at time t ∈ [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10
    − t)i + (t^2 − 10t)j + sin tk.

    i. Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.

    ii. Find the curvature of the path, at t = 2π ≈ 6.28 seconds.

    iii. Find the angle between the path (at start and end-points) and the k-direction.



    And this one (obviously sketching might be tough)

    A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t + 1)i + (4 − t^2 )j.

    i. Sketch the path of the particle.

    ii. Find the value of t* at which the particle has greatest distance from (0, 0, 0).

    iii. Show that at position r(t* ), the particles velocity is not perpendicular to r(t*).
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  2. #2
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    I had a go at your second question.

    ii)
    The distance between your particle and the origin |r| and its square |r|^2 will be minimum at the same time t*

    <br />
|r|^2 = (2t+1)^2+(4-t^2)^2<br />

    I multiplied this out to get t^4 + ...

    Differentiate the result and set equal to zero to get 4t^3 + ... = 0

    Find the three roots of this equation (one is t = an integer)

    Ignore the negative root as negative time is silly.

    Find |r| for each of the other two roots, one of these is t*.

    iii) Find r(t*) from part ii)

    Find v(t) = 2i - 2tj by differentiating r(t)

    Find the dot product of v(t*) and r(t*). By my calculation the answer is zero proving that they are perpendicular. Could the question be wrong?
    Last edited by Kiwi_Dave; May 31st 2008 at 08:37 PM.
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  3. #3
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    Quote Originally Posted by jimmy View Post
    Hi there i need help with these two questions before my exam next week, if anyone can help that would be great:

    firstly this one:
    A moving particle at time t ∈ [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10
    − t)i + (t^2 − 10t)j + sin tk.

    i. Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.

    ii. Find the curvature of the path, at t = 2π ≈ 6.28 seconds.

    iii. Find the angle between the path (at start and end-points) and the k-direction.



    And this one (obviously sketching might be tough)

    A particle’s path, in two dimensions, is described by its position vector (in metres and time t ∈ [1, 2] seconds) relative to point (0, 0, 0) by r(t) = (2t + 1)i + (4 − t^2 )j.

    i. Sketch the path of the particle.

    ii. Find the value of t* at which the particle has greatest distance from (0, 0, 0).

    iii. Show that at position r(t* ), the particles velocity is not perpendicular to r(t*).
    The first question has been asked (by another member) and replied to in this thread: http://www.mathhelpforum.com/math-he...tml#post158518
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