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Math Help - particle movement, partial differentiation

  1. #1
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    particle movement, partial differentiation

    Can someone explain how to do this question?

    "A particle moves such that its position at time t is given by r=(t,t2,t3).
    Find the rate of change of the distance of the particle from the origin."

    I don't understand the way r is described, does it mean the x,y and z coordinates of r? If so, how do you use the chain rule to differentiate this?

    Thanks
    Laura
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  2. #2
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    Re: particle movement, partial differentiation

    Hi

    We cannot explain something either you haven;t study yet or you forgot it...
    Please have a look first here
    http://www.ittc.ku.edu/~jstiles/220/...n%20Vector.pdf

    Position (vector) - Wikipedia, the free encyclopedia

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  3. #3
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    Re: particle movement, partial differentiation

    Oh ok, that makes a bit more sense. Still struggling with how to differentiate it though.
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  4. #4
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    Re: particle movement, partial differentiation

    First there is just the one independent variable, t, so this is NOT "partial differentiation", just ordinary differentiation.

    This is a vector equation: r= <x, y, z>= <t, t^2, t^3>. Do you know how to differentiate t with respect t? What about t^2 or t^3.
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    Re: particle movement, partial differentiation

    For a vector valued function \displaystyle \begin{align*} \mathbf{r}\,(t) = \left( x(t), y(t), z(t) \right) \end{align*}, its rate of change is \displaystyle \begin{align*} \mathbf{r}'(t) = \left( x'(t), y'(t), z'(t) \right) \end{align*}.
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    Re: particle movement, partial differentiation

    I get an answer of 1+2t+3t^2, but that is not the correct answer
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    Re: particle movement, partial differentiation

    That's because your answer is also supposed to be a vector. Read my last response.
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    Re: particle movement, partial differentiation

    I am still obviously not fully understanding this and need more of an explanation.

    What I have understood so far is that r is a vector with coordinates (x(t),y(t),z(t)). Where x(t)=t, y(t)=t2 and z(t)=t3.
    In order to find the rate of change of r I must find the derivatives x'(t), y'(t) and z'(t) which are 1, 2t and 3t2, respectively. I also know you can write the coordinates in terms of unit vectors: r= t(i)+t2(j)+t3(k). Is this all correct? What happens to the unit vectors if that is the case?
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    Re: particle movement, partial differentiation

    Quote Originally Posted by lauramorrison93 View Post
    I am still obviously not fully understanding this and need more of an explanation.

    What I have understood so far is that r is a vector with coordinates (x(t),y(t),z(t)). Where x(t)=t, y(t)=t2 and z(t)=t3.
    In order to find the rate of change of r I must find the derivatives x'(t), y'(t) and z'(t) which are 1, 2t and 3t2, respectively. I also know you can write the coordinates in terms of unit vectors: r= t(i)+t2(j)+t3(k). Is this all correct? What happens to the unit vectors if that is the case?
    They stay there.
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