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Math Help - Problem Finding the Speed of Particle (Vector)

  1. #1
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    Problem Finding the Speed of Particle (Vector)

    Hey everyone,
    I am having a bit of difficulty solving this problem. It says:
    A particle has position function \ r(t) = (2\sqrt{2})i + (e^{2t})j + (e^{-2t})k. What is its speed at time t?

    Well I first took the derivative of the function. Then I took the magnitude to find the length which corresponds to the speed. I got my answer to be
    \sqrt{8+4e^{2t}+4e^{-2t}}, is this answer correct? If so, how would I further simplify it? If it is incorrect, what am I doing wrong?

    Any help and feedback is greatly appreciated, thanks.
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  2. #2
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    Re: Problem Finding the Speed of Particle (Vector)

    Hello, Beevo!

    A particle has position function: \ r(t) \:=\: (2\sqrt{2})i + (e^{2t})j + (e^{-2t})k.
    What is its speed at time t?

    Well, I first took the derivative of the function.
    Then I took the magnitude to find the length which corresponds to the speed.
    I got my answer to be: \sqrt{8+4e^{2t}+4e^{-2t}}.

    Is this answer correct? . Yes!
    If so, how would I further simplify it?

    It can be simplified . . .


    \sqrt{4e^{2t} + 8 + 4e^{-2t}} \;=\;\sqrt{4(e^{2t} + 2 + e^{-2t})} \;=\;2\sqrt{e^{2t} + 2 + e^{-2t}}

    . . . . . . . . . . . . . =\;2\sqrt{(e^t + e^{-t})^2} \;=\;2(e^t + e^{-t})

    Thanks from topsquark and Beevo
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  3. #3
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    Re: Problem Finding the Speed of Particle (Vector)

    Quote Originally Posted by Beevo View Post
    Hey everyone,
    I am having a bit of difficulty solving this problem. It says:
    A particle has position function \ r(t) = (2\sqrt{2})i + (e^{2t})j + (e^{-2t})k. What is its speed at time t?

    Well I first took the derivative of the function. Then I took the magnitude to find the length which corresponds to the speed. I got my answer to be
    \sqrt{8+4e^{2t}+4e^{-2t}}, is this answer correct? If so, how would I further simplify it? If it is incorrect, what am I doing wrong?

    Any help and feedback is greatly appreciated, thanks.
    Looks good to me, unless you have to use hyperbolic functions to simplify.

    -Dan

    PS okay I should have seen that one coming Soroban! Thanks.
    Thanks from Beevo
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  4. #4
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    Re: Problem Finding the Speed of Particle (Vector)

    Thanks for the feedback guys. The simplification part just kind of threw me off.
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  5. #5
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    Re: Problem Finding the Speed of Particle (Vector)

    Okay, I'm confused! Where did that "8" come from? With the position r(t)= 2\sqrt{2}\vec{i}+ e^{2t}\vec{j}+ e^{-2t}\vec{k},I get the velocity to be 2e^{2t}\vec{j}- 2e^{-2t}\vec{k} and so the speed is \sqrt{4e^{4t}+ 4e^{-4t}}= 2\sqrt{e^{4t}+ e^{-4t}}
    Thanks from topsquark
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  6. #6
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    Re: Problem Finding the Speed of Particle (Vector)

    Quote Originally Posted by HallsofIvy View Post
    Okay, I'm confused! Where did that "8" come from? With the position r(t)= 2\sqrt{2}\vec{i}+ e^{2t}\vec{j}+ e^{-2t}\vec{k},I get the velocity to be 2e^{2t}\vec{j}- 2e^{-2t}\vec{k} and so the speed is \sqrt{4e^{4t}+ 4e^{-4t}}= 2\sqrt{e^{4t}+ e^{-4t}}
    My fault, there should be a 't' after \2\sqrt{2}
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