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Thread: First Order Error Analysis for O.D.E.

  1. #1
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    Question First Order Error Analysis

    Given that the velocity of the falling parachutist can be computed by,

    v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})

    Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

    I'm having a tough time starting this one. Can someone nudge me in the right direction?

    EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?
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  2. #2
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    Quote Originally Posted by jegues View Post
    Given that the velocity of the falling parachutist can be computed by,

    v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})

    Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

    I'm having a tough time starting this one. Can someone nudge me in the right direction?

    EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?
    Click the traingle with ! in the middle and type in the forum it should be in and hit submit.
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  3. #3
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    Quote Originally Posted by jegues View Post
    Given that the velocity of the falling parachutist can be computed by,

    v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})

    Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

    I'm having a tough time starting this one. Can someone nudge me in the right direction?

    EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?
    Start by expanding as a Taylor series (in c) about c=12.5 and truncate after the first term.

    CB
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  4. #4
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    Question

    Quote Originally Posted by CaptainBlack View Post
    Start by expanding as a Taylor series (in c) about c=12.5 and truncate after the first term.

    CB
    So,

    v(t) = f(t) + R_{n}

    v(6) = g(50)(1-e^{\frac{12.5 \times 6}{50}}) + R_{n} = -1706.03 + R_{n}

    How do I get the error from this?
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by jegues View Post
    So,

    v(t) = f(t) + R_{n}

    v(6) = g(50)(1-e^{\frac{12.5 \times 6}{50}}) + R_{n} = -1706.03 + R_{n}

    How do I get the error from this?
    After the first non-constant term.

    CB
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  6. #6
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    Quote Originally Posted by CaptainBlack View Post
    After the first non-constant term.

    CB
    Here's what I came up with.

    I'm pretty sure I must be misunderstanding something because I dont see any nonconstant terms.

    We know the values for every variable so we should just end up with a number, no?

    Where did I go wrong?

    EDIT: Don't worry about the -1706.03 in the first line, it's not supposed to be there.
    Attached Thumbnails Attached Thumbnails First Order Error Analysis for O.D.E.-q4.18cont.jpg  
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  7. #7
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    Quote Originally Posted by jegues View Post
    Given that the velocity of the falling parachutist can be computed by,

    v(t) = \frac{gm}{c}(1-e^{(\frac{c}{m})t})

    Use a first-order error analysis to estimate the error of v and t = 6, if g = 9.8 and m = 50 but c = 12.5 + or - 1.5.

    I'm having a tough time starting this one. Can someone nudge me in the right direction?

    EDIT: I just realized I posted this is the wrong section, it's clearly not an ODE can someone please move it?
    v(t,c=12.5+\varepsilon)=\dfrac{gm}{12.5+\varepsilo  n}(1+e^{(12.5+\varepsilon)t/m})

    Now expand the right hand side as a series in \varepsilon

    CB
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  8. #8
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    Quote Originally Posted by CaptainBlack View Post
    v(t,c=12.5+\varepsilon)=\dfrac{gm}{12.5+\varepsilo  n}(1+e^{(12.5+\varepsilon)t/m})

    Now expand the right hand side as a series in \varepsilon

    CB
    Okay so,

    v(t) = \dfrac{gm}{12.5+\varepsilon}(1+e^{(12.5+\varepsilo  n)t/m}) + \dfrac{gm}{12.5+\varepsilon}(1+\frac{(12.5+\vareps  ilon)}{m}e^{(12.5+\varepsilon)t/m}) + ... + R_{n}

    What's next?
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  9. #9
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    Quote Originally Posted by jegues View Post
    Okay so,

    v(t) = \dfrac{gm}{12.5+\varepsilon}(1+e^{(12.5+\varepsilo  n)t/m}) + \dfrac{gm}{12.5+\varepsilon}(1+\frac{(12.5+\vareps  ilon)}{m}e^{(12.5+\varepsilon)t/m}) + ... + R_{n}

    What's next?
    Is that a power series in \varepsilon?

    CB
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  10. #10
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    Quote Originally Posted by CaptainBlack View Post
    Is that a power series in \varepsilon?

    CB
    I'm confused as to what you want me to do.

    Do you mean this,

    e^{12.5 + \varepsilon } = \sum \frac{(12.5 + \varepsilon)^{n}}{n!}

    The sum starts from n=0 and goes to infinity, I can't seem to put it in.
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