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Thread: Differential Equation involving mechanics

  1. #1
    Junior Member
    Mar 2009

    Differential Equation involving mechanics


    I need help with the following question. I have attached my question. I need help with part b)

    this is what i have managed to do so far, but i don't think i am doing it correctly.


    dividing both sides my m



    \dot{r}=\frac{MG}{r^{2}}t+C<br />

    any help is appreichated


    Attached Thumbnails Attached Thumbnails Differential Equation involving mechanics-capture.gif  
    Last edited by cooltowns; May 12th 2010 at 11:27 AM.
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  2. #2
    MHF Contributor chisigma's Avatar
    Mar 2009
    near Piacenza (Italy)
    This problem can be solved as in...

    Setting for semplicity sake M G = \mu the DE becomes...

    r^{''} = -\frac{\mu}{r^{2}} (1)

    Now because is...

    r^{''} = \frac{d r^{'}}{dt} = \frac{d r^{'}}{dr} \frac{d r}{dt} = r^{'} \frac{d r^{'}}{dr} (2)

    ... we obtain from (1)...

    r^{'} dr^{'} = -\frac{\mu}{r^{2}} dr (3)

    ... i.e. a separable variables DE the solution of which is...

     r^{' 2} = \frac{2 \mu}{r} + c_{1} (4)

    ... just like in Your textbook. The further step could be finding r(t)... a little less confortable step ...

    Kind regards

    \chi \sigma
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