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Math Help - Projectile Motion Using Differential Equations

  1. #1
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    Projectile Motion Using Differential Equations

    Hi everyone. It seems I've forgotten how to solve this one. I was wondering if someone could help me remember how.

    Think projectile motion, straight up/down no resistance. I point out the part I have a problem with near the end. (this is all in accordance with a book on applied mathematics.)

    d2x/dt2= -g initial conditions are dx/dt=v0 x(0) = 0
    x(t) = -1/2*g*t2 + v0 *t

    The book makes says that I can solve for t to get
    t=v0/g - how?
    and then realize the next equation without any explanation.
    Xmax= v02/2g - how?

    I would be very grateful for help and direction on this one. Thanks.
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  2. #2
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    Re: Projectile Motion Using Differential Equations

    Since we are given:

    x(t)=-\frac{1}{2}gt^2+v_0t

    then differentiating with respect to t, we find:

    v(t)=-gt+v_0

    Equating this to zero (at the apex of the trajectory which is x_{\text{max}}, the velocity will be zero), we find:

    -gt+v_0=0

    t=\frac{v_0}{g}

    Now, using this value for t, we find:

    x_{\text{max}}=x\left(\frac{v_0}{g} \right)=-\frac{1}{2}g\left(\frac{v_0}{g} \right)^2+v_0\left(\frac{v_0}{g} \right)=

    -\frac{v_0^2}{2g}+\frac{v_0^2}{g}=\frac{v_0^2}{2g}
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  3. #3
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    Re: Projectile Motion Using Differential Equations

    "A pattern so grand and complex..."

    Can you explain v(t) from x(t)

    Thanks Mark.... BTW I'm a HUGE Rush fan.
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Projectile Motion Using Differential Equations

    By definition, we have:

    v(t)\equiv\frac{d}{dt}x(t)

    This simply means that velocity is the time rate of change of displacement, or position.

    So, if you have the displacement, you may differentiate it to get the velocity.

    I was saddened a bit the other day to hear "Fly By Night" used in a commercial to sell Volkswagens.
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  5. #5
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    Re: Projectile Motion Using Differential Equations

    LOL... to all of your last post... I must have missed that because it is late/early here... Can't believe I didn't see Calc I.

    Yeah, that commercial is a bit... off.

    Thank you again Mark. I can get back to studying in the morning. Good to know that there are great people on this sight with great answers.
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: Projectile Motion Using Differential Equations

    By the way, I was raised in Evansville...I really miss the Fall Festival there!
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  7. #7
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    Re: Projectile Motion Using Differential Equations

    The colors are wonderful this year.
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  8. #8
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    Re: Projectile Motion Using Differential Equations

    BTW, how are you generating the equation graphics? They look great.
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  9. #9
    MHF Contributor MarkFL's Avatar
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    Re: Projectile Motion Using Differential Equations

    Those are generated using \LaTeX. Do a search here and online for LaTeX usage, and you will soon be making nice looking expressions too!
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