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Mechanics - Projectile Motion

  1. Nov 2, 2011 #1
    Problem statement is here: http://www.phys.uri.edu/~gerhard/PHY520/wmex139.pdf

    My approach:
    [tex]y = -\frac{gt^2}{2} + v_0 cos( \alpha) t + h[/tex]

    Projectile hits ground at:
    [tex] t_{gnd} = \frac{v_0 sin( \alpha ) + \sqrt{ v_0^2 sin^2 ( \alpha) + 2gh}}{g}[/tex]

    Now compute derivative of x and solve for alpha:
    [tex]x_{max} = v_0 cos( \alpha) t_{gnd} [/tex]
    [tex]\frac{dx_{max}}{d \alpha} = 0[/tex]

    This last step where you solve for alpha is what buggers me. I get a huge quartic expression in alpha that does not simplify, I suspect the problem setter expects you to make some approximation but I can't figure it out.
  2. jcsd
  3. Nov 3, 2011 #2


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    Homework Helper

    In the first equation (the one for y), it should be sine.

    I see where you're trying to go with your approach. And I think it would work, but maybe you should try a different approach.

    Are you familiar with Lagrange multipliers? Because you've got y=0 as the constraint and x as the function to maximise. The functions x and y are each functions of both alpha and t, so the problem looks well suited to the method of Lagrange multipliers.
  4. Nov 3, 2011 #3
    Worked like a charm, good idea.
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