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Projectile Motion

  1. Oct 15, 2007 #1

    danago

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    Gold Member

    A missile is fired at a target from the origin O, with the velocity vector, t seconds after it was fired, given by [itex]\overrightarrow v (t) = [u\cos \theta ]\overrightarrow i + [u\sin \theta - gt]\overrightarrow j[/itex], where u, theta and g are constants. The target is moving with velocity [itex]v\overrightarrow i[/itex] and at the instant the missile is fired, the target is at position [itex]h\overrightarrow j[/itex].

    Prove that for the missile to hit the target [itex]u^2 \ge v^2 + 2gh[/itex]


    Alright, from the information given, ive come up with the following set of displacement equations:

    [tex]
    \begin{array}{l}
    \overrightarrow r _{missile} (t) = \left( {\begin{array}{*{20}c}
    {ut\cos \theta } \\
    {ut\sin \theta - 0.5gt^2 } \\
    \end{array}} \right) \\
    \overrightarrow r _{t\arg et} (t) = \left( {\begin{array}{*{20}c}
    {vt} \\
    h \\
    \end{array}} \right) \\
    \end{array}
    [/tex]

    For the missile to hit the target, both components of the motion must be equal for the same value of t; that is:

    [tex]
    \begin{array}{l}
    ut\cos \theta = vt \\
    ut\sin \theta - 0.5gt^2 = h \\
    \end{array}
    [/tex]

    Now, the first equation is only true for t=0, unless [itex]u\cos \theta = v[/itex], which i interpreted as a requirement for the collision to occur. From the second equation, the time when the vertical components of displacement are equal is give by:

    [tex]
    t = \frac{{u\sin \theta \mp \sqrt {u^2 \sin ^2 \theta - 2gh} }}{g}
    [/tex]

    Now its here where im not really sure what to do. A hint would be greatly appreciated :smile:

    Thanks,
    Dan.
     
  2. jcsd
  3. Oct 15, 2007 #2

    danago

    User Avatar
    Gold Member

    Ahh, the second i posted this i realised what to do. Since [itex]u\cos \theta = v[/itex], it can be shown that [itex]\sin \theta = \frac{{\sqrt {u^2 - v^2 } }}{u}[/itex], and then i just sub that into the quadratic discriminant and then set it to be greater than or equal to zero. Sound right?

    Thanks anyway :P
     
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