Proving Missile Collision Conditions for Target Hit

[danago]

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

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


Alright, from the information given, I've come up with the following set of displacement equations:

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

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

<br /> \begin{array}{l}<br /> ut\cos \theta = vt \\ <br /> ut\sin \theta - 0.5gt^2 = h \\ <br /> \end{array}<br />

Now, the first equation is only true for t=0, unless u\cos \theta = v, 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:

<br /> t = \frac{{u\sin \theta \mp \sqrt {u^2 \sin ^2 \theta - 2gh} }}{g}<br />

Now its here where I am not really sure what to do. A hint would be greatly appreciated

Thanks,
Dan.

 

[danago]

Ahh, the second i posted this i realized what to do. Since u\cos \theta = v, it can be shown that \sin \theta = \frac{{\sqrt {u^2 - v^2 } }}{u}, 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