Crazy cannons

Two cannons are fired at one another and the cannonballs collide... what can you deduce?

Problem

Two cannons both fire balls at speed $100 \mathrm{ms}^{-1}$. One of the cannons is fixed at an angle of $45^\circ$ to the horizontal and the other is fixed at angle $30^\circ$ to the horizontal. The cannons are set up facing each other at a distance $D$ from each other. One cannon is fired and then, $T$ seconds later, the other cannon fired.

The balls subsequently strike each other in mid air.

Assuming that the cannon at an angle of $45^\circ$ is fired first, show that that $T$ must lie within a range of values and that both $T$ and $D$ must be greater than zero. 

What if it was the cannon at an angle of $30^\circ$ which was fired first? 

Did you know ... ?

The mathematics used in this question, along with an understanding of how gravity works at large distances, is sufficiently complex to send rockets to the moon and the planets of our solar system.