It is rather like a robust, lightly constructed Trebuchet at the end of a sling.   I will draw a sketch for you to consider until I get time to get one on the board.   It is a fun curiosity that never really found favor with me as the Pro/Con list balanced out with no real advantage to be gained in the end.   But a nice curio it is.....

here's my idea on the physics of a sling with an elastic portion:

In an overhand throw, the stone is swinging from in front of you to behind you, with the strings taut the whole time. Then, when the stone is at about 6 o'clock or before, you twist your torso and throw your arm and shoulder forward, thusly pulling the stone with you and accelerating it. When the stone is at 3 or maybe 2 o'clock is when the greatest tension is on it. If there was a piece of elastic in the cord, then at that point the band would stretch due to the tension. From there I'm not sure what the result would be. It seems like it might propel the rock faster - imagine pulling a loaded trailer with a truck. If its connected with a chain, that's like a conventional sling. However, if the connection was a stretchy cable, the trailer would *snap* forward when the tension in the elastic cable becomes too great. I think the same effect would happen with the sling.
What do you think?

I almost forgot.....welcome to the forum WildAtHeart.   To quote you:

"I wish I knew some physics equations to help me figure this out. I have the worst physics teacher ever".


I wish you did too, WildAtHeart.    I wish you did too.....      I have always been a bit 'reversed' when it comes to equations.....

Techstuff:  All I can say is "Wow."  That's quite the device.  Do both missiles go forward? 

Wildatheart:  Welcome!  I think, with elastic cords, they wouldn't so much snap forward as they would continue to extend and then snap back into shape after the release.  If you accelerate your swing at a constant rate, centripetal (O.K., centrifugal, although my high-school physics teacher says there's no such thing) force will continue to increase until you release -- more force, more speed, and more extension of the cords until you let go.  Also, since the cords start off as shorter, you're robbing yourself of leverage through the initial portion of the swing.  If you could make a sling go from longer to shorter, you'd get more speed/power/distance, but with elastic cords I think you'll be going from shorter to longer, giving yourself less speed/power/distance.  I don't have any equations for you, and my last physics course was some time ago, but that's what makes sense to me.

Wow indeed! That's quite the sling! I hope you can dig up an example and share it... maybe even a video? 

This is at right angles to what I was thinking about, so I'll carry on this weekend (I'm out of the city with room to throw!) and let you guys know how it pans out.