The problem with slingshots is the dependence on highly corruptible materials. And crossbows have that awkward -- cross-bow thingee. It seems like we could get the same power from coil springs. Here is a rough suggestion:
(I hope the final image is a little larger than that. If not, there is a full-size copy in my private gallery -- please have a look.)
Carbon filament springs would be lighter than steel springs, and the modern materials are very strong. If this weapon worked, it could potentially be quite small -- almost like a pistol.
The question is whether the springs would impart the speed required. In the mechanics of a bow, the more it straightens, the faster the string is drawn. This roughly corresponds to the desired acceleration of the arrow. But a coil spring might be the opposite -- the closer it gets to its desired shape, the more it relaxes. Thus, I thought it might need a little compound action with the double pulleys.
If anyone has any experience with a similar design, success or failure, please let me know.