Similar to this question except not about using a machine: What payloads and launch speeds could a sling launcher get using modern materials on the Moon?
How fast could a stone be thrown using a sling while standing on the moon? Could you launch a rock into lunar orbit with a very long slingshot? Google has nothing about this.
Edit: I only said 'of the type whirled around above the head' to clarify that I didn't mean a catapult. Don't assume that that the solution needs to involve anything whirling above the head.