At what point does a projectile leave a slingshot?

Question

Assuming a frictionless / "perfect" environment, and given a ball held in an elastic sling (like a hand-held catapult) where the pocket is lighter than the projectile itself, what is the point at which the ball separates from the sling pocket? At the start of the shot, or as the sling-pocket passes through its at-rest position?

My thoughts -- please confirm or refute:

• Hooke's Law tells us that the force applied to the sling-pocket is directly proportional to the distance from the rest position.

• Given the pocket's lower weight, I suppose then that it's velocity (resulting from the applied elastic force) would be greater than that of the projectile itself; thus it should be pushing the projectile (and thus be in contact with it) until such point as we pass the rest position for the first time (assuming sinusoid motion of the band as it vibrates)

• In passing the rest position, the force now begins to be applied in the opposite direction and thus reduces the forward velocity of the pocket to be less than that of the projectile, which is travelling at a constant, maximal velocity.

I am a programmer, not a physicist, so your patience is appreciated. Also note that I am trying to keep certain factors, eg. momentum, out of the picture here, as it is not necessary for what I am building (due to certain shortcuts taken). It is really just force, acceleration, mass and velocity I am concerned with.

Answer 1

The ball is being accelerated by the pocket of the sling, so it will stay in contact with the pocket as long as the pocket is accelerating i.e. as long as the rubber retains some tension.

If you assume the sling is described by Hookes law the pocket will start decelerating as it passes through its "at rest" position, so at this point the ball will separate.

In a real sling, of the sort I made as a kid, the rubber bands go slack before the ball reaches the sling, and they will remain slack as the ball and pocket pass through the arms of the sling and for some distance beyond that. The exact point at which the two separate will depend on how fast the pocket and ball are slowed by air resistance. In practice I'd guess the two will separate as soon as the rubber is slack, or at worst very shortly afterwards. I guess this because the pocket is much lighter than the ball and it's dragging the (now slack) rubber bands.

Answer 2

The ball leaves the slingshot when the speed of the ball is greater than the speed of the slingshot. Specifically, in this case, where the slingshot begins to decelerate.

Imagine the ball and slingshot are moving together (let's just call the cup of the slingshot the slingshot), with the slingshot accelerating because of some elastic strings attached to it. The ball also feels this acceleration because the slingshot is pushing it. Once the slingshot has reached the point where the bands are loose, the ball and slingshot are still together, travelling at a constant velocity. Once the strings tighten, the slingshot is slowed down, but since the ball isn't connected to them, it doesn't feel this force, so it carries on. This is where they separate.

Let's say there's no point where the bands are loose, we will instead say there's a point where they're at their loosest. Immediately after this point, they will start pulling the slingshot back, but not the ball. This is where they separate.

In both these cases, it was where the slingshot reached its maximum velocity, just as it was about to be decelerated.