Weight (heavy object, hook,...) on a rope, cable, string

long time no see. I had to dig out this account from past as i haven´t been actively blending for past six years.

Anyway, as i’m getting back into 3D graphics more actively, in my new project, i have ran into a problem. It seemed straightforward at first, as it is theoretically really simple, but after two days of furious googling and trying stuff out, i am nowhere near finding solution.

So - I am creating a simple animation, that includes a kids toy gun. Something like this:

And i want simply to shoot the cork (bullet) from the nozzle, while it’s tied to string, other side of string still fixed inside the barrel of the gun. String should follow the cork, stop it when it reaches its limits and the cork should fall down and dangle like a pendulum on the string.

I tried different approaches - softbodies with hooks or cloth sims with pinning combined with rigid bodies, manual animation of the cork, bones, nothing that would work. I have found tutorials solving only one part of the problem. It seems that i am unable to recreate simple relation between these three objects - string is following the cork and is attached to fixed base. Cork is simulated rigid body, its movement limited by the length of the string. Closest i got, was when i attached string of cloth to the cork via pinning group and parent and had it also attached via hook modifier to the base. But the cork is falling to infinity and cloth is stretching with it. Here is the blend file. https://www.dropbox.com/s/hsu0smldd4h8m9r/rope_weight.blend?dl=0

Any help with a solution would be greatly appreciated. As i said, this seemed really simple, am i missing something?