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Hear me out
Trebuchet
I would definitely like to see a ship propel itself using a trebuchet and a windlass. Not enough to try building one, but still.
I bet you could get a bunch of money for a CGI prototype of your new green cargo ship.
Doesn't this defy the laws of physics?
A trebuchet primarily transforms downward motion (of the counterweight) into forward motion, so it would actually work - the trebuchet doesn't push the ship back as much as it pushes its load forward. This is particularly so if your trebuchet has wheels and you have room on your ship to accommodate it rocking back and forth when firing.
Edit: Thinking about it this technique would work even with something that does impart equal backwards force on the ship when firing a projectile, because there is considerably more force involved in winching the ship towards the anchor than what is involved in actually moving the anchor. You aren't pulling against the inertia of a free floating anchor after all, you're pulling against the ground the anchor has hooked into.
Not only that, you're using the friction between the anchor and the seafloor, so even the recoil of a propulsive ballista will allow you to perform work.
Indeed, I just realised that point - the force of propelling the anchor is tiny compared to what you can exert on the ship once the anchor is hooked.
I've got a really wild idea: what if you change the shape of the anchor so that it's actually getting resistance against the water itself instead of having to hit the sea floor. You'll have to pull it out if the water once you're done moving it and then drop it in again. Maybe put it on the end of a big stick. I'm thinking about calling this type of anchor an "oar."