So, what makes this problem fun and also challenging? If you just threw a grappling hook (or ball) upwards without a string, it’s just plain projectile motion (assuming you ignore air resistance). However, with a rope the hook will have an extra interaction with the different parts of rope as it gets pulled off the ground.
Differential Equation from the Momentum Principle
Let’s start off with the simplest case — a hook that is launched straight up. Here is a diagram of the hook and rope at some point after the toss.