Modeling the Motion of a Grappling Hook With a Rope

Photo: Rhett Allain. A ball launched with a weighted string

What happens when a ball is launched upwards, but it also has a rope attached to it? How do you model the motion of an object like this? I attempted to model this kind of motion when I analyzed a recent Paramount Plus commercial. You can see that post here.

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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.