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I was explained in a lecture that if lets say, I jumped from height h and grabbed onto a vine, I would reach y height at the tip of the swing. But if I were to jump from 2h, I would still reach the same height (y) when swinging. This doesn't really make all that much sense to me... Won't my velocity increase if I jump from a greater height, thus making me swing higher?

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It might depend on the total length of the vine. Let's say the vine only has length $y$ and you swing all the way to the top when you jump from height $h$. Then, when you start with more energy because you jump from height $2h$, there's no higher you can go when you're still holding on to the vine, so you'll also make it to height $y$ but with a greater kinetic energy than the original scenario.

Otherwise, if $y$ is less than the maximum height the vine can reach, you are correct that jumping from higher should get you a higher swing on the vine. The thing that may be consistent between the two scenarios is the amount of time it takes to do one full swing (because that only depends on the length of the vine and the strength of gravity).

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