I hate that it was that simple ahahah thank you heaps. Also a discussion would be hella neato and interesting
Sweet! Okay, so it's pretty much about symmetry. Picture the path as defined in the question - It spends a certain amount of time (call it \(T_1\)) being accelerated downwards on its way up, which sets the velocity to 0 at the peak of motion. Then it spends another amount of time (call it \(T_2\)) being accelerated downwards to reach a final velocity.
In your mistake, you swapped the scenario - You
started at the bottom, and you wanted to land on the cliff. To do that, you would be tracing your path in reverse. But to make sure you get to the same peak, you'd need to start with the vertical velocity you
finished with in the above scenario, since you know that is how much will be taken away on the ascent. You spend \(T_2\) time getting accelerated the same as before, but now it acts against you - So to end up at \(v_y=0\), you need to start with that higher value of velocity. Then it is similar on the way down - Before, you were stopped after \(T_1\) seconds, so now doing it in reverse, you end up with the same velocity you started with in the first scenario.
Okay, that was actually tough to explain without a diagram - Maybe it made some sense? Maybe not... Haha
if anyone wants to jump in and clarify feel free