Physics: How to Find the Final Height of a Moving Object

Thanks to the principle of conservation of mechanical energy, you can use physics to determine the final height of a moving object. At this very moment, for example, suppose Tarzan is swinging on a vine over a crocodile-infested river at a speed of 13.0 meters/second. He needs to reach the opposite river bank 9.0 meters above his present position in order to be safe. Can he swing it? The principle of conservation of mechanical energy gives you the answer:

At Tarzan’s maximum height at the end of the swing, his speed, v₂, will be 0 meters/second, and assuming h₁ = 0 meters — meaning that he started swinging from the same height as the tree branch he's swinging to — you can relate h₂ to v₁ like this:

Solving for h₂, this means that

Tarzan will come up 0.4 meters short of the 9.0 meters he needs to be safe, so he needs some help.