If you throw a ball up into the air describe what happens to the potential energy as it rises?

As other have stated, the only state where it has no kinetic energy is E, UNLESS the ball is considered to only move in the vertical direction (in which case, the ball has no kinetic energy at B).

As for potential (gravity) energy, if considering the ground as the "absolute" level 0, then it has none, but, obviously, if someone was to dig a hole under the ball, it would fall... which indicates that it DOES have potential energy (but that the energy cannot be converted, as the ball cannot fall), rather than having none at all. However, this "approximation" is being used by most physicists, so we can let it slide.

All in all, the book seems to be garbage in the way it explains things, even if the author seems to know what he's talking about. Better dump it in the trash bin, and look for a better one.

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When the ball is thrown, chemical energy in the muscles of the person is transferred to the ball as kinetic energy. As the ball rises, kinetic energy is transferred to potential energy as the ball slows down and gets to be further from the ground. When the ball reaches its peak height all of its energy is in the form of potential energy, which as the ball begins to fall back down, coverts into kinetic energy as the ball speeds up.