# Where did potential energy disappear in this case?

Just a new thought experiment. We know that the kinetic energy depends on frame of reference as do work and velocity. Now we can consider a falling ball. It converts potential energy to kinetic energy as it descends. Let us observe the same ball from a different frame reference one that is freely falling with the ball in such a way that the ball appears to be at rest. That means, zero kinetic energy. But shouldn't the ball still possess potential energy by virtue of its height. Or more precisely the earth ball system has potential energy. And as the ball descends downwards there is a continuous decrease in the potential energy of the system. So where did the potential energy go. Did it get converted into kinetic energy of earth or something?

You have to be consistent in your choice of system. In the Earth-ball system the potential energy is converted to kinetic energy, i.e. $$0=\Delta T+\Delta U.$$ Even though the ball is at rest in it's rest frame, the rest frame of the ball is not at rest with respect to the Earth.
If you choose as your system the ball, then the ball's rest frame is a non-inertial frame and is in free fall. By the principle of equivalence there is no gravitational field in the ball's rest frame. Thus the ball experiences no change in either potential energy or kinetic energy, and again $$\Delta E=0$$.