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?
2 Answers
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 $\Delta E=0$.
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$\begingroup$ Ok. So we have to consider equivalence principle and so there is no gravitational field, only that ground is accelerating from this perspective. Is that what you mean? $\endgroup$ Commented Apr 9 at 13:16
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$\begingroup$ Well, I am really getting at your choice of mechanical system, this is important, are you simply changing frames of reference but always working in the Earth-ball system, or do you sometimes consider the ball as a system unto itself? $\endgroup$ Commented Apr 9 at 13:18
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$\begingroup$ I did consider the frame of falling ball as a mechanical system too $\endgroup$ Commented Apr 9 at 13:19
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$\begingroup$ So I guess we have to apply equivalence principle to be consistent in our choice $\endgroup$ Commented Apr 9 at 13:21
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$\begingroup$ Well, the point is that, you must choose a particular system or the analysis turns out different. See how there are two parts of my answer, one part is Earth-ball and agrees with the answer given by the other user, the second part of my answer is done for the ball alone as mechanical system. $\endgroup$ Commented Apr 9 at 13:27
In this thought experiment, your frame of reference is accelerating. Hence you should account for that by considering the acceleration of the falling ball (since it is stationary in your frame of reference which is accelerating!) Hence, the potential energy still gets converted to kinetic energy of the ball in frames of reference other than than ball (from an observer's viewpoint)
