Thanks.
KE is the square of a vector - that is, if the velocity of something with mass m is
in some frame, its kinetic energy
in that frame is
. As you can see it's positive definite - zero only if all the components of v are zero, otherwise positive.
Ah, right. Of course, I was forgetting the squaring stops any negative energy (except in alternative therapy

).
When you change reference frames to one moving at velocity
in the original frame, the velocity in the new frame of something with velocity v in the original frame is
. That changes the KE, obviously, and not linearly. For example, if we choose
, the KE in the new frame is zero. In general, the new KE is
.
Nevertheless the laws of physics are identical in the new frame - in particular the (total) energy is still independent of time. That's why you can pick any frame you want, do an analysis, and then transform back to the frame you're interested in.
Ok. So
particular objects will have different energies in different frames, just as they have different velocities. The air in the room, and the cart on the treadmill, from the frame of the Earth (and indeed tread
mill body) has no KE (ideally), right?
Presumably, you'd also say that it does have (mv
2)/2 from the frame of reference of the tread? If so, it seems that humber's KE discussion is no more and no less than a mathematical repeat of his English failure to see things from that correct perspective of the tread? - since he keeps arguing that the air and the cart have no KE on the treadmill - he's just seeing it from Earth-frame.
I'm still a bit confused about this conversation you had with humber, though:
What question - why there's no subscript? For precisely the reason you said before you contradicted yourself in your own post. You can pick any frame you want, and the value of the KE depends on that choice. In particular you can always make it zero by choosing a frame that moves with the object in question. Sometimes it's convenient to do so, sometimes it isn't.
What I mean is that I would expect that there
should be a subscript saying what a velocity, acceleration or KE is relative to, but obviously this might be no more than a convention. You can choose any frame you want, as you say, but that will alter the KE. That would seem to be the only reason a value would need a subscript, to say what frame we've used, although it could be put once and then taken as read. I wonder if you and he are genuinely at cross purposes there, and he's arguing that the lack of one shows that there's only one absolute frame for KE measurements (the centre of the humberverse), hence no qualifying subscript is needed. KE just is what it is. D'you see? Obviously that means that somehow he accepts boosts as they relate to velocities, but not KE, which kind of screws up (mv
2)/2.
Oh who cares. Thanks again, I appreciate the lesson. I'm just going to ignore the word 'entropy' for the time being. I can't chase every white rabbit that runs past.
