Re your solution to my problem: I think you've misinterpreted what I've tried (badly no doubt) to say. (The problem is probably not worded as well as it could be - I got it from someone else but rewrote the description from scratch here.)<snip>
I'm not comparing energies between the different reference frames - but instead looking at whether energy is conserved within each frame separately. In the first frame, I end up needing mgh = mv^2/2 (if total energy is to be conserved). That admits non-trivial solutions - for each positive value of h (and positive m also!), we can find two possible values for v (corresponding to the cart moving left or right) and so everything looks reasonable. However, with the second reference frame, there is no similar solution to the equation that I show at the end. Does that make "the mystery" any murkier?
You seem to have a typo after doubling speed to 2v: the KE goes to 2mv^2, not 2mv/2 as you've (accidentally no doubt) written. But even after that minor correction, I'm still not bothered! I am aware you can't compare KE values (or differences) across different frames and generally expect anything to line up.Okay here is an even worse one for you if that one bothers you. Again two different systems an object is going left to right at v it has a rocket motor on it and doubles it speed to 2v. Its KE was mv^2/2 and now it's 2mv/2, a difference of 3mv^2/2. Our second system is moving with the object at the speed of v. Again the object changes its speed by rocket motor adding v to its speed (which was zero in this frame). Now its change in KE is mv^2/2. The first system seems to have gained more energy![]()
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