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Suppose I roll a ball (without friction) down the passage and up a ramp at the end. If I release the ball with speed $v$ it acquires kinetic energy $T = (1/2) M v^2$ and, by conservation of energy. when it reaches the ramp it rises to a height $ h = T/Mg = (1/2g)(v_{initial}^2 - v_{final}^2 )$ where $v_{final} = 0$ .So far so good.

Putting some numbers into this, I impart a speed of $2 m/s$ and $g = 9.81 m/s^2$. Then the ball will rise up approximately $h = 0.2m$ on the ramp.

What I didn't mention is that I am actually doing this in a train (if trains were good enough for Einstein then they're good enough for me). I'm rolling the ball in the forward direction and the train is moving at $20$ m/s. So now when I calculate the rise I get $ h = (1/2g)(22^2 - 20^2 ) = 4.81m$

I can see that I probably need to impart more energy to the ball to accelerate it from $20 $ to $ 22 m/s$ than from $0 $ to $ 2$. What bothers me is that this seems to contradict the equivalence of inertial frames, and the speeds involved are hardly relativistic.

Problem solved here: https://physics.stackexchange.com/q/315526

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    If you are doing this on a train, you would still only see 2 m/s as the velocity of the ball since you are stationary w.r.t. the train. – 2017-02-28
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    @Triatticus. Yes, but how high does the ball rise ? – 2017-02-28
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    Same as in the stationary case – 2017-02-28
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    @Triatticus. OK, so can you put that in terms of energy conservation ? – 2017-02-28
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    You already did in your first calculation, the calculation is identical in both cases. In other words $T_0 = U_f \Rightarrow \frac{1}{2}Mv_0^2 = Mgh$ – 2017-02-28
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    This question looks like it belongs on physics.stackexchange.com instead. – 2017-02-28
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    It most definitely does – 2017-02-28
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    @amd Up to a point I'd agree, but physicists tend to gloss over the finer points ! – 2017-02-28
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    Nevertheless, there’s hardly any mathematical content per se to it. – 2017-02-28
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    The vast generalization of Physicist - tut tut! We never gloss over the finer points, we assume all things are spherical.. – 2017-02-28
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    @Chinny84 Maybe a bit harsh - what's your well-rounded opinion ? – 2017-02-28
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    Not sure why you tagged general relativity or relativity since we have neither effects in this local, and low speed model. None of your model makes sense - Einstein was trying to see if there is a concept of relative motion that extends to that of light which a 2m/s and 20m/s just doest cut it. At least in my humble opinion :). – 2017-02-28
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    @Chinny84 It seems to concern equivalence of laws in inertial frames, and I couldn't find a tag for special relativity. My initial assumption was that I'd made some basic error in the maths, but so far I can't see it. If correct, this seems to suggest one could determine a preferential orientation in space, which I find hard to believe. Still looking for an answer. – 2017-02-28
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    take a look at this equation [http://physics.stackexchange.com/questions/51220/kinetic-energy-with-respect-to-different-reference-frames](http://physics.stackexchange.com/questions/51220/kinetic-energy-with-respect-to-different-reference-frames) – 2017-02-28
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    There is no paradox here. There's simply the fact that kinetic energy is *frame-dependent*. Just like position and velocity and momentum and ... There are a lot of fundamental quantities which are not invariant in the Newtonian (or really Galilean) sense. – 2017-02-28
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    @Chinny84 thanks for the reference. It isn't immediately obvious though it would resolve the issue of how high the ball rises when observed by myself (who doesn't realise the train is moving) and by my friend (with extremely good eyesight) standing on the platform ? – 2017-02-28
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    @Tom Collinge: Why don't you post the same question on Physics Stack Exchange (possibly with a link to the question here), and see what they say. Then maybe post a link here to your Physics Stack Echange question. That way, we get a (hopefully cooperative) multi-forum interaction. – 2017-02-28
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    @quasi OK I've done that (I thought I'd get a better answer from the mathematicians) http://physics.stackexchange.com/questions/315526/problem-with-kinetic-energy-in-inertial-frames – 2017-03-01
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    @Chinny84 (P.S. I added a special-relativity tag and removed the general relativity reference). – 2017-03-01

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