energy
How could you store the kinetic energy of a soda-bottle using a rubber band so that it unwinds after being rolled across a flat surface?
Basically, how can you make a bottle roll back after being pushed across a flat surface.
(preferably using a rubber band)
The person before has put an answer which might be more complex and answer your entropy question. however to answer your magnet issue in a simpler way concentrating on your metal ball might be more fruitful for you.
What you have is 2 fields here at work both with similar properties gravity and magnetism. let me ask you this, why does your ball fall when you raise it 6 inches and let go? gravity. the earth (and conversely the ball) attract each other. Your ball wants to always fall to the lowest energy state it can and the closer the ball and the earth are to each other the lower the potential energy as you rightly said. lift the ball and you spend your energy to move it further away and give it the gravitational potential energy. Release it and that energy is converted back into kinetic energy of movement as the ball falls to the lower gravitational potential energy state ie. closer to the earth.
your magnet is much the same. the magnet attracts the ball as the earth does, except it is an electromagnetic force not a gravitational one and it is stronger than gravity. your ball wants to fall to a lower magnetic potential energy state too and this is closer to the magnet. So therefore pulling the ball away from the magnet your providing energy to move it to a higher potential energy state and it wants to be in a lower one next to the magnet, hence attraction. now this is not as intuitive as gravity is always there and your magnet is not. but really when your magnet is not there its in the higher potential state it just cant get to a lower one until the magnet is present.
combine the two together and in opposite directions ie gravity pulling the ball down and the magnet pulling it up it will go towards the strongest. why? well because it will loose more magnetic potential energy than gravitational because the magnet is stronger. so whilst you see it fly 6 inch off the table its actually in a lower energy state than being six inches closer to earth.
hope this helps :)
Newton's second law
As you probably know, Newton thought that energy is linearly proportional to velocity: the Latin terms vis [force] and potentia [potence, power] were used at that time to refer to what today is called energy. The second law's original formulation reads: "Mutationem motus proportionalem esse vi motrici impressae" = "any change of motion (velocity) is proportional to the motive force impressed".
This law, which nowadays is wrongly interpreted as: $F = ma$ (there is no reference to mass here) simply states states: $$[\Delta/\delta v]( v_1-v_0) \propto Vis_m$$ and in modern terms is sometimes (illegitimately) also interpreted as impulse, sort of : $$\Delta v \propto J [/m] \rightarrow \Delta p = J$$. But mass is not at all mentioned in the second law (as the original text shows) but only in the second definition, where we can see a definition of momentum as 'the measure of [quantity of] motion'
Quantitas motus est mensura ejusdem (motus) orta ex velocitate et quantite materiæ conjunctim = 'quantity of motion' (modern 'momentum') is the measure of the same (motion), originated conjunctly by velocity and 'quantity of matter' (total mass)
and, moreover 'motive force' (vis motrix) is used, like all other scholars of the time, referring to the yet unknown kinetic 'force' that made bodies move, which Galileo had called 'impeto' and Leibniz 'motive power'. The interpretation of this formula as the definition of force in modern usage is an ex post facto historical manipulation, done against the author's own will: he knew about this interpretation proposed by Hermann and refused to adopt it in the final edition
The historical facts
It was Gottfried Leibniz, as early as 1686 (one year before the publication of the Principia) who first affirmed that kinetic energy is proportional to squared velocity or that velocity is proportional to the square root of energy: $$ v \propto \sqrt{V_{viva}}$$. He called it, a few years later, vis viva = 'a-live/living' force in contrast with vis mortua = 'dead' force: Cartesian momentum ([mass/weight =] size * speed: $m *|v|$). This was accompanied by a first formulation of the principle of conservation of kinetic energy, as he noticed that in many mechanical systems of several masses $m_i$ each with velocity $v_i$,
$\sum_{i} m_i v_i^2$
was conserved so long as the masses did not interact. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction or in elastic collisions. Many physicists at that time held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum: $\,\!\sum_{i} m_i v_i$ was the conserved kinetic energy.
The concept of PE played no role, it did not exist yet, nor did the concept of mechanical energy to which you refer (E = U + V), but Leibniz, in this first paper, uses the term potentia motrix/ viva [motive power] to refer both to the energy a body acquires falling from an altitude and to the force necessary to lift it to the same altitude (mass/weight * space: $F*s$) which are considered equal. Some scholars see, wrongly, here a first definition of PE, but that is simply one of the axioms of Galileo.
The principle to which you refer: $E_{mech} [KE + PE] = k$ in astrodynamics is called vis viva equation in his honour. Leibniz stated the conservation of KE per se besides the conservation of all (kinds of) energy in the whole universe. We need to underline this amazing stroke of genius.
His theory was strongly adversed by Newton[ians] and DesCartes-ians because it seemed to contrast, to be incompatible with the conservation of momentum. In Newton there was no distinction (as shown above) between speed, motion, momentum and energy but quantitas motus (momentum) was the prevailing concept and it was proven to be conserved in all situations, therefore Leibniz' vis viva was considered a threat to the whole system. Only later it was acknowledged that both energy and momentum, being different entities, could be conserved (by Bošković and later (1748) by d'Alembert).
We can thank Émilie du Châtelet for the modern..understanding of kinetic energy – user121330
There is no energy formula ..in the discovery of conservation of energy are Joule and... – Ben Crowell
That's overlooking historical facts (Joule was not concerned with KE): soon after Leibniz' death, the quadratic relation was confirmed by experiments independently by the Italian Poleni in 1719 and the Dutch Gravesande in 1722, who dropped balls from varying heights onto soft clay and found that balls with twice speed produced and indentation four times deeper. The latter informed M.me du Châtelet of his results and she publicized them. Two centuries later, after Joule had shown that mechanical work can be transformed in heat, Helmholz suggested that the lost energy, in inelastic collisions, might have been transformed in heat.
Thomas Young is thought to have been the first to substitute the terms 'vis viva/ potentia motrix' with 'energy' in 1807 (from the Greek word: ἐνέργεια energeia, which had been coined by Aristotle on the stem of ergon = work, therefore: energeia [= the-state-of-being-at-work]). Later (1824-1829) Coriolis introduced the current formula and the terms 'work' and 'semi-vis viva'; this concept and the consequent theory of conservation of energy was eventually formalized by Lord Kelvin, Rankine et al. in the field of thermodynamics.
The formula of kinetic energy
The question is much more complex than it appears, as there are at least four formulas involved here, and each issue is complex in its turn:
I did not want to make this post too long, but I'll take the suggestion from the bounty and address the issues in separate answers. Just a brief note here to make this post self-contained: the formula of KE was not derived from work, as it may seem: it's the other way round. $W = F * d$ and $F = m * a$ were by-products of the KE formula. Once the quadratic relation had been verified and universally accepted: $E \propto v^2$, any coefficient (0.2, 0.5, 2..) could be added as an irrelevant and arbitrary choice that depended only on the choice of units.
The only avalaible (and precisably measurable) source of KE at the time was gravity and the Galilean equations were too strong a temptation, as they included, too, a [0.5] quadratic relation: it seemed a stroke of genius to make the energy of the unitary mass at unitary (uniform) acceleration coincide with space. In this way energy was simply the integration of [m] $g$ on space.
Conclusions
Tying energy to gravity, that is, to acceleration and in particular to constant acceleration was not a wise idea, it was a gross mistake that tied, confined newtonian mechanics in a strait-jacket because it was in this way unable to deal with the more natural situations when KE is related to velocity and when there is just a transfer of energy: the concept of impulse was just an ad hoc awkward attempt to deal with that.
Tying work-energy to space and not to the mere transfer of energy was an insane decision that had irrational, catastrophic practical consequences. But consequences were even more devastating on the conceptual, theoretical level because explaining and identifying KE with the acceleration gave the illusion that the issue of motion-KE had been understood, and prevented further speculation.
Leibniz invented the concept of (kinetic) energy, prefigured and discovered its real formula $E = v^2$ resisting the Siren of gravity, suggested the right way of integration and established the universal principle of 'conservation of energy' as prevailing on/independent from 'conservation of momentum' (transcending Huygens' principle of 'conservation of KE').
He engaged in passionate controversies until his death but was opposed and overwhelmed by obtuse/ignorant Newtonian contemporaries. He was vulnerable as he could not account for the loss of energy in inelastic collisions. He lost, and newtonian integration on space produced: $\frac{1}{2}$ mv2 which is not the formula, but just one of the possible formulas of KE: the newtonian formula. Had he won, instead of the joule, now we would use the 'leibniz' (= 1/2 J) and we would have a different, probably deeper, insight into the laws of motion and of the world.
You can find additional information on work here
Best Answer
Glue one end of the rubber band to the bottom of the bottle (on the inside) and the other to the inside of the cap. (It may be easiest to cut the bottle open for this and later tape it back together, but with sufficient dexterity and suitable tools, you could do this through the neck of the bottle.)
Attach a weight to the middle of the rubber band, in such a way that the weight can't rotate around the rubber band without twisting it. Make sure the band is short enough that the weight hangs suspended when the bottle is horizontal.
When the bottle is horizontal, gravity will keep the weight below the rubber band, and will thus prevent the middle of the band from turning. Thus, when you roll the bottle on the floor, the band will twist, storing energy. When you let go, the rubber band unwinds, making the bottle rotate backwards and thus roll back to where it started.
Of course, there's a limit to how much energy you can store this way. If you roll the bottle far enough, the tension in the rubber band will eventually overcome gravity and set the weight spinning (or the band will snap).