Every high school student can easily recall the formula for kinetic energy: K=m*V^2/2. Nevertheless I assert that hardly anyone understands what that formula really means, and very few care.
What is velocity? Can velocity be defined in the universe comprised of a single point-like body? Apparently that cannot be done operationally (i.e. without resorting to the elusive notion of absolute space of Newton). Indeed, it does not make sense to speak of motion in the universe that contains only one object to begin with. Then how many point-like bodies do we need in our universe in order to make introduction of the notion of velocity possible? We need at least three point objects. Can you see why two is not enough?
Let us put aside for the moment the difficult question of what inertial mass is. In order to quantify the intuitive notion of kinetic energy we need at least three point masses in our universe. We take any two of them with inertial masses m and M. Let these two point objects approach each other with a relative (to each other) constant velocity V. Now, it seems reasonable to assert that:
1. Body m has a relative kinetic energy m*V^2/2 with respect to the body M.
2. Body M has a relative kinetic energy M*V^2/2 with respect to the body m.
3. The combined kinetic energy of the two bodies is equal to (m+M)*V^2/2. This total energy can be said now to be absolute because the speed V at which they approach each other does not depend on any particular reference frame.
I am asking you now:
1. What is the correct way to define the notion of kinetic energy: as a relative value, or as an absolute value?
2. If the notion of kinetic energy is to be defined as relative value, it does not seem reasonable to speak of conservation of energy. Wouldn't you agree?
3. If the notion of kinetic energy is to be defined as absolute value, i.e. as a measure of some kind of relationship between two bodies (very much like the notion of potential energy), then where does kinetic energy reside exactly? In the body M? In the body m? In both? Or, perhaps, somewhere in between? Does it make sense at all to ask: where does kinetic energy reside exactly?
4. If the notion of kinetic energy is to be defined as relationship thing, can we rely on the formula (m+M)*V^2/2? If we cannot, then what is the correct formula for kinetic energy?
1. Motion and Inertia by Wenzel Hofmann, 1904
2. On the Relativity of Accelerations in Mechanics by Hans Reissner, 1914
3. The Possibility of Fulfillment of the Relativity Requirements in Classical Mechanics by Erwin Schrödinger, 1925
These papers can be found (partially) in Einstein Studies, Vol. 6, Mach’s Principle: From Newton’s Bucket to Quantum Gravity, 1995, Editors: Julian B. Barbour and Herbert Pfister.
Online version of the book is available at http://bookfi.org/book/713858
Here is a quote from The Variational Principles of Mechanics by Cornelius Lanczos, 1949, pointing out Hertz's insights into this conundrum:
While Newton is an obvious advocate of a more mechanical view he was only the most noteable of a widespread view that empirical data must inform philosophical musings. Particularly the religious establishments promotion of patent untruths concocted by revered philosophers such as Aristotle, led to dissatisfaction with the old consensus.
Thus force, life , celerity were all metaphysical and philosophical concepts for the empirical manifestation of the work of the gods, spirits or supranatural .
Aristotle's great investigation into the cause of everything written in his Metaphysics was very influential . It established several descriptions of motion and how motion itself could be considered as a synonym for a great being which caused without moving.
Without going too far into it , this Newton developed as an a reference frame within which all relative motion occurs by what Newton called Motive.
In addition to this Metaphysics Newton developed his philosophy of Quantity, described in the Astrological Principles.
Newton tackled the difficulties of knowing, what can be known or distinguished, and how measurement of quantity should be refined and redefined in this process.
It has taken time, over 200 years to switch fom the religious paradigms to the semi religious Newtonian paradigm. . Then it quickly moved to a more anti religious formulations.
Currently the paradigms are in disarray. More observers challenge the established view.
To measure one requires a Metron and a memory. Thus we cannot exclude the human processing from any system. It is the human processing that sets the relativity of the reference frame. It is the Pythagorean measure of the right angled triangle that establishes so called conservation in dynamic systems. However human processing has to apply this instantaneously. So much of what we say is observed is only instantaneously observedL that is tantamount to saying it is not conserved at all Dynamically!
Certainly in the bigger picture we now know that so called " chaos" pervades all! At any time our concept of Pythgorean measure being inviolate could be shattered by space as we have measured it warping as we have not accounted for it!
Panta Rhei .
Think of a boy who loves his dog. Where does this love reside exactly? In the boy? In the dog? Maybe in both, or somewhere else in between? The fact is: it resides nowhere, or better said, the question is meaningless. Love is a relationship, not a thing, and as such it does not make sense to speak of the place of its residence in space. Same is true with kinetic energy - it is not a thing, it is a relationship. After all, it never occurs to anyone to ask: Where does velocity reside? Why is that? Because it is plain that velocity is not a thing - it is a relationship. Then, shouldn't it be obvious as well that kinetic energy must necessarily be a relationship, and not a thing, for the simple reason (if not for anything else) that the magnitude of kinetic energy is tied to the magnitude of velocity which is a relationship thing!
Now, if kinetic energy is a relationship, then the formula for it, i.e. K=(m*V^2)/2, seems very odd, because we see the mass of one and only one body in that formula while any relationship requires, at least, two objects! Where is the second object? Why don't we see the mass of this second object in the formula? Besides, relative to what do we measure the velocity V in that formula?
I contend that the classical formula for kinetic energy K=(m*V^2)/2 is simply an asymptotic residue of a more fundamental formula. This asymptotic formula refers to the relationship between two bodies of mass m and M in the limiting case of m/M → 0, while V is the relative velocity between these two objects. In other words, the classical formula for the kinetic energy implies the presence of a second body, but the mass of the second body, M, simply drops out from the formula due to the nature of the degenerate case: M/m → ∞, i.e. M is incomparably larger than m.
And what is the exact formula for the relationship between two objects, which we call kinetic energy, when these objects' masses m1 and m2 are comparable? I assert that unabridged formula for kinetic energy should have the form:
In the next comment I shall endeavor to prove that any other formula would contradict classical mechanics of Newton. But here is a fundamental question: Is this formula in complete agreement with common sense?
When the two bodies collide they will press against each other with ever increasing elastic deformation of both bodies until, at some point, they reach a state of complete stillness relative to each other. This is the point of complete exhaustion of the kinetic energy relationship of these two bodies for they are not moving with respect to each other at this very moment. Therefore, to find the formula that we are looking for, all we need to do is to find the combined potential energy of elastic deformation for both bodies at this critical juncture. Now, that's an easy task in the framework of Newtonian mechanics. Denote the absolute velocity of bodies at the critical moment by v, then in accord with the conservation of energy, we have the following expression for the formula we are looking for:
K=m1*v1^2/2 + m2*v2^2/2 - (m1+m2)*v^2/2
Now, recall the conservation of momentum:
m1*v1 + m2*v2 = (m1+m2)*v
Excluding v form the above equations, we finally get:
K = m1*m2*(v1-v2)^2/[2(m1+m2)],
and that concludes the proof.
Now, let us examine this formula for robustness. Consider the degenerate case: m1=m, m2=M, v1=w, v2=0, and M >> m. Then, in the limit of M/m → ∞, we get the classical formula for the kinetic energy:
Consider another degenerate case: m1=m2=m, and v1=-v2=w. Now we have:
Why didn't we get the familiar expression for the kinetic energy in this particularly simple case? As a matter of fact we did, if we imagine that in this completely symmetrical case it can be thought that the energy relationship K=mw^2 is simply being split, as it were, between the two bodies whereby each getting "half of the relationship", so to speak.
The fundamental question still remains though: Is the suggested formula for the kinetic energy relationship between two material points in complete agreement with common sense? We'll delve into this question in the next comment.
Suppose two perfectly elastic bodies of mass m1 and m2 are moving along two different straight lines with given constant velocities (u1, v1, w1) and (u2, v2, w2), respectively, in the inertial reference frame of fixed stars. Next, assume that these lines intersect and, in addition, initial positions of the bodies are picked in such a way that they both reach the intersection point at the same time. Then, after the ensued collision, they get glued to each other and the combined body of mass m1+m2 acquires constant speed with components (u, v, w). In this more generic scenario the expression for the kinetic energy relationship looks like this:
K=m1*(u1^2+v1^2+w1^2)/2 + m2*(u2^2+v2^2+w2^2)/2 - (m1+m2)*(u^2+v^2+w^2)/2,
while the conservation of momentum takes the form of the following three equations:
m1*u1 + m2*u2 = (m1+m2)*u,
m1*v1 + m2*v2 = (m1+m2)*v,
m1*w1 + m2*w2 = (m1+m2)*w.
After simple algebra, we get the generalized formula for the kinetic energy relationship:
(1) K = m1*m2*[(u1-u2)^2+(v1-v2)^2+(w1-w2)^2]/[2(m1+m2)].
Now, what are the logical problems with this formula I have spoken above? First of all, let us recall the assumptions we have made in deriving it. We have assumed that the straight lines intersect each other, and the bodies moving along these lines reach the intersection point simultaneously. What if either of these conditions is not met? Then it is manifest that the bodies won't collide (assuming the sizes of the bodies are infinitesimally small), and if they don't collide, all the calculations that went into the derivation of our formula (1) are void! What does happen to the kinetic energy then, does it disappear without a trace? Or, perhaps, it is nonexistent to begin with? The assumption that the kinetic energy relationship between two bodies, provided their fate is never to collide, is void does not sit very well with common sense. Then how do we go about defining the concept of kinetic energy relationship between two material points and deriving a formula for it if collision is never to happen?
That's one problem. There is another problem, which is more subtle, and we'll examine it next.
(1) K = m1*m2*[(u1-u2)^2+(v1-v2)^2+(w1-w2)^2]/[2(m1+m2)].
Note that the absolute velocities of the bodies per se do not matter any more for we have managed to get a formula that depends on the relative velocity of one body with respect to the other, which is invariant in all reference frames. Therefore one is tempted to entertain the idea that we have succeeded in deriving a truly relational formula for kinetic energy, and we can safely dispose of the notion of inertial reference frames, which are necessarily pinned to the fixed stars. Unfortunately the situation we are in is more complicated and subtle than that.
The subtlety is in realizing that relative velocity of body 1 with respect to body 2, i.e. (u1-u2, v1-v2, w1-w2), is not the same thing as the rate of change of the separation of the bodies. Relative velocity is a vector quantity, while the rate of change of the separation of two material points is an algebraic one, i.e. a signed quantity with plus sign if the separation is increasing, and minus sign if the separation is decreasing.
It is true that the rate of separation change can also be defined as a vector quantity, but in order to do so we need something tangible to pin the reference system to: either we need to keep the fixed stars or, if we chose to dispose of the fixed stars, to identify some other objects within the system itself to anchor the coordinate system to them. Mathematicians routinely imagine and work with coordinate systems without having any compunction, or any obligation to identify material objects to tie their reference systems to, but the physicists do not have that kind of luxury provided they want to remain realistic in their constructs.
We are lucky to have the fixed stars, not so much because of their "fixed" nature, which makes them a reliable anchor for our reference systems, but mostly because of the fact that no one really knows what would happen to the notion of inertial reference system if the fixed stars were suddenly to disappear.
An easy way to see the stark difference between the notions of relative velocity on the one hand, and the notion of rate of change of the separation between two bodies on the other, is to think of two material points of equal mass m moving along two separate, but parallel lines in opposite directions with equal speed u. The relative speed of one point with respect to the other is then a constant equal to 2u. What about the rate of change of the separation of these points? It is neither equal to 2u nor even constant!
An observer, located at material point 1 and deprived of the fixed stars background for orientation in space, would see the material point 2 approaching him with deceleration until the separation of the points gets equal to the distance between the parallel lines along which these points are moving. At that moment, the rate of change of the separation of the moving material points is zero, i.e. it would seem to the observer that the material point 2 has reached a point of rest for a split of a second!
We can see now how the contours of a second logical problem start to emerge: we cannot dispose of the context of fixed stars and keep at the same time our formula (1) for kinetic energy, which seemed perfectly relational at first glance.
A relational formula should be able to do away with the fixed stars altogether and it should be expressible in truly relational terms, i.e. in terms of the configuration of the system and rates of change of separations between the material points that comprise the system without any reference whatsoever to the fixed stars, or a notion such as inertial reference frame.
Are there any suggestions or ideas towards the resolution of the logical difficulties described here and in the previous comment? The answer is yes and it has to do, not surprisingly perhaps, with the concept I am trying to advance - the concept of material point with rotation.
It is good enough that we get approximate results from our calculations.
However our arrogance has made our theoretical models king over reality! So we expect nature to obey our laws!
LaGrange I believe said there is no time only velocity! Indian Sages have also said matter is an illusion so mass by all accounts is illusory.
Thus kinetic energy is a formal illusion that we mentally construct . In addition we are constructing a measure that requires a calculation of more fundamental measures. What we measure is motion. That motion we experience whatever happens to the measure.
Kinetic energy resides in our heads. All we observe is the effect of motions as our senses convey that to our internal processes.
Is this formal measure useful? Yes in its approximate results, because it focuses our attention on some observable relationships that we can learn by rote. However like boiling water these calculations break down when conditions are not met. In that case a new empirical investigation is required to construct a more general calculation .
The relationship between the external things and their reflections in human brain is the subject of epistemology. There are certain requirements to the formation of these images if they are to be effective and accurate in the anticipation of future events, so that we may arrange our present affairs in accordance with such anticipation.
In the introduction to The Principles of Mechanics Hertz sets the stage for those requirements:
The epistemology of what we know is what I mean by it is in our heads. I am, like Hertz an empiricist, which as he describes means I adopt a foundational other "reality". But this is a Socratic or Platonic choice.
Socrates and Plato played a game. The game has to do with the philosophy of ideas/ forms. The participants in the game have to choose one side or the other: are forms independent of our conscious perception or are ideas the progenitor of forms?
Of course you do not have to choose, but then you are left wondering.
If you choose you are still left in the position of asserting your reality! That is the nature of the game. Everything in your reality flows from that choice.
So I choose to accept some form and relations and dynamics I call work or energy. But all I perceive are forms in motion and the impact of those forms on others including my senses. The models we make of energy are constructed from measures we design and define. In this precise metrical sense these concepts are in our heads. Instead what we experience we usually call force or heat or electric shock or radiation.
So the question about where kinetic energy might reside is a difficult conceptual question.
Your analysis reworks the way we construct the calculated measure is a worthwhile exercise. Because as Hertz says, the best fit should be the model we use!
K(1;2) = m1*m2*(V1 – V2)^2/[2(m1 + m2)], (V1 and V2 are vectors here!)
can be generalized still further to account for the kinetic energy relationship between any number of material points. The most natural way of doing this suggests itself at once.
Let there be any number of material points m1, m2, m3, ... each moving uniformly with absolute velocity Vi, i = 1, 2, 3, ... correspondingly. Now, consider the collision of material points m1 and m2 forming a single material point of mass m1 + m2 and velocity (m1*V1 + m2*V2)/(m1 + m2). Next, consider the collision of this newly created object with material point m3 forming a single material point of mass m1 + m2 + m3 and velocity (m1*V1 + m2*V2 + m3*V3)/(m1 + m2 + m3), and so on until we end up finally with a single material point of mass M = m1 + m2 + m3 + ...
The combined kinetic energy relationship between all material points then obviously boils down to the following sum:
K(1; 2; 3; ...) = K(1; 2) + K((1, 2); 3) + K((1, 2, 3); 4) + ...
If we do the math right, we'll get a result that can easily be proved by mathematical technique called induction:
K(1; 2; 3; ...) = Σmi*mj*(Vi – Vj)^2/(2M),
where summation is to be done pairwise for all j>i, and M = m1 + m2 + m3 + ... is the total mass of all material points.
Note that our final result does not depend on the particular order in which we arrange the material point collisions that lead to the formation of a single object. That's the way it should be, of course, otherwise the generalized concept of kinetic energy advocated here would make no sense.