# Thread: Contextual Relationships Between Momentum, Energy, and Force.

I've had long running difficulties with these classical concepts. Not to say I can't apply or use them. I've taken three different mechanics courses (due to bureaucratic reasons) and gave excellent performances each time. However, I still maintain a feeling of unease as to what I'm actually dealing with.

All the instruction I've ever had gave defininitonal based explanations: (net) F = ma, momentum = mv, and energy is work and work is energy (this is also a point of confusion). 'These things are what they are because that's how they are defined.' I don't want definitions, I want derivations. Not in a strict mathematical sense but conceptually. I understand these are rather nebulous demands, so I will demonstrate my current train of thought.

Noticing that Newton defined Force in terms momentum, I like to use momentum as the base for all the other concepts. I'll start with the assumption: Within a closed system, the amount of mass and the amount of velocity associated with that mass never change. Closed system in this case defined as a selection of particular masses that are imagined as the only masses in existence (no 'outside masses act on the 'inside' masses in any way). I love this starting point because it's simple and intuitive. Perhaps you could argue against the phrase "amount of velocity" but I feel it is sufficient.

Because of the use of "associated" with our assumption, velocity is added only as many times as there is mass that holds that velocity and vice-versa. So a quantity of velocity is added m times, mathematically stated as m*v. We have now arrived at the textbook definition of total momentum: mtotal*vassociated total = constant. Good job guys.

If our closed system were to be breached and momentum altered without adding mass, (classically speaking) velocity is the only quantity subject to change. The derivative of velocity is acceleration, therefore the change in momentum would be m*a and is given the name "Force". This brings into question statements of 'forces' being applied but canceling out.

If force is the change in momentum, can a force exist if there is no change? Momentum can be separated into known components because that is our starting point; our initial condition's are the masses and their velocities. It is from there we derive notions of momentum. Our reasoning doesn't allow for reverse engineering, starting from momentum and then deriving masses and their velocities. Now apply that idea to force. Given simply just force (or momentum) allows for infinite possibilities. So the statement "a force is applied" is false, we can only say "a force is noted". Force is a derived term. In a closed system of a ball and the earth, if the ball were to be at rest on the earth (ignoring internal changes) textbooks would say there are two forces: gravity down and normal force upwards, canceling out. Why can we say that? Under that logic, couldn't there be an infinite combination of forces? Maybe one leftwards and one identical in quantity rightwards. My current reasoning allows for forces to exist only when there is a change in momentum. Text books, too, only define net force, anything other doesn't seem to technically exist. What I want from you, Internet, is a logical thought process that satisfies current conventions.

Furthermore, energy mathematically seems to be the integral of momentum with respect to time m*v to (m*v2/2). But energy is also constant in a closed system which doesn't hold when momentum's constant is integrated with respect to time. Maybe I should have started with energy as the base concept and momentum and force as the derived concepts, I don't know. Also work seems to be the change of energy, via force (W = integral of f*dx). Force represents a type of change in energy and momentum but I'm not to sure on how to explicitly state this.

These are my problems with classical mechanics. I suppose the heart of my difficulties lies in what are fundamentally derived concepts and what are initial statements, assumptions, or definitions. I struggle with momentum, energy, and force because they all seem so tauntingly related but I can't quite glue them together myself.

I am grateful for any enlightenment

2. Originally Posted by Cameron Blake

I've had long running difficulties with these classical concepts. Not to say I can't apply or use them. I've taken three different mechanics courses (due to bureaucratic reasons) and gave excellent performances each time. However, I still maintain a feeling of unease as to what I'm actually dealing with.

All the instruction I've ever had gave defininitonal based explanations: (net) F = ma, momentum = mv, and energy is work and work is energy (this is also a point of confusion). 'These things are what they are because that's how they are defined.' I don't want definitions, I want derivations. Not in a strict mathematical sense but conceptually. I understand these are rather nebulous demands, so I will demonstrate my current train of thought.

Noticing that Newton defined Force in terms momentum, I like to use momentum as the base for all the other concepts. I'll start with the assumption: Within a closed system, the amount of mass and the amount of velocity associated with that mass never change. Closed system in this case defined as a selection of particular masses that are imagined as the only masses in existence (no 'outside masses act on the 'inside' masses in any way). I love this starting point because it's simple and intuitive. Perhaps you could argue against the phrase "amount of velocity" but I feel it is sufficient.

Because of the use of "associated" with our assumption, velocity is added only as many times as there is mass that holds that velocity and vice-versa. So a quantity of velocity is added m times, mathematically stated as m*v. We have now arrived at the textbook definition of total momentum: mtotal*vassociated total = constant. Good job guys.

If our closed system were to be breached and momentum altered without adding mass, (classically speaking) velocity is the only quantity subject to change. The derivative of velocity is acceleration, therefore the change in momentum would be m*a and is given the name "Force". This brings into question statements of 'forces' being applied but canceling out.

If force is the change in momentum, can a force exist if there is no change? Momentum can be separated into known components because that is our starting point; our initial condition's are the masses and their velocities. It is from there we derive notions of momentum. Our reasoning doesn't allow for reverse engineering, starting from momentum and then deriving masses and their velocities. Now apply that idea to force. Given simply just force (or momentum) allows for infinite possibilities. So the statement "a force is applied" is false, we can only say "a force is noted". Force is a derived term. In a closed system of a ball and the earth, if the ball were to be at rest on the earth (ignoring internal changes) textbooks would say there are two forces: gravity down and normal force upwards, canceling out. Why can we say that? Under that logic, couldn't there be an infinite combination of forces? Maybe one leftwards and one identical in quantity rightwards. My current reasoning allows for forces to exist only when there is a change in momentum. Text books, too, only define net force, anything other doesn't seem to technically exist. What I want from you, Internet, is a logical thought process that satisfies current conventions.

Furthermore, energy mathematically seems to be the integral of momentum with respect to time m*v ? (m*v2/2). But energy is also constant in a closed system which doesn't hold when momentum's constant is integrated with respect to time. Maybe I should have started with energy as the base concept and momentum and force as the derived concepts, I don't know. Also work seems to be the change of energy, via force (W = integral of f*dx). Force represents a type of change in energy and momentum but I'm not to sure on how to explicitly state this.

These are my problems with classical mechanics. I suppose the heart of my difficulties lies in what are fundamentally derived concepts and what are initial statements, assumptions, or definitions. I struggle with momentum, energy, and force because they all seem so tauntingly related but I can't quite glue them together myself.

I am grateful for any enlightenment
Try reading the first 10 or pages of Goldstein's Classical Mechanics. That should show you what is assumed (Newton's three laws), and what is derived (everything else). You can argue a bit over whether the second law is really a law of mechanics or a definition of force, but I don' t think that is where your issues lay..

3. Originally Posted by Cameron Blake
All the instruction I've ever had gave defininitonal based explanations: (net) F = ma, momentum = mv, and energy is work and work is energy (this is also a point of confusion). 'These things are what they are because that's how they are defined.' I don't want definitions, I want derivations. Not in a strict mathematical sense but conceptually. I understand these are rather nebulous demands, so I will demonstrate my current train of thought.
There are three basic forms of quantities/entities and forms of knowledge in physics. The first of these are concepts which remain undefined but which are relayed to people through description from everyday experience as well as from use. Three of these come to mind. They are space, time and energy. There exist no proper definitions of these three concepts. Oh, you’ll see people define them but I can promise you that they’re flawed in some way or another. Then there are those things which are defined and then those things which are proved and then those things which we obtain from observing nature. Quite often it’s a combination of these. An example is force. The idea of force comes from the law of inertia which states that an object at rest stays at rest and an object in motion stays in motion unless acted upon by a force. This hinges on the concept of an inertial frame and the inertial frame hinges on the law of inertia. This is a well-known circularity. Mass is defined such that the quantity mv is conserved for a closed system. You’re supposed to know what a closed system is, i.e. its an undefined term … I think. . Momentum is then defined as p = mv and force is quantitatively defined as F = dp/dt. Newton defined “quantity of motion” to be mv where v is speed. We now know that only works best when v is a vector. After that you’re given laws of physics that are relationships between physical quantities that are assumed to hold true under certain conditions. For example; Newton’s third law states that where there is a force there is always an equal and opposite force. For instance if a body A exerts a force F on body B then body B exerts the force –F on body A. For this rule to be exact and hold in all cases the forces must be contact forces.

Then there’s the definition of potential energy V such that F = -grad V.

Then there’s the conservation of mass of a closed system. Things get more involved in relativity and quantum mechanics.

I hope that’s good enough for a start

4. Originally Posted by PhyMan
There are three basic forms of quantities/entities and forms of knowledge in physics. The first of these are concepts which remain undefined but which are relayed to people through description from everyday experience as well as from use. Three of these come to mind. They are space, time and energy. There exist no proper definitions of these three concepts. Oh, you’ll see people define them but I can promise you that they’re flawed in some way or another. Then there are those things which are defined and then those things which are proved and then those things which we obtain from observing nature. Quite often it’s a combination of these. An example is force. The idea of force comes from the law of inertia which states that an object at rest stays at rest and an object in motion stays in motion unless acted upon by a force. This hinges on the concept of an inertial frame and the inertial frame hinges on the law of inertia. This is a well-known circularity. Mass is defined such that the quantity mv is conserved for a closed system. You’re supposed to know what a closed system is, i.e. its an undefined term … I think. . Momentum is then defined as p = mv and force is quantitatively defined as F = dp/dt. Newton defined “quantity of motion” to be mv where v is speed. We now know that only works best when v is a vector. After that you’re given laws of physics that are relationships between physical quantities that are assumed to hold true under certain conditions. For example; Newton’s third law states that where there is a force there is always an equal and opposite force. For instance if a body A exerts a force F on body B then body B exerts the force –F on body A. For this rule to be exact and hold in all cases the forces must be contact forces.

Then there’s the definition of potential energy V such that F = -grad V.

Then there’s the conservation of mass of a closed system. Things get more involved in relativity and quantum mechanics.

I hope that’s good enough for a start
Interesting perspective, but quite irrelevant to the question posed by the OP which is directed toward classical (Newtonian) mechanics. In that case force is more or less defined by the second law (rather loosely defined at that), but given the notion of force, energy is well-defined in terms of work, as is momemtum if one accepts knowledge of mass (which in the Newtonian sense is what you measure with a laboratory balance).

There is no question that in this context momemtum is mv and v is ALWAYS a vector. So your statement that "Newton defined “quantity of motion” to be mv where v is speed. We now know that only works best when v is a vector." is somewhere between confusing and nonsensical.

It is probably best to start with understanding the context of the question, in this case classical Newtonian mechanics.

5. Originally Posted by DrRocket
Interesting perspective, but quite irrelevant to the question posed by the OP which is directed toward classical (Newtonian) mechanics.
Since my responses were exclusively with respect to Newtonian Mechanics your comment makes no sense.
Originally Posted by DrRocket
In that case force is more or less defined by the second law (rather loosely defined at that), ..
That is incorrect. Newton's second law contains a well-known problem in circular logic. It requires the first law that in turn requires the definition of an inertial frame. However the inertial frame is defined in terms of the lack of force acting on a body.

Sir Arthur Eddington famous parody on Newton's first law reads Every body continues in a state of rest or uniform motion in a right line in as far as it doesn't.

For those of you who are interested in this fact please see On force and the inertial frame by Robert W. Brehme. Am. J. Phys., 53, 952 (1985)
Abstract - The logical difficulty surrounding the definition of an inertial frame and Newton’s first law of motion can be circumvented by defining the inertial frame in terms of its spatial and temporal properties, as embodied in the special theory of relativity. Whether or not these properties apply can be determined, in principle, by experiment. The mass of a body is measured through a completely inelastic collision with a body whose mass is taken as the standard. Rather than attempt to define force by a means not involving Newton’s second law, we use the second law as the definition.
Originally Posted by DrRocket
..energy is well-defined in terms of work, ....
That is incorrect. There is nothing about the definition of energy that defines energy. Work has units of energy and the work done on a particle changes the kinetic energy K of the particle. However there is a huge difference between that and the concept of energy. There is a theorem that if force is defined in terms of a potential energy function V which is not an explicit function of time then the quantity E = K + V is a constant. However that is merely the total mechanical energy and not energy itself. All the forms of energy are quite well defined. The sum of all forms then being conserved for a closed system. But what it actually is, beyond a number which remains constant (i.e. a bookkeeping system) is unknown and undefined.

Richard Feynman does a great job explaining all of this in his Lectures V-I section 4-1. You'd have to read the entire section to understand what I'm about to quote so for those who comment on the quote without reading it is not doing a complete job. I only quote this part to motivate you to read what Feynman says about it. You can then freely ignore it all after that.

From The Feynman Lectures on Physics Volume I section 4-1
It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and we add it all together it gives “28” - always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.
There are very good reasons for what he concludes here. A.P. French reaches the same conclusion in his text Newtonian Mechanics The MIT Introductory Physics Series pages 367-368. For those interested in what French says about this please let me know in PM since I’ve lost interest in this thread.

I can e-mail this to anyone who wishes to read it.

Originally Posted by DrRocket
There is no question that in this context momemtum is mv and v is ALWAYS a vector.
Umm .. I already said that so I don’t see the need to repeat me. Perhaps you didn’t read my post closely where I clearly wrote Newton defined “quantity of motion” to be mv where v is speed. I.e. that is the way Newton used it. I didn't say anything beyond that besides we now know that v is a vector quantity and in this way mv is a conserved quantity. For those of you who want to know more on this point then I refer you to Comments on ‘‘Newton revisited on ‘Quantity of motion’’’ by A.P. French, Am. J. Phys., 53, 499 (1985)

Originally Posted by DrRocket
So your statement that "Newton defined “quantity of motion” to be mv where v is speed. We now know that only works best when v is a vector." is somewhere between confusing and nonsensical.
That's probably because you didn’t read my post closely enough.

6. Originally Posted by PhyMan
Since my responses were exclusively with respect to Newtonian Mechanics your comment makes no sense.
If that were true there would have been no (inaccurate) comments on quantum mechanics and relativity regarding conservation of momentum.

Originally Posted by PhyMan
That is incorrect. Newton's second law contains a well-known problem in circular logic. It requires the first law that in turn requires the definition of an inertial frame. However the inertial frame is defined in terms of the lack of force acting on a body.
No it is not incorrect. Newton's second law is a much a definition of force as it is anything else. No one has suggested that Newton's laws are a mathematically tight system of axioms. In fact physics itself has defied axiomatization since the problem was first proposed by Hilbert.

Originally Posted by PhyMan
Sir Arthur Eddington famous parody on Newton's first law reads Every body continues in a state of rest or uniform motion in a right line in as far as it doesn't.

For those of you who are interested in this fact please see On force and the inertial frame by Robert W. Brehme. Am. J. Phys., 53, 952 (1985)
Irrelevant

Originally Posted by PhyMan
That is incorrect. There is nothing about the definition of energy that defines energy. Work has units of energy and the work done on a particle changes the kinetic energy K of the particle. However there is a huge difference between that and the concept of energy. There is a theorem that if force is defined in terms of a potential energy function V which is not an explicit function of time then the quantity E = K + V is a constant. However that is merely the total mechanical energy and not energy itself. All the forms of energy are quite well defined. The sum of all forms then being conserved for a closed system. But what it actually is, beyond a number which remains constant (i.e. a bookkeeping system) is unknown and undefined.
That is just plain wrong.

In the context of Newtonian mechanics, once a reference frame has been selected in which to measure energy the notion of kinetic energy is defined in terms of the work performed and conservation of both energy and momentum fall out of the laws rather easily.

Originally Posted by PhyMan
Richard Feynman does a great job explaining all of this in his Lectures V-I section 4-1. You'd have to read the entire section to understand what I'm about to quote so for those who comment on the quote without reading it is not doing a complete job. I only quote this part to motivate you to read what Feynman says about it. You can then freely ignore it all after that.

From The Feynman Lectures on Physics Volume I section 4-1

There are very good reasons for what he concludes here. A.P. French reaches the same conclusion in his text Newtonian Mechanics The MIT Introductory Physics Series pages 367-368. For those interested in what French says about this please let me know in PM since I’ve lost interest in this thread.

I can e-mail this to anyone who wishes to read it.
Feynman is right. But you are wrong again. You have taken Feynman out of context.

Feynman is not working within the confines of Newtonian mechanics, but rather is speaking to the entire body of knowledge of physics, and he is presenting a valid philosophical stance in that context.

However, in the more confining space of Newtons laws and their implications aka Classical Mechanics the notion of energy is neither so pervasive (note the Feynman mentions heat energy, elastic energy, electrical energy, chemical energy, radiant energy, nuclear energy, and mass energy).

Originally Posted by PhyMan
Umm .. I already said that so I don’t see the need to repeat me. Perhaps you didn’t read my post closely where I clearly wrote Newton defined “quantity of motion” to be mv where v is speed. I.e. that is the way Newton used it. I didn't say anything beyond that besides we now know that v is a vector quantity and in this way mv is a conserved quantity. For those of you who want to know more on this point then I refer you to Comments on ‘‘Newton revisited on ‘Quantity of motion’’’ by A.P. French, Am. J. Phys., 53, 499 (1985)

That's probably because you didn’t read my post closely enough.
Incorrect.

Read the Principia and you will find that Newton defines momentum as a vector, albeit in somewhat older language -- remember that elementary calculus had barely been invented.

7. Originally Posted by DrRocket
If that were true there would have been no (inaccurate) comments on quantum mechanics and relativity regarding conservation of momentum.
That is incorrect. In Newton's presentation of the concept in the Principia he effectively defined quantity of motion as mass times speed. This was the precursor of what we now call momentum and defined as mass times velocity, i.e. a vector quantity. This is all explained in the article by A.P. French in the American Journal of Physics. In fact that's why he wrote the article, i.e. to explain what I just tried to get across to you.

(snipped similar responses and assertions)

DrRocket - Please clarify something for me if you please. I have chosen to provide all these references in order to clarify and expound on, clarify and in some cases give the source of whatever point I’m making. I do this in order to do my best at explaining the physics to people. If you’ve decided beforehand not to read anything I reference in any instance then I will thank you to do me the kind favor of stating that now so that I may save myself a great deal of work in the future.
Originally Posted by DrRocket
In fact, on this point, I not only read the part which I'm talking about but I went beyond that and read A.P. French's article in the American Journal of Physics and you'd understand what I was explaining to you if you chose to read it as well.

8. From Newtonian Mechanics A.P. French, The M.I.T. Introductory Physics Series, W.W. Norton & Company, Inc., (1971). Chapter 10 Energy conservation in dynamics; vibrational motions, pages 367-368
[quote]
Of all the physical concepts, that of energy is perhaps the most far-reaching. Everyone, whether a scientist or not, has an awareness of energy and what it means. Energy is what we have to pay for in order to get things done. The word itself may remain in the background, but we recognize that each gallon of gasoline, each Btu of heating gas, each kilowatt-hour of electricity, each car battery, the wherewithal for doing what we call work. We do not think in terms of paying for force, or acceleration, or momentum. Energy is the universal currency that exists in apparently countless denominations.
The above remarks do not really define energy. No matter. It is worth recalling once more the opinion that H.A. Krammers expressed: “The most important and fruitful concepts are those to which it is impossible to attach a well-defined meaning.” The clue to the immense value of energy as a concept lies in its transformation. It is conserved – that is the point. Although we may not be able to define energy in general, that does not mean that it is only a vague, qualitative idea. We have set up quantitative measures of various specific kinds of energy: gravitational, electrical,

As an example to illustrate French, Feynman and my point, i.e. what it means for physicists not to know what energy is, I'll give a clear example: consider the electric and magentic field of the electron. The electron is basically a point charge which has a magnetic dipole moment. If one were to calculate the Poynting vector it would show that there was a circular flow of energy. The energy density is constant though.

It's for reasons like this that nobody can logically define what exactly it is that is flowing. The only thing that can be done is to give it a name and examples to clarify the point.

This is analogous to the term set in mathematics. That too is a term that is undefined.

9. "energy is work and work is energy (this is also a point of confusion)".

No, work is a measure of change in energy... the work done ON or by.. (of course work used in *breaking" ,eg breaking the bolts, isn't then found in the kinetic energy of the system.. )

"Within a closed system, the amount of mass and the amount of velocity associated with that mass never change"

FALSE

Its the SUM OF mass * velocity , that never changes.
Each velocity is a vector, and the sums and additions are vector operations, the result is a vector too !

The difference between a scalar (eg weight) and vector ( eg velocity) is that direction of each is important.

You may break these down into x y and z dimensions (if needed), so that the velocity is a real number for each, and then the rule is true for each dimension too, the sum will remain constant.

Energy is the sum of the 1/2 m speed ^2 (speed is a scalar), momentum is the sum of mass times velocity ( velocity is a vector)

Thanks to the other people arguing about tangents way above the OP's scope....

10. Originally Posted by Isilder
"energy is work and work is energy (this is also a point of confusion)".

No, work is a measure of change in energy... the work done ON or by.. (of course work used in *breaking" ,eg breaking the bolts, isn't then found in the kinetic energy of the system.. )
Yep. Quite correct.

Originally Posted by Isilder
"Within a closed system, the amount of mass and the amount of velocity associated with that mass never change"

FALSE

Its the SUM OF mass * velocity , that never changes.
Each velocity is a vector, and the sums and additions are vector operations, the result is a vector too !
If you're ignoring fields and talking only about particles then yes, the sum of the relativistic masses of a system is constant.

Originally Posted by Isilder
Energy is the sum of the 1/2 m speed ^2 (speed is a scalar), momentum is the sum of mass times velocity ( velocity is a vector)
I think that you're referring only to the mechanical energy of a particle. There are many forms of energy but energy itself is undefined. There is also electromagnetic energ, rest energy and thermal energy to name but a few more. We give the name energy to what is in circular flow around an electron due to the Poynting vector calculated from the electrons electric field and its magnetic dipole moment. But nobody reallly knows what it is. As I quoted Feynman as stating above. i.e.

It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount. It is not that way.
Underline is mine. He explains in the text itself why there is no definition that is satisfactory as does A.P. French.