Homework Help: Existence of the 3rd law

1. Jan 12, 2005

UrbanXrisis

how does the impulse momentum theorem and conservation of momentum show the existence of the third law of motion?

impulse momentum theorem- the force of an object a that hits object b will create a force. The force that object b exerts back on object a should be the same as the force object a exerts on object b. So the impulse should be equal in magnitude.

conservation of momentum-since the sum of the initial velocities of both objects are equal to the sum of the final velocities. Therefore, there is the same amount of force exerted back on each objects.

did I explain this right?

2. Jan 12, 2005

UrbanXrisis

any suggestions?

3. Jan 12, 2005

vincentchan

Force is the derivative of momentum....so..... conservation of momentum has no different from newtons 3rd law

4. Jan 12, 2005

quasar987

5. Jan 12, 2005

Curious3141

What Vincent is saying but more formally.

Consider a closed system of two bodies A and B, with no other forces acting on the system. Let the momentum of A be $\vec{p_A}$ and that of B be $\vec{p_B}$.

Now let us say the two bodies interact such that the momentum of A changes by $\Delta \vec{p_A}$ and that of B by $\Delta \vec{p_B}$.

By the law of conservation of linear momentum,

$$\Delta \vec{p_A} + \Delta \vec{p_B} = 0$$ since no net force acts on the system.

Hence $$\Delta \vec{p_A} = - \Delta \vec{p_B}$$

Take the first differential with respect to time.

$$\frac{d}{dt}(\Delta \vec{p_A}) = - \frac{d}{dt}(\Delta \vec{p_B})$$

But by Newton's second law, the derivative of momentum w.r.t. time is equal to force.

Hence $$\vec{F_A} = - \vec{F_B}$$

and we've proved Newton's third law.

6. Jan 12, 2005

UrbanXrisis

what about the impulse momentum theorem?

7. Jan 12, 2005

dextercioby

Don't be mislead.U cannot prove the thord law...It is an axiom.The theorems regarding the conservation of total energy and momentum in isolated systems are proved via the three laws.So what the previous posters tried to show is conceptually and logically garbage.

Daniel.

8. Jan 12, 2005

vincentchan

What is axiom? and what is fundamental law?
if you DEFINE conservation of momentum is more FUNDAMENTAL than 3rd law.. you can say 3rd law can PROVED by conservation of momentum.. however, if you DEFINE 3rd law is more FUNDAMENTAL.. sure, conservation of momentum can proved by 3rd law..
In mathematics, this relation is called equalvance.. none of them are more fundamental, they are the same thing which appears differently..
In physics, we usually called the conservation law is more fundamental..

9. Jan 12, 2005

dextercioby

I surely as hell resent those pleonastic expressions.Things are fundamental and that's it.No more,no less.
Conservation laws are not fundamental.They appear in ideal situations,usually unencountered in nature.I'm sure u weren't aware of that,else u would have deleted your post.
While principles/postulates/axioms,well,they are the basis.Newton's laws go along with the principles of QM,SR,GR,Thermodynamics,Statistical Physics,QFT into the axiomatic structure of physics,of major theories in physics.

One cannot have equivalence between a consequence seen a fortunate particular case of an axiom and the axiom...Elementary logics...I believe science is based on logics...

Daniel.

Last edited: Jan 12, 2005
10. Jan 12, 2005

UrbanXrisis

impulse momentum theorem:
I=Ft
I/t=F
F=F

This shows the relationship between the 3rd law and the theorem correct?

11. Jan 12, 2005

Curious3141

It's not "garbage" as you so eloquently put it. There is a logical dependence of the third law on the law of conservation of momentum and the second law. Of course, you can argue the other way, saying the three laws came first and conservation of momentum is a consequence. But that's just the chicken and the egg.

There are no true axioms in physics, because it is an empirical system. Physical "laws" are statements we make based on inductive reasoning to try and explain the observations of many, many repeated experiments. Just because it is a "law" does not mean it cannot be disproved by a completely unexpected observation 10 years down the line. Contrast this with a mathematical axiom which is truly unfalsifiable because it is simply defined that way and the logical structure is built from the ground up. Your trying to equate physical law to an axiom is fallacious.

12. Jan 12, 2005

dextercioby

No.I'll say it again and will say it whenever necessary:there are rules imposed by logics.The only valid methods would imply determining the laws of momentum and energy conservation starting with the fundamental laws of Newton...

Daniel.

13. Jan 12, 2005

Curious3141

And why should the conservation laws not be the "fundamental" ones and the laws of Newton derive thereof ?

14. Jan 13, 2005

dextercioby

It's true.There is:the three laws imply everything at classical macroscopical level.

Let's leave analogies and collateral discussions to the phylosophers,okay??They have to make a living,right??

False is the conclusion,true is the hypothesis...I would advise you to read more.Science needs theory.Theory is logics and mathematically based.Axioms are a part of both....

True,however modern theoretical physics doesn't have the empirical basis anymore.Yet it's unanimously accepted,simply because it lack the only the only way of falsifying...

True,science in general is full of theories overturned...

Really??Compare that to the fact:by definition,force is the time derivative of momentum.Is there any difference??Definitions are in a sense axioms... You mean physics does not have "logical structure (...) built from the ground up"??? :surprised What planet have you been living on??

Nope.It's irrefutable.If experiment falsfies the results of a theory,then its axioms were wrong.However,we can find a new theory...New axioms,new theorems,new sylogisms and this time confirmation of results...

Daniel.

15. Jan 13, 2005

dextercioby

Logics??One of them is a consequence of the other,seen a restriction???Particular case,perhaps??I believe i said that before...Can u derive Schroedinger's equation using the equation of continuity for the localization probability density???

Daniel.

Last edited: Jan 13, 2005
16. Jan 13, 2005

Curious3141

OK, derive the principle of conservation of energy from Newton's three laws of motion.

Not just conservation of kinetic energy, but of all forms, including the first law of thermodynamics.

Conservation of energy IS a fundamental proposition in Physics. It is not logically dependent on Newton's three laws for its truth.

So you see, there are other fundamental propositions upon which Physics rest, it's not just Newton's three laws.

BTW, "fundamental" is a rather nebulous word, especially when we're trying to grade it. Conservation of momentum is to me, just as fundamental as Newton's three laws. As I think I have shown, conservation of momentum is actually equivalent to the 2nd and third law of Newton. The major reason for splitting up the law into two in Newton's formulation is to define the concept of a force as a rate of change of momentum. Force is no more fundamental than momentum, the former is just the first differential of the latter wrt time.

We ARE discussing the philosophy of science. You opened the door, remember ?

A mathematical axiom is FUNDAMENTALLY DIFFERENT from a physical "axiom". A mathematical axiom can exist with totally *no* reference *whatsoever* to the "real world" of physical objects. A physical "axiom" or law or principle absolutely NEEDS to say something about something in the real world. Don't you see this very basic difference ?

Of course it does !!! Just because QM involves a lot of fancy mathematics doesn't mean the conclusions can stand unsupported by empirical verification. Why do you think so much money is spent building humongous particle accelerators and colliders, if not to verify the fruits of mathematical searching ?

Even the most abstruse theory, like the ones that talk about superstrings and what not, are constructed to lend mathematical coherence to a greater model that CAN be verified in the real world. If a theory makes absolutely no predictions that can either directly, or through ramifications thereof, be falsified, then it is not a scientific theory and has no merit.

1st proposition : F = dp/dt

2nd proposition : 1 is a natural number.

The 1st proposition defines Force. It depends on the concept of momentum for its definition. However, forces can be measured by physical instruments, as can momentum, so they are both "real world entities". It is remotely possible that 50 years from now, an extremely subtle physical effect becomes apparent that changes this relationship. This would negate the proposition.

This is not so far fetched. If one were to look back to the 19th Century and question the validity of "F = ma = m(dv/dt)", one would be made a laughing stock. Of course, now we know that isn't correct, since mass alters with motion. Even though F = dp/dt is still correct at the moment, can you really rule out another paradigm shift in thought in the future ?

The 2nd proposition is Peano's first axiom. It relates two concepts : "1" and "natural number". Neither of these has any real meaning in the "real world". Although we can define the cardinality of a real world set using the naturals, the mathematical definition does not depend on the nature of the physical world for its veracity. The whole Universe can dissolve away tomorrow, leaving only one self aware particle and the particle can still cognize the successor of one, two, even though it cannot find anything to count using this new number.

Let's not assume airs here. Physics is much more bread and butter than mathematics, and is far more pragmatic. This need not be pragmatic in the sense of inventing new machines to do stuff, but can also be the pragmatism of needing to tie up observations of the physical world into a coherent theory. Mathematics can exist solely for the pleasure of an arbitrary game of the imagination.

You've just proved my point ! Your physical "axioms" are simply tentative truisms that can be falsified ! Mathematical axioms can NEVER be falsified. They can be undecidable in a theoretical framework (like the Axiom of Choice in ZF set theory) but cannot be proven "wrong", simply because they are by definition "right". Always and forever.

I'll get right on it. In the meantime, read more formal logic.

Last edited: Jan 13, 2005