Two balls of masses m1 and m2 approach each other at velocities v1 and v2 relative to a stationary observer. They collide and bounce, so that they are now moving apart at velocities w1 and w2. What is known is that total momentum is conserved. Suppose we do not use the conservation of energy as a fact, what other condition must be true so that we can compute w1 and w2, given m1, m2, v1, v2? Can this be worked out without using energy at all, kinetic or otherwise?
In other words, how did Isaac Newton, the father of classical mechanics, do this?
Update:
All action is colinear, to keep things simple.
Update 2:
Questor, Isaac Newton was able to do that himself using his own laws. So, how did he do it? He never used the concept of kinetic energy or conservation of energy.
Update 3:
First Grade Rocks!, there is a fine English translation of Newton’s Mathematica Principia. Look up Andrew Motte.
Update 4:
The answers I’ve received here so far are some of the best responses to any of my Y!A questions. I thank all of you enormously, because I wanted to see the different ways how the concept of energy, however extremely useful it is, can be replaced with something else. At least for elementary classical mechanics. Now, one of the answers here is very much like how Isaac did it, but I won’t say which until later.
Update 5:
As an extreme example of “replacing one concept with another”, some of you might have heard of John Barbour’s “End of Time”, where he reformulates general relativity that makes no use of time at all as a variable. He then argues that “time is illusionary”. I argue that we have a choice of using time as a parameter, or not. Neither is “more valid”, because–after all–he did demonstrate that both views are mathematically equivalent.
Update 6:
Several answers here have at least suggested two conditions that must be true, if no mention of energy is to be made. 1) The motion of the center of mass of the system remains uniform, 2) results are same under time-reversal. In fact, conservation of energy is a consequence of “symmetry under time” as per Noether, which ties in with 2), but Newton (and perhaps even early Einstein) was not aware of the connection between symmetry and conservation laws. Also per Noether, conservation of momentum is the consequence of “symmetry under translation”.
Update 7:
Zo Maar, precisely because of difficulties like inelastic collisions it took centuries after Newton’s Principia before physicists got around to perfecting the concept of energy and its conservation. So, we’ll try to limit this discussion to ideal collisions.
Update 8:
It is a paradox that Newton successfully tackled curvilinear motion in a force field long before there was a methodology to handle ideal collisions Unless you read his book through, you’d never guess that from reading school texts on “Newtonian Physics”. It’s a bit of a misnomer.
Update 9:
It is very tempting to argue that “things ought to look the same going either direction in time”, i.e. time reversibility, as a primitive version of Noether’s Theorem. Then the next argumentive step is to say that balls rebound from the meeting point which is the center of gravity of the system in exactly reverse manner of the approach. That would work. However, it’s difficult to expand this argument for more general cases, which is the problem with this approach. Newton didn’t use this argument.
Update 10:
In Newton’s Principia, he worked entirely in the context of force between bodies, which is solely dependent on the distance between them. The necessary condition is that the force field be conservative, i.e., dependent only on distance and not vary in time. This is the stronger constraint, closer to Noether’s Theorem. Newton never treated the problem of “colliding billiard balls” in fact, but two objects coming from infinite distance with a conservative repulsive force present between them would be all he needed to work out the rebound. I know that this perfectly describes an elastic collision, and I did say that this thread should be limited to such ideal collisions.
Update 11:
By integrating this force over time, we end up with the conservation of momentum law. By integrating this force over distance, we end up with the conservation of energy law, even though that was never mentioned in the Principia. But Newton’s contemporaries had noticed that. Nonetheless, it took well over a century before physicists took the idea of energy conservation seriously for more general cases such as inelastic collisions and thermodynamics.
Update 12:
Zo Maar, I see your additional comments. I’m about ready to pick your answer as BA because you’ve gone farther in analyzing this question, but I’ll hold of a bit longer to see what else comes up between anyone.
Update 13:
I should also add that a conservative force between bodies depends only on the distance between them, and not where the bodies happen to be. This is the translational symmetry that leads to conservation of momentum.
Update 14:
Final note: Not only Newton’s Cradle was not invented by Newton, but by Edme Marriote, a contemporary, it doesn’t even demonstrate conservation of momentum and energy, as it is underdetermined for different sized balls. Classical mechanics of colliding balls is deterministic ONLY if at most 2 balls are in contact in any collision. However, I have to wonder if Isaac Newton had started at such a “Mariotte’s Cradle” trying to figure out why it should behave so.
Update 15:
“stared at”, not “started at”
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✅ Answers
? Favorite Answer
The other condition might be the time reversal symmetry (as far as elastic collisons are concerned).
Changing the sign of time, t -> -t, changes the sign of velocity, v -> -v. The acceleration a remains the same, a -> d(-v)/d(-t) = dv/dt = a. From the second Newton’s law, ma = F, it follows that the forces of interaction are also the same.
Consider the collision in the coordinate system, where the total momentum is zero. Balls move towards each other with velocities u and -ku, where k is the mass ratio of the balls. After collision, the balls will move with velocities -v and kv. Now, we can reverse time and consider collision of balls moving towards each other with velocities v and -kv. From the symmetry reasoning it follows that u=v. Then we can recalculate the resulting velocities relative to a stationary observer.
Actually, we have to use the conservation of energy implicitly, by assuming that after collision the balls and environment do not change much, so that the time-reversed process occurs under more or less the same conditions as the direct process. However, the explicit concept of energy is not required.
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EDIT:
The major assumption here is that the collision can be described using only two degrees of freedom, which are coordinates of the ball centers x1 and x2. Then we have two dynamic equations
d p1/dt = F1 and d p2/dt = F2,
with the interaction forces F1 = – F2. This is not true in a general case. The balls produce sound during collision, they can get heated etc, which requires excitation of some internal degrees of freedom. Once we agreed that all the other degrees of freedom are insignificant, the solution is straightforward. We can integrate above equations over time, as d/dx+d/dy+d/dz has suggested. This however does not give us new information. We only reconstruct conservation of the total momentum, which is supposed to be known from the very beginning. To complete the solution, we can multiply the equations on d x1/dt and d x2/dt respectively and perform the integration. This can be done, because the interaction force is a function of x1-x2, and it gives conservation of the energy. However, this is not necessary. Using symmetry arguments, we see that in the coordinate system related to the mass center the balls bounce off with their initial velocities.
Clearly, time reversal arguments cannot be applied to inelasic collisons. Inelastic collisions involve other degrees of freedom, which are not described by our equations. Thus, the initial conditions for the “time-reversed” collision will be different from the original condition.
Fifi, you can reverse time by putting rigid walls at sides. The balls will reflect from these walls and go back to collide. Something like this (Questor surely can make better pictures):
|……………|
| O -> <- o |
|……………|
|……………|
| <- O o -> |
|……………|
|……………|
| O -> <- o |
|……………|
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EDIT 2:
Right, Remo. And how are you going to prove that:
“both particles rebound off of the center of mass with exactly the same velocity as they had before but in the opposite direction”?
Remo, this is true but this not a proof. We know only that the total momentum is conserved. Nothing is known in advance about the redistribution of the total momentum between the particles. Scythian asks how to figure it out without explicitly using the energy conservation.
Fifi, I never thought it was sarcasm. I just tried to make my arguments simpler. Collision with a big solid wall produces the same effect, at least temporarily, as time going in reverse.
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EDIT 3:
I would add to Remo’s announce that “relying on the conservation of energy” should not be understood too literally. The principle of conservation of energy describes some feature of reality. We are asked to use alternative concepts to describe this feature (see Scythian’s comments about “End of Time”), and it might be impossible to say which of these concepts is more fundamental. Strictly speaking, we rely not on the conservation of energy, but on the feature of reality, which is commonly called the “conservation of energy”. Of course. This is what the question asks us to do.
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Scythian, I have no idea how Newton solved this problem, but it is quite possible that some rudimentary relations between symmetries and conservation laws were observed before Noether. Just the same as Archimedes found the sphere volume whithout formally using the integral calculus.
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Scythian, the order of developments was not so paradoxical in my view. It was more difficult to handle ideal collisions than the curvilinear motion in a force field. In the latter case, the interaction was determined by the gravitational force. Plugging this force into Newton’s second law and solving the resulting equation is a mathematical task. Newton had enough skills to do it. On the contrary, the interaction force between colliding balls was unknown until Hertz solved the contact problem for rigid bodies. The only way to handle elastic collisions, was to prescribe some properties to the interaction force. Since Newton’s cradle had been designed, someone could come across the idea to reverse time or to state that the force depends only on the distance between the ball centers. The principle of conservation of energy came later, because a large amount of factual material was required to realize that existence of first integrals reflects a law of nature rather than peculiar properties of some mechanical systems. The Noether theorem is a step further. It became possible only after discovering that dynamic equations satisfy the variational principle.
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The actual values of w1 and w2 depend on the elasticity of the collision. Since you have two unknowns, w1 and w2, you must have two constraints. One of these is conservation of momentum, or equality of impulses (from Newton’s third law) which is equivalent to this. We need another one.
We treat perfect elasticity by using conservation of kinetic energy. I don’t know how Newton did it.
I don’t think time reversal would do the trick. Consider the time reversal of an inelastic collision. This is a physically impossible process, but we invoke the fact that it implies an increase in kinetic energy to show this.
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Wow Zo Maar’s explaination made me smile, I love the way his mind works! Seriously.
Without reversing time, perhaps if we use easy explaination of braking it down into units and without explicitely using energy concept, but rather simply the frame of reference, then the balls are accelerating towards each other.
Lets say the balls move towards each other at 1m/s then covering individual distance 0 to 1, m1 and m2 are covering 1m at 1m/s each and 2m/s towards each other, when individual distance covered is 2m but relative to each other 4m are covered and so on….untill they collide and split in an equal and opposite direction. Momentum is conserved velocities are the same individually but relatively to each other there is an acceleration.
This seems so simple that I do not dare to post but still will!
Are we allowed to use the forbidden word now?
Edit: Oh and I assume you will now ask, what if one of the ball is at rest? Then the moving ball acts as a force on the other ball, accelerating it, momentum is still conserved.
Forbidden word is transferred, as force time distance covered, from ball 1 to ball 2.
Anyhoo, the same is true in either case, both balls helped each other overcome their inertia.
Edit again: Locust, I am aware that to you this is extremely basic physics, but consider that Newton percieved and gave a meaning to these extremely basic ideas, where would we be if no one did just that?
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Zo Maar, absolutely no sarcasm was nor is intended towards you and your answer.
Why should we not percieve inelastic collision in reversed time as an ideal explosion?
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Friday AM: Dear Zo Maar, TY for your kindness, your model seems to fit inelastic collision in reversed time also, if both objects cancel each other’s momentum, like an ideal explosion in which object is separated in 2 in our time, but seen in reverse.
Although, I am not sure to totally grasp your model, except in the context of some kind of “elastic spatial path”, the walls substracting distances when the path is compressed and adding space when path is stretched, but would that not add and substract mass to the objects by disturbing space?
Also that you would need to compress the path enough to brake the speed of light limit?
What about 2 balls of same mass, in outer space that are coming from opposite directions at the same velocity but their respective path is parrallel by “R” they start orbiting each other their common center being R, there is no collision (unless a little worm hits the brakes).
PS: I gave everyone a well deserved TU, great question and great answers!
Everyone have a great weekend =)
Source(s): http://blog.lib.umn.edu/carls064/freealonzo/apples…
You are giving us a thing to think of outside of Newton’s Laws? We’ll see if we could come up with something.
The nearest I could think of without using conservation of energy is the theory of particle reflection on a surface. The Law of Reflection would apply in which the angle of the “incident ray” would be equal to the angle of the “reflected ray” on the reflecting surface. And it will have to be Specular Reflection in all cases of perfectly elastic collissions.
In general, systems used in the model that gave us the equations in Physics are “closed systems.” It is premised on the provision that no energy has been created within the system, or introduced from without. This condition of “Conservation of Energy” is the ‘sine qua non’ of it all. Without “conservation” the perimeter of the event will be open, and therefore given to indefinite results or unquantifiable quantities. In Mathematics, we have to specify the validity of a particular function to be true only within a specific domain.
We can recall in Quantum Mechanics, the uncertainty regarding photons, that in the process of observing the photon’s state, a beam will have to be radiated on the photons, and therefore, unnecessarilly introduces external energy to the system, thus negating the chance of obtaining the true values.
With “conservation” a perimeter is established, limiting everything back to zero. This would give rise to the question: “How about all these positive matter that came to existence? It did, but there is anti-matter to level it off. There is the Big Bang and then the Great Crunch thereafter, resulting to zero.
“Every equation equates to zero.”
Edit:
It seems you are referring to the Laws of Newton, before the Law of Conservation of Energy is derived.
We have the Law of Action and Reaction. and then F = ma.
If two bodies hit each other, on a frontal collission, there will be a force exerted on each other, determined by F = ma. Now, the result will be a rebound by one body, then the other body will make an opposite reaction which will be also a rebound in the opposite direction. This results in both bodies bouncing off each other. The masses will be a factor in the forces generated.
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FGR, good reasoning. Everyone here has good ideas. I gave TU to everyone, except myself, Yahoo did not let me.
You mentioned the apple, well it figured in the fall of Adam and Eve, then it caused the Trojan War, then it caused all these gravity problems in school. Maybe it’s about time Obama bans this fruit.
I looked for Scythian’s answer in order to give him TU, but I forgot he was the asker, hehe.
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The solution can be obtained by considering impulsive forces (Integral Fdt). This route is equivalent to conservation of momentum (F=dp/dt), but need not explicitly mention momentum.
consider it this way: you roll down a ball on the horiontal ground towards an incline. it hits the incline at its bottom. now if you know the frictional co-efficient(μ) of the ground, then the force is F(1)= μmg and then calculate the velocity(v) it’ll hit the incline with. now, when it strikes the incline, there’ll be torque against rolling over the incline, that’ll be (mgr sinθ). now, lets assume it ‘ll feel a Force F due to thurst. it must equate {2mgsinθ +μ(incline)mgcosθ} to atleast create a tendency to roll over. if it is less, it wont, and if it’s greater it will.
Here’s what I have on my ‘cheat sheet’ I use when answering momentum ?s:
Vel of CM: Vcm = (m1v1*m2v2)/(m1+m2)
Relative velocity Vr = v1 – v2
Velocity relative to CM: u1 = Vr(m2/(m1+m2))……u2 = Vr(m1/(m1+m2))
Knowing that the u values are equal and opposite after the collision, all I have to do is say
w1 = Vcm – u1 and
w2 = Vcm + u2
Energy not required…….
La valde locusta, unlike you, I have never read Newton’s principalia. I suspect its the latin. But this is how I would do it:
I would work with the center of mass frame of reference. In perfectly elastic collisions, both particle rebound off of the center of mass with exactly the same velocity as they had before but in the opposite direction. In non-elastic collisions, they stick together, and everything else is in between. Works great in a 1D system. Momentum is always conserved because it nets out as zero in the com frame of reference.
Personally, I try to solve all billiard ball equations in this frame of reference. It allows me to avoid using energy. It also allows all the math to be first order and thus avoid having to solve the quadratic. Dirt simple. You can also easily adapt it to 2D and 3D analysis. (Collisions causing rotation and angular momentum can make it a slog, but they work too)
I’m not sure if Newton used this method, but if he were alive today, I would try to convince him that he should. But I have heard that he was a bit stubborn and if he had a method that work (aka in calculus), he would use it even if it was to his disadvantage.
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Edit: Having read Questor and Zo Maar analysis, they both have different facets of the same idea: That you have perfect reflections from the frame of reference of the center of mass.
Fifi, of course, was as ever illuminating. We need our muses.
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I should add that in the com frame of reference, for an elastic collision, 1% of the momentum of ball 1 is transferred to ball 2 and 1% of ball 2’s momentum is transferred to ball 1. No reflections, no reversal of time necessary. 😉
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I was about to announce a new insight, but on closer reading, I will simply announce that I agree with Zo Maar and expound a bit.
The limitation of any methodolgy rely implicitly on the conservation of energy. This applies to a reverse time analysis, or an analysis based on center of mass. The principal of conservation of energy, at least with respect to Newton and billard balls, represents the limits of the maximum post-collision momentum a particle can have in relation to the center of mass. And this maximum is equal to the pre-collision momenutm.
Source(s): If the apple hadn’t hit Newton’s head, would it have smashed to bits in its collision with the ground?1