In the post below, John D. gave a long version of what I think was Dan
Crowes response and is what I think is an appropriate answer to the
question.
However, I am a little perplexed by John's statement in a prior post
where he stated that he didn't see what was "unreasonable" about the
result.
Certainly it is not unreasonable in the sense that that is how it works.
However, we have identified a situation (sticking collision) where a
result is independent of the mechanism causing the sticking. I.e. this
factor of 1/2 as analyzed in the lab frame is extremely robust. I find
this amazing and a bit perplexing, though true. And I think that is
where the question is coming from. Most of the time when I calculate
the energy lost to a dissapation mechanism, the energy lost is highly
dependent upon the mechanism in question, hence the amazement regarding
the collision under discussion.
Of course, the answer is that most of the time when I'm calculating the
energy lost in a dissapative mechanism I don't have the constraint
imposed by our problem. And with that constraint, Dan C.'s or John D.'s
(as well as other similar responses) provides an appropriate
explanation.
________________________
Joel Rauber
Department of Physics - SDSU
Joel.Rauber@sdstate.edu
605-688-4293
| -----Original Message-----
| From: Forum for Physics Educators
| [mailto:PHYS-L@list1.ucc.nau.edu] On Behalf Of John Denker
| Sent: Wednesday, December 07, 2005 10:57 AM
| To: PHYS-L@LISTS.NAU.EDU
| Subject: Re: Momentum Again
|
| Rick Tarara wrote:
| > I still think most people are missing the reason for the question
| > about exactly 1/2 of the KE being transferred in a
| perfectly inelastic
| > collision between same mass objects, one at rest.
| >
| > The question, as I read it, is HOW is exactly 1/2 transferred?
| > Restating the math of momentum conservation or stating that no
| > real-life situation actually behaves this way, is not, if I
| interpret
| > the question properly, an answer. Here is what I imagine a
| student thinking:
| >
| > "A ball is moving along at speed v, has momentum mv, and KE
| .5mv^2.
| > Now there is this collision, and after the collision, the math says
| > that only half of the energy is KE. But where did the rest of the
| > energy go? Heat, sound, deformation--OK, but HOW does
| exactly 1/2 of
| > the original energy find its way into these other channels, none of
| > which are very well defined nor necessarily exactly the
| same from collision to collision."
| >
| > Again, I offer that the problem here is the reification of
| > energy--considering it to be 'contained' in the first ball
| and somehow
| > 'flowing out' into these other channels during the
| collision. In that
| > model, it is natural to ask 'how does exactly 1/2 remain in the
| > kinetic channel and half moves into the others?'
|
| I don't see it that way at all.
|
| To my ears, the question of "why" the system loses half of
| its KE is closely analogous to asking why the hypotenuse of a
| 45/45/90 triangle is ~41% longer than the other legs. Why?
| Because if it were any bigger or any smaller it wouldn't fit
| the gap between the other legs, i.e.
| it wouldn't satisfy the stated conditions of the problem.
|
| To apply this analogy: If the KE(after) were any larger or
| any smaller than half of the KE(before), it wouldn't satisfy
| the known laws of physics and the stated conditions of the
| problem ... namely conservation of momentum plus the stated
| requirement that the objects *stick*. If they didn't lose
| energy, they wouldn't stick.
|
| Indeed this is one of the factors that made my thesis
| experiment possible:
| when a hydrogen atom collides with a hydrogen atom, it does
| not stick, for the simple reason that it has no way to
| conserve both energy and momentum. Sure, there is a huge
| attractive potential, but that is nowhere near sufficient to
| cause things to stick. A comet is strongly attracted to the
| sun, but after falling into the potential well it shoots back
| out again ... as it must, because there is not much dissipation.
|
| Rail cars collide and stick because they *do* have dissipation.
| http://www.google.com/search?q=draft-gears
|
| By way of contrast, consider what would happen in the
| following example, which is fairly typical of what happens
| when there is not enough dissipation.
|
| We have a 100kg cannonball intially moving with a speed of 100m/s.
| We also have a 100kg boxcar initially at rest. The boxcar
| has a special trapdoor, such that the cannonball freely
| enters the boxcar but can never leave; it just bounces back
| and forth, fore and aft. (For simplicity assume a
| one-dimensional geometry.)
|
| So the picture looks something like this:
|
| ________________________________
| | |
| | |
| | O --> |
| | |
| | |
| |________________________________|
|
| and then after the next bounce
|
| ________________________________
| | |
| | |
| | <-- O |
| | |
| | |
| |________________________________|
|
|
| Assignment: draw the world-line of the boxcar. The slope of
| this line indicates the speed of the boxcar as a function of time.
|
| Also: Calculate the long-term average speed of the boxcar.
| Compare it with the min and max speeds. Does this have
| anything to do with the dreaded "half" in the statement of
| the original question?
|
| From this we learn that it is simply not true that when
| objects collide, they "magically" (or otherwise) lose half
| their KE. It is perfectly possible to have other types of collisions.
|
| The key word in the problem is the requirement that they
| collide *and stick*.
| Things don't stick unless there is dissipation!
|
| =========
|
| Keeping track of the energy (and momentum) via the
| conservation laws -- which are in a very precise sense *flow*
| laws -- makes this problem easier, not harder.
| Energy flows. Momentum flows.
| http://www.av8n.com/physics/conservative-flow.htm
|
| Also perhaps
| http://www.av8n.com/physics/euler-flow.htm
|
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