Ahh, now your question in the other post makes a lot more sense to me so I know better how to respond to it. I suppose I should have read this first. But, I didn't - so there.

Your engineer friend could either be right or wrong depending on what he's arguing isn't right in your take. I can't speak to his view, but I can offer you mine.

It isn't exactly transient in a specific collision. This is a degrees of freedom kind of thing. For each collision, there is some ultimate coefficient of restitution. Just like for each braking situation, there's some ultimate coefficient of friction. It isn't transient, per se; it's just not a set number *until the event happens*. We frequently say that "the coefficient of the road" is thus and such.

This isn't exactly right. The road, standing alone, doesn't have its own coefficient of friction. The road just has certain properties which contribute to the ultimate CoF *between* the tires and it. This implies that the tires have their own component to contribute. But neither alone has an absolute value - it's always in relation to the other thing it's in contact with - and the nature of that contact.

The case with e is related, but notably different.

Each vehicle has its own, unique ability to deform. But the ultimate mitigation of eccentricity is going to be a function (I'm not using the word function here in the mathematical definition) of the maximum deformation of the "softest" vehicle. The softest vehicle should be able to absorb more force than its harder other vehicle. So, if we had a static object to ram these into, we would expect the softer vehicle to change velocity less violently than its harder counterpart. This is assuming its the same thing we're ramming these both into - a control.

But, in a crash, they're ramming into each other. The downside is that we can never put the cars back the way they were before the collision and test their deformation capabilities. So, the water is muddied a bit as the ultimate eccentricity is a function of some combination of both of their coefficients of restitution. How much to ascribe to which isn't obvious, nor is the expected result terribly predictable.

In that sense, the ultimate deformation of all involved vehicles is variable (transient) dictated by the actual cars involved. We can never know this until after it's happened, which is precariously the thing that makes it unknowable because we get there afterwards. Using exemplar cars isn't necessarily helpful except for in extremely rare cases because the life of each car is dramatically different based upon myriad factors: terrain it's used on, salinity of the air, how that batch of material aged as compared against a different batch of material used to make a similar car at the same time and so on. This muddies the water even more.

As far as it being an integrable function goes, I'd have to see some hard mathematics to even start to agree with that. Recall: all functions (in the mathematical sense) meeting the mathematically required definitions are differentiable, but not all can be antidifferentiated (integrated). So, assuming the three conditions are met and the function is differentiable, there's no reason to assume that the derivative is necessarily integrable. I'm not saying I'm right, but I'm just not seeing how it'd work itself through.

I'm not really sure how best to describe this in words, but . . .
Recall that: if f is continuous in [a,b], and g(x) = int {a,x}[f(t)]dt, then g'(x) = f(x) from the Fundamental Theorem of Calculus (part 1). This essentially says that if f is some derivative function which we integrate, we should get some antiderivative function, g. If we then take g and differentiate it, we should get out f. So, they undo each other. That's assuming that we can construct the antiderivative. Not all derivative functions, however, have an antiderivative.  The reason this is relevant is that velocity is the derivative of position, and acceleration is the derivative of velocity.  Or, working the other way up the chain, velocity is the antiderivative of acceleration and position is the antiderivative of velocity.

For a concrete example, recall that the sine integral function does exist and is the derivative of something. But, it cannot be integrated as it has no antiderivative (consider the indefinite integral). So, to find its integral, a numeric approach is used (consider the definite integral). This holds true for a whole host of functions.

 

So, while its true that some particular function may not have an indefinite integral (the antiderivative), it isnt safe to assume that the function cant be integrated using other means.  Most notably is technology. We can now find definite integrals it would have 100 years ago taken a lifetime to approximate in a relatively short period by using the home computer.  This is simply because a computer can choose x more quickly than we can.  Also, it can keep track of a greater degree of precision than we can simply because we can only record information so fast.  But, we have to consider that computers are really bad at math, and humans are extremely good at math.  Computers are extremely good at calculations and humans are notoriously not good at that.  The difference, as was told to me during my math education and as I tell my students, is that the human brain is uniquely wired to understand what a number is, and what the math is telling us.  Computers are extremely efficient (and accurate) at computing numbers, while humans are notorious for making arithmetic errors.  But the computer has no understanding of what the result is.  Context is everything. 

 

While its true the computer wont let you divide by zero because its logic circuits take at face value that its not allowed, only the human mind has a concept of what that means.  Also, our brains are excellent at understanding area which is the central problem of the mathematical level were currently discussing.  If you want to test that, cut a piece of cake into 2 pieces of unequal size and ask a child which one he wants.

 

So, I'm not convinced that there's a discretely integrable function to define e in a particular case. I think a better way to go would be statistical analysis, polynomial regression, or some numerical method of exhaustion. Since that's my take on it, I don't think it'd be a reliable method against which to base someone's liberty given that it will necessarily have a degree of error, a range of possible values which are all roughly equally likely to be true. And taking into account the relevant standard deviations and how those would translate into the range of speeds - all of which could be true - would leave a lot of ambiguity in the work. This isn't workable given the mathematical training of *most* collision experts.

Maybe I'm just jaded from seeing the grossly egregious conclusions drawn up by many investigators. This considering the relatively low level of mathematics known to a great majority of collision investigators. If we're getting conclusions grossly inaccurate with high school algebra, you can only imagine the problems that would arise having to use graduate level mathematics . . .

Maybe it's doable. I don't claim to have the right answer for that. But I do think that as a general premise, it's more problem than it's worth particularly when there are already reliable techniques which yield rather accurate results when properly applied.

I do, of course, agree with you that it isn't a number we should pull out of the air and use in a formula.

Now to address the technological approach. I think it would be rather embarrassing to put officers in a position to testify in court about something mathematical they can't show. I also think that it could lead to a lot of misrepresentation as well. Because as long as the data inputed doesn't have a parody error for some reason, you'll get an answer. But it might have nothing to do with anything that actually happened. And since the officers (indeed even most engineers) won't have the mathematics background to check, it'll just be one of those having to take it at face value that it's right. That hardly seems like it's of any evidential value.

As for me, I don't put a great deal of stock into crush analysis anyway. There are far superior ways to account for momentum and kinetic energy than assuming particular metallurgical behavior of a car that's been through god only knows what in the 10 years it's been on the road.
 Ok, Ive finally had a nice sleep and reread your post.  Youve talked about the program PCCRASH, which you admittedly dont use.  I dont either.  In fact, beyond diagrams, I use very little technology when I reduce the mathematics because, well, I dont necessarily trust it.  Its a program which can only yield results based on whats entered.  When Im actually in the mathematics of something, its entirely more obvious whats going on as there isnt any distance from whats being done.  Thats me. 

 

But you bring up the concept of optimization of the function, which is indeed a moderately difficult thing to do mathematically.  Having taught mathematics, its one of the particular topics which most students struggle to master a lot.  And thats just with functions in one or two variables which doesnt begin to compare to the complexity inherent in actual science in industry.  If youre interested in some reasonably simple optimization problems which I think well delineate the thrust of this, go to the home page of Dr. Annalisa Crannell at Franklin and MarshalCollege and look at her class projects and the excellent solutions.  She really has a gift in my humble opinion.  I mention her name because one of her writing assignments to her students uses watered down collision investigation techniques to prove whether a murder occurred, or not.

 

I will say that your algebra was correct if your axiom is accepted.   So, sure the conclusion is *valid* mathematically, but it doesnt follow that the resultant information is worthwhile to have.  Assuming that all the relevant factors have been taken into account, and the mathematics is done correctly, then I dont see why one couldnt come up with this so-called e.  But I dont see the point.  If theres sufficient information to otherwise resolve the functions with such a degree of precision that we can come up with what must have been the coefficient of restitution, then I think the need to do so is moot as the collision can otherwise be solved with a high degree of accuracy.

 

My issue is with resolving the collision based on so-called crush analysis.  That is to say having some suspected value(s) for e and deducing the minimum required force to deform the material to the extent we see doesnt seem a good use of time.  And heres why.

 

In these programs youve mentioned, the computer comes up with what must have been e given the parameters of what you tell it happened.  Its a degrees of freedom issue.  In any particular crash, there are many, many variable issues which we whittle down from evidence we find.  And yes, there has to be some value for e thats derivable.  The computer does this by having as possible values all possible values between 0 and 1.  Something is alternately completely elastic, completely inelastic, or somewhere in between.  The 2 extreme cases have never been found to be true, but are still theoretically possible.  The computer doesnt know this; it just knows that its a potentially true outcome.  So, by whittling away at what cant be the case, the computer deduces what could be a workable value for e.  Since this deductive reasoning is entirely predicated on the information entered, the fewer variables we can eventually quantify, the greater the range of potential values is.  Since were dealing with a range of values which are equally likely to be true, it seems to me that statistics (actual statistics, not the stuff the media try to pawn off as statistics) is better suited for these analyses.  Even if the all of the values the computer can come up with arent equally likely to be true, statistics is still equipped to handle that.

 

Then there are the moments to deal with.  The moments at different points of the vehicle will be different (though they can be the same due to fluke).  As we went over in the other thread, different parts of the car have different forces operating on them.  So, were having to look at not e, but the mean of e among all the contributing subparts.  This again seems better suited to statistical models in my mind.  While it is possible to have complex monitoring programs to find the mean of it, we dont typically have these on board cars. 

 

Take the rather common example of an ECG (EKG) instrument.  It monitors of the electrical activity of the heart and reduces that to graphs which we can interpret depending on the angle from which we view it.  Does it really tell us the precise electrical activity in the heart?  No.  It operates by looking at the mean electrical axis of the heart much like we tend to only focus on the center of mass and what its doing.  For gross analysis, this is generally a good idea as its the generalized thrust we are concerned with.

 

For studying the dynamics of how a vehicle crushes and how that relates to delta v, that kind of generalized overview is subject to being skewed.  This is because different structures in the vehicles fail under different force thresholds, and those alter the geometric orientation of the resultant vectors in the reciprocal zones.  We cant account for that without instrumentation.  I dont suggest that if bolt x fails before weld y that the result is significant.  It may or may not be depending on the relative degree of force required compared against the total force present.  I just suggest that we cant know after its happened if those sheer forces are relevant.  Crush analysis would be more sensitive to this than the more traditional methods we use.

 

I can say that this would be an extremely interesting topic of study, and I wouldnt mind taking part in one, but Im unaware of anyone whod be willing to fund this study particularly given the current economy.  But itd be interesting for sure.  Any good grant writers out there? <wink>

 

You said, AND it can have a greater effect upon the preimpact speeds than the coefficient of friction, and thus can be misused and misrepresented to validate an invalid analysis.   That is situationally true and not true.  The less oblique the angles the greater minor errors effect the conclusions.  The more oblique cases are less sensitive to error because of the nature of the underlying trigonometry as a definitional matter.  In the old forum, I had a rather lengthy post on source error as a function of angles.  Sadly, it seems to be lost forever now and Im completely unwilling to type it out again.

 

By the by, can we call it something other than e?! That numbers already taken!

 

Anyway, those are my first thoughts on it.  Let me know what you think as Im finding this to be a very stimulating discussion.


-- Edited by ashman165 at 01:13, 2008-08-15

My apologies to the natural logarithm.  Thanks, Johnathan, for such a thoughtful and excellent response.  You have expressed several of my thoughts exactly, although I probably was not perfectly clear about them in my first post.  You have a way of stating mathematics concepts clearly and accurately.

I happen to agree with you that conventional damage analyses are, in general, not very rigorous.  Using a single data point and an assumed onset of crush speed would never be a sufficient number of data in most other scientific studies (with the possible exception of sky-is-falling-type global warming studies).  Rarely do we get more than two points because it is just so expensive to crash automobiles. 

FYI, the company I work for is located relatively close to Karco Engineering and at times we have had them crash multiple vehicles in order to establish more data points; however, most legal cases do not warrant the expense of multiple crash tests.  That said, sometimes a Campbell damage analysis is the only means available to provide an indication of the severity of a collision.  Although the analysis lacks the rigor we'd really prefer, having a weak opinion can be better than having none, especially for first approximations or during the initial consultation process.  I also agree with your point that a 5-year old car from Chicago is probably going to be much more worn and rusted than a 10-year old car from Barstow (assuming they are parked and driven similarly), but I would not want to throw out all damage calculations, they just require some qualification and context for their data and results. 

The example I like to use in order to establish a reasonable perspective about the precision (and sometimes the rigor) of speed calculation methods in general is that of your vehicle's spedometer.  Most vehicle spedometers are accurate within ±2.5 mph or so.  If the tire sizes have changed then this range could be very much larger.  But even in the best of cases, most people drive according to the nearest 5 mph increment.  In accordance with our ability to perceive the needle's position and the precision of the speedometer, it is of no consequence to a driver if he or she is traveling at 56.75 mph, because they would probably call that 55 mph.  I would like to think that a person's freedom would never be in jeopardy over anything as irrelevant as 5 (or even 10) miles per hour estimated from a oversimplified work-energy calculation, but I've seen worse.  (Unless they are driving a heavy commercial vehicle - those drivers SHOULD be held to a much higher standard of care.)

Sometimes, I get a real kick out of experts that carry their calculations out to 5 or 6 significant digits because it is so silly and without any relationship to the real-world task of driving.  All  of this is really a tangent to the original discussion, anyway, but I wanted to express my agreement with your opinions about damage calc's.  I also agree with you about programs like EDCRASH and PCCRASH and the like.  They can be great tools for certain specific things, but in general, I agree that they lack the ability to choose the best tasting piece of chocolate cake.  Further, they can be very dangerous in the hands of a likeable, impressive, and well-spoken expert who is experienced with manipulating such programs.  You can hide a lot of garbage deep in the input file of an EDCRASH run, and I think there is even more room for "flexible" data in PCCRASH.

I suppose I wasn't exactly clear what I meant about the CR (is this a better abbreviation than e?) being transient in a given collision.  What I meant to say is that it is variable from one collision to the next according to myriad circumstances, which include the mechanics of the impact as well as the material properties of the vehices among others.  Many of these circumstances are too insignificant to measure or estimate individually, but collectively they affect the elasticity of the impact.  After reading your post, and doing a little more reasearch on the subject, I am starting to think that the best way to estimate the CR in a collision is by finite element modeling.

It has been a while since Calculus, but I understand what you are getting at with a function being differentiable but not integrable, but I just can't quite get my head around how this applies in a physics sense.  I think I got that definition for CR from Limpert's book, but I'm not sure at the moment.  It does stand to reason that the CR is going to be different for the same collision at different speeds because of the vehicle parts involved.  (This is one of the reasons I'd like to see a better damage-energy model developed that uses curved or multiple regressions, which change slope drastically at certain crush depths when different parts come into play.)

Since CR is a unitless ratio, does it have a physical analogue?  For example, what is frequently referred to as "drag factor" would be analogous to deceleration rate and we could conceptualize a real-world phenomenon that relates to drag factor, even though it really has no units until we include the gravity of whatever planet we are sliding on.  Does CR have such an analogue?  The only one I can think of is that it is a measure of the colliding objects' ability to restore their shapes (thus, restitution).  But it is also a measure of the energy that is stored (as potential energy) in comparison to what is released (as kinetic energy) during this restoration process.

I was thinking that one could plot CR as a function with respect to time, but now that I think about it a little deeper, I'm not so sure.  I was thinking that it would start at zero, increase sharply, then decrease to a level more than zero but less than one at the point of separation.  However, I think this is incorrect because there is nothing to provide this ratio until after maximum engagement.  Until then, we would be missing a critical factor in the ratio - the separation velocity.  Not only that, but it would seem that the entire collision impulse is required since it would be an irrelevant number until the intervehicle collision forces were no longer present. 

Hmmmmmmmmmm.  If a vehicle slows to a stop normally, the impulse is very gentle because the delta-V occurrs over a prolonged delta-t.  In a collision, the delta-V occurrs over a very short delta-t, thus the high accelerations that usually result in injuries.  Theorizing a completely elastic central collision, energy (other than what is converted to sound or heat) is conserved.  So a pool ball that has a very high CR can strike another and bounce backward, sometimes significantly if the right spin is introduced.  (Which brings up another factor influencing CR - angular velocity.)  But in any case, the combined kinetic energy of both balls after impact can never be more than the kinetic energy of the bullet ball before impact.  Without spin, one would expect the bullet ball to stop dead and send the target ball exiting the impact at the exact same speed the bullet ball entered (ideally with a CR of 1).  In a completely inelastic collision, we would expect both balls to exit the collision together at 1/2 the preimpact speed of the bullet ball.  If the target ball was glued to the table so that it could not move, the bullet ball would bounce back without any speed loss for a CR of 1.  For a CR of 0, it would be damaged until all of its kinetic energy was dissipated then it would stop (or completely fail if KE was greater than the damage that could be dissipated).  Perhaps it is worthwhile to think of it as a measure of the efficiency of the collision(?)  I think I am starting to get a grasp of the concept, but when we start to include it in a formulation, it becomes more like squeezing Jello.

You have given me fresh insight into the concept and I will have more to add later when I get a bit more time.  Until then, thanks again for your response.  I was hoping to generate such thoughtful discussion.

I must bow to your superior geekness, David. *Genuflects*

I'm sorry if my thoughts aren't as linearly ordered as your first post. But I'm just kind of writing this as I scroll back up to your response to my reply. So, it's really whatever paragraph my eye lands on that I write about after I finish a thought. Sorry. ^_^

Thanks for the kind words, sir!

I guess I should explain a little about my background so you'll know why I tend to focus on the mathematics in my explanation rather than physics. I'm a mathematician! Tada - surprise!

And being that I'm one, I'd like to offer a partial explanation of how we get this way. You know what I mean - when math profs use language similar to "this obviously shows", "this clearly implies" and that kind of thing it's because we frequently neglect to remember that the obscurity of the issue is directly proportionate to how new it is to you. Also, it lets us sound smug and superior while simultaneously intimidating our students not to ask us questions we probably can't answer on the fly.

The only reason I chucked in that bit about the Fundamental Theorem of Calculus (part 1) is because you'd made mention in your post that there was something to do with integrating the impulse moment - or something to that effect. That got me to thinking of how difficult, if not altogether impossible, it woud be to actually come up with a generic equation we could slap some bounds on and integrate to get the CR (much better than e). Impliedly I was kind of trying to get across that even if we have some function, or series of functions, which would model a generic case for the event, I think that it would be sufficiently complex such that we could never construct its antiderivative. Since we can't construct that antiderivative, it follows that we can't have an indefinite (generic) integral to work from because it simply can't exist.

And then I went on to point out that even if we can't construct the antiderivative (the indefinite integral), a function can still be integrated - just not in a traditional way. I put that in mainly to refresh you if calculus was a while back in your past, and to have that caveat in there because some of colleagues and even students are occasioned to read my posts here. And I'd hate to get chumped for not stating the caveat.

As far as its role in physics is concerned, physics - indeed all true sciences, is simply an extension of mathematics to applications. Mathematics is the foundation; science is the building. Remember that Newtonian Calculus (fluxions as he called it) was merely a footnote in a physics treatise he wrote. So, there was this lengthy, scholarly dissertation on physics with a footnote laying the foundation for the science. (I wish I could come up with an outright idea that is worth half the importance of his mere footnote)

Also in my statement is the implication that since there isn't a workable solution to find some actual integral in the classical ways, we'll have to use something like numeric integration or some other method of exhaustion, it doesn't seem like a good expenditure of resource to approach it from that direction. Since we're already in the numeric method, we might as well continue in that way in a section of math which is keenly attuned to that sort of thing: statistics. Once again, by statistics I don't mean the crap printed in the local rag, or pontificated on high in what we laughingly call the "news". I mean serious, mathematically rigorous statistical analyses. That is how much of industry already works anyway, so why not use it?

Just to clarify - and I agree with you fully - that a rough speed is about the best we can hope for. So and so was traveling at 55(ish) mph. We should, however, attempt to make that rough approximation as accurate as possible. But it has to have a local implication, not a global one. The difference of 2 mph can be extreme if we're discussing the difference between 2 and 4 miles per hour. In fact, there the disparity would account for a difference in kinetic energy by a factor of 4. The difference between 56 and 58 miles per hour is much less significant because that change as compared to the global energy of the system is minute. Obviously, we don't reconstruct the 2-4 scenario, but I'm a mathematician, not a physicist. And I have the jokes to prove it!

When you gave your clarification of what you meant by transient, you in short summed up the my number 1 concern. Namely that there are far too many lurking variables for which we can't account which when taken individually aren't overly important, but stacked together do create substantial sources of error.

Yeah, I've seen the ridiculous use of multiple significant digits in various analyses. But a far worse crime to my mind is the way people use ranges. It isn't that a range of speeds is used, it's the range of speeds used I take issue with.

Because the relationship between speed and energy isn't linear, it doesn't make sense to use the same range in each case as that range of values has relatively greater or lesser significance based entirely upon the speeds at issue. The energy between 35 and 45 is a much more substantial portion of the total energy in the system than is the energy between 55 and 65. This is, as I've pointed out, why I think collision reconstructionists should be required to take higher levels of mathematics so that they can get a feel for scale.

I also think a better understanding of physics should be required before someone is deemed an expert. Sadly, the requirements to be qualified as an expert aren't exactly high. There's a lot of simple recitation of what people learn in traffic classes without the benefit of true understanding. Take impulse as a case-in-point. It's an oft used concept and term in court, but it's rarely expanded upon in any in depth way. When I did consulting work, these were the things I'd sit down and teach to the attorney so that he could really keep an ear out for buzzwords and force the witness to prove in open court whether or not s/he had the knowledge the word expert implies s/he should have.

And the questions didn't even have to be very in depth before many expert witnesses started to sound less and less expert. You can bring up really basic stuff from the equations they used in their analyses. Things like, say, "And what exactly is cosine?". Or, "What, pray tell, is a moment?" These are questions which experts should be able to readily explain to a jury, or anyone else really.

This is one of the reasons I've been working on writing a mathematics manual specifically geared for collision investigators. It isn't that the questions I mentioned above are tricks to discredit an expert's qualifications, it's that these are the things anyone who reconstructs collisions needs to know. I put a post up here with an open solicitation for what people would like to see in such a book - be it something they want to know more about for themselves, or be it a topic they've noticed others in the field seem not to have a good handle or something in between. Any feedback would be greatly appreciated.

I'd like to make a point about your pool ball example for the sake any layperson who reads this. The only reason that it's true that we'd expect them each to travel forward at half the speed of the first ball is because pool balls have equal, or roughly equal, mass. That wouldn't hold true if a marble hit the 8 ball, or the 8 ball hit a marble. And yes, pool balls are an example I use a lot (and it's the way I think about it as well.) I also think about about silly putty hitting a wall, an egg hitting a wall as opposed to being thrown at a hanging sheet. These, in my mind, are wonderful ways to demonstrate elasticity, impulse, kinetic energy and momentum.

About the degree of rigor we use, it's woefully inadequate to qualify as scientific inquiry. Here, we often use the 85th percentile, which confuses me. Why such a low standard? In scientific endeavors, I can't recall seeing anything taken as serious when it's even the 90th percentile. While I don't think that we would as a matter of course need to restrict ourselves to the 99th percentile inasmuch as people's lives aren't at risk in trials for these types of things (like say they are in a drug study for a new medication), I think the 85th percentil is a little low because people do stand to lose substantial sums of money, their personal liberties, the right to vote, run for office and a whole litany of other things.

About these various programs and the likeable witnesses who use them . . . here in Washington a couple of years ago a man was convicted of vehicular homicide based on a reconstruction of the crash in PC-CRASH which, claimed the investigator, showed that Sipin had been driving the vehicle. You can read about it if you'd like in Stave V. Sipin: . The slip opinion is actually somewhere in my old e-mail files as I get e-mailed all appellate and supreme court opinions when they're published. But I'm far too lazy to go looking for it. But you can find it at that website I googled.

This is, of course, an extreme case of how using a program can lead to goofy - to say the least - results. Most of the misuses of these technologies are likely to be lesser in severity than this case in that the programs are generally being used for their designed purposes. But that isn't to say that even if one is using something within the confines of its explicit purpose that playing with the numbers doesn't lead to results this egregious. I digress.

Does CR have a physical analogue? I'm not sure, but the first thing which charged into my mind is Hooke's Law. Even though we're talking about the "springiness" of the materials, the analogue isn't exactly apt.

I'd have to give think about it for a while to come up with an answer one way or another.

Incidentally, I googled you earlier and read an article you wrote in a magazine (journal?) about what exactly collision dynamics is - and what we think about when reconstructing scenes. Nicely put, David.

I look forward to your response. ^_^

<ed note:  I just noticed that in my first reply here all of my apostrophes are missing.  I swear I typed them in so I don't know where they went. :( >

-- Edited by ashman165 at 18:27, 2008-08-15

Well, I think our framing of the issues (David and mine, and then yours) are different. I'm generally loath to speak for another, but I think David will agree with my understanding. David and I, while aware that the investigative efforts are eventually resolved by lay people in the form of a jury, and that the legal system is heavily involved, it has been our focus in this discussion per se.

We've been mulling over the science and maths bits without much concern over the jury. Sure, I've made a few token references to court, but it was never with respect to the any jurors, it was merely to the qualification - or lack thereof - of some expert witnesses.

So, I just wanted to point that out. Our first course in scientific endeavors can't be how it'll be understood by the public. Though they're the ultimate consumer of the data, they aren't relevant to it.

That said, let me see how I can parse this up.

Assumptions are an integral part of the scientific process. Although we'd idealy like to minimize the number of assumptions made, they are inevitable. That it's an assumption doesn't necessarily mean a conclusion is without merit. The crux is the nature of the assumption.

The quality of what's assumed is generally related to the inductive and deductive reasoning skills of the person holding that opinion. And this is where I think my strongest argument for requiring more math and physics be taught to Officers during their training. The more they understand the topics, the better their assumptions generally will be.

To the program topic I can say that the programs are "aware" of the rules of physics. And they'll accept as valid *any* possible situation that doesn't violate the laws of physics. So, there's a big onus on the operator to have lots of integrity, as well as a keen understanding of the relevant topics.

I'm not sure that vagueness here is dispositive. Oftentimes in academia, we can only prove something by what we don't know. Namely, it's often easier to prove what didn't or couldn't have happened - or what is extremely unlikely to have happened - and narrow the field until we get to what likely happened.

Now for the court part. That's the entire nature of our adversarial system. What you call confusing the arbiters, we call the fine art of argumentation. The idea is to persuade them that your view is the correct one, or that the other side's view isn't.

Yes, it's a nuanced thing when one works in science. Some of it is extremely subtle. But I have to disagree about it having a limited relevance. The devil's in the details.

Johnathan, et al,

Nice quotation.  It's a good thing people don't speak with such strict respect for the rules or we'd never be able to understand anyone.

Yes, that was part of the assumption, that the pool balls have the same mass.  I think I edited that sentence out before I posted because I was getting a little long-winded.  As for silly putty, when is the last time you through some against a wall?  It's been many years for me, but from what I remember, it bounces pretty darn well!  I think modeling clay or mud are better examples of things that take part in inelastic collisions.

You bring up a very good issue that is worthy of its own topic on the board, that of police officer training.  Personally, I think we ask far too much of our policemen and women already to expect them to be physicists or at least well-versed in higher order mathematics.  I have spent a lot of time with traffic collision reports from all over the nation, and I have developed some conclusions about the very nature of accident investigation and the legal system that are not very reassuring for accident victims.  Could you imagine, if in addition to the knowledge you have in mathematics, you also were required to have an understanding of criminal law, crime scene investigation, traffic direction, management of difficult people/situations, search and seizure procedures, constitutional law, and still be able to drive a police unit like you stole it?  I think it is completely understandable how the police have evolved into the primary accident investigators/recconstructionists for nearly all communities in the U.S., but that does not mean there isn't a better way.  Most police officers do not get into that sort of work so that they can diagram crash scenes and work up time-distance calculations.  Unless they get "bitten" by the reconstruction bug (which some of them do) and are fiercely self-motivated, most police officers do not regard traffic accident reports as an activity they want to do on a daily basis.  And if there is little potential for criminal charges, they are even less likely to put in the extra work. 

Please don't misunderstand, I don't want officers to become better reconstructionists at the expense of being better police officers.  When I call 9-1-1 because somebody is breaking into my house or stealing my car, I want an officer who can catch the bad guy and put him in jail, not teach me math.  My father was a deputy for 33 years (his last assignment was seargent in charge of the county crime lab) and he would be the first to tell you that it is a good thing that most cops do not investigate traffic accidents.  There are always exceptions (like the former engineer turned police officer), but I just don't think you are going to get many officers to attend a calculus class because they specifically chose police work in order to avoid that sort of thing.  This lack of math/science skills is not a phenomenon restricted only to police officers, right now on my desk I've actually got the deposition of a PhD who testified to some things that would make even the most jaded and cynical police officers blush.  I still haven't figured out if he is acting like a hired gun, or if he actually believes his own opinions.

All of this is why I think the police should not have exclusive access to a crash scene or the vehicles immediately following an incident.  But, like I said, that is a topic for another conversation altogether (but one I'd like to have).

I have been bouncing this thing around in my head for a while now, and it occurred to me over the weekend that there must be more to CR than the properties of the materials involved in the collision.  Here is my reasoning:

If we focus on the "bounce" aspect of restitution, and we assume that restitution of materials is the source of this bounce, then it would seem logical that an object that returns to its preimpact state should return all of the energy lost in its deformation back to the system.  But this is not true.  The classic bouncing ball problem invalidates this premise.  If a ball is dropped from any appreciable height to an immovable suface, it will never return to the same elevation from which it was dropped even though there is no permanent damage to the ball.  That must mean that there is another force involved, whether it be friction, sound, heat, or something else, because there is energy lost between one bounce and the next.  I remember reading that the coefficient of restitution, in this case, can be defined by the ratio of the ball's speed when it contacts the ground to the speed of the ball when it lifts off again.  Since this coefficient is somewhat independent of the preimpact and postimpact energies (for a homogenous ball), it would stand to reason that, according to Newton's first law, restitution (or lack thereof) is the only reason why the ball would not bounce forever.  Which leads to the really important question here:  Does restitution remain constant even if the objects collide in deep space?  Since inertia and momentum are not affected by gravity per se, would restitution remain the same?  I think so, and any differences are due to forces other than restitution.

I agree that most every driver, definitely every commercial driver, and all computer graphics animators should be taught the difference between accelerated motion and constant motion with respect to distance traveled, acceleration, and energy.  This is not to say they need to be able to take the derrivative of the distance function or anything, but they need to understand which of these are parabolic functions, and which are linear.  I don't know how many animators I have worked with that do not understand why you cannot put a vehicle at the start of its acceleration and another at the end and simply have the computer fill in the middle frames evenly.  Hell, we can't even teach drivers in California that the left lane is for passing and the right lane is for driving, so the idea of teaching them the difference in kinetic energy between their car at 20 and their car at 50 is pretty ambitious.

Anyway, I'll get back to you again on this topic.  Thanks for all your help.