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Time, distance, and speed are related by [math]d=v*t[/math], so if you know any two of the three you can compute the third. There are indeed two unknown quantities. They don't necessarily have to be distance and speed; you could, for example, solve for time and speed, leaving distance as a derived quantity. markus, please write some of the relevant equations so we can give you a hand. Our policy here is to help people do their homework rather than doing it for them.

Numerical solution of ODEs, in other words. This can get very involved, but can also be quite simple (conceptually). For a system modeled as a collection of point masses, the typical approach is to numerically integrate Newton's second law, yielding a "3-DOF" (3 degrees of freedom) simulation (or in your case, a 2-DOF sim). If there are a small number of particles you can afford (computationally) to use a more accurate ODE solution technique. For a lot of particles (or for a project you need to get done in a semester), I'd recommend starting with symplectic Euler integration and moving up to the Verlet (or velocity Verlet) method. Do not use the basic Euler technique. Ever. One exception: You need to understand the basic Euler technique to understand any of the more advanced techniques. Once you have learned it, forget it. It almost always yields very bad results.

Tom, you are the one making unsupported claims. Yes, quantum computers do exist today. Why, they are even capable of factoring numbers. If the number is 15, that is. One company, D-Wave Systems, claims to have built a quantum computer that can solve sudoku puzzles. Many key quantum computing theorists rejected that claim (here, here, and here). That was last year's news. Where is D-Wave now? From the Dwave presentation at SC08 (Supercomputer conference), they indicated that the current 128 qubit adiabatic quantum computer has sub-PC performance but they project by next year to reach that level with whatever number of qubits are operational then. Tom: A challenge: Find a commercially available (or even a one-off in a research lab) quantum computer that exists today that can solve anything but toy problems and do so faster than a PC one can buy at any local computer store.

The original poster asked how there can be any non-zero net force acting on an object since, according to Newton's third law, forces always come in balanced pairs. This question represents a fairly common misperception of Newton's third law. It certainly doesn't help that primary and secondary teachers do not understand Newton's third law. Teachers often portray the downward gravitational force exerted on a person by the Earth as a whole and upward normal force exerted on a person by the Earth's surface as an example of Newton's third law. It would be nice if the educational systems required primary and secondary teachers to know something about the subject matter they are teaching. Lacking that, the way to overcome this misperception is to properly describe Newton's third law as being the same force mechanism acting on two different bodies. The above example, gravity and normal forces forces acting on a person, obviously are not third law pairs because (a) these two forces are acting on the same object, and (b) these two forces arise from very different mechanisms. Tom, you have to gauge your help for the audience. When someone has a misunderstanding of some very basic concepts of physics, delving into more advanced concepts such as energy or relativity or quantum mechanics or Lie algebras is not offering help.

Solipsism and science do not mix well. Science inherently assumes the world out there is real. Sophistry and science do not mix well, either, particularly when the sophistry is based on a faulty assumption.

I agree with you there. If you feed a single twenty+ sigma outlier into a Bayesian update you will get a prior estimate that is completely out-of-whack because the Bayesian update algorithm implicitly assumes that there is no such thing as a twenty+ sigma outlier. To be a bit too anthropomorphic, the Bayesian update bends over backwards to accommodate an apparently nonsensical value. So, simple, use a rule of thumb that rejects such outliers. You never use them to update the estimate. This is fine sometimes. Sensors (or people) can and do give completely false readings. A high order zero can be misread as a one on transmission, in which case throwing out the twenty+ sigma outlier is exactly the right thing to do. Or the sensor just had a glitch, which it will correct (possibly with a twenty+ sigma outlier in the other direction). Once again throwing out the outliers is the exactly right thing to do. Or the sensor failed, in which case throwing out every subsequent reading from that sensor is exactly the right thing to do (there hopefully are some redundant sensors). Or the system just underwent an un-modeled state change and the twenty+ sigma outlier is in fact very close to representing the true state, in which case throwing out the twenty+ sigma outlier is exactly the wrong thing to do.

Almost certainly not, and we certainly don't see any. The Hubble flow at 250 million light years (the distance to the Great Attractor) is over 5 thousand kilometers/second, which is nearly an order of magnitude greater than the peculiar velocities toward the Great Attractor. In other words, at the distances to which jeff Mitchel was referring, the Hubble flow dominates peculiar motion. All of the galaxies "on the other side" of the Great Attractor are red shifted.

Congrats, Klaynos and mooey! Congrats to iNow for scooping the official notification!

You sir, are a terrible liar. Swansont did not say "True". He said The question you presented is a false dichotomy, a logical fallacy. You are apparently well versed in the use of fallacies. Use of such is not appreciated at this site.

That is exactly what I meant about being sloppy.

Here is where I have to depart from what fredrik and resha have written -- somewhat. To the contrary. Have you ever tried to compute the inverse of an nth difference equation? Calculus is, IMHO, much more powerful and much simpler than are finitary methods. A lot of mathematicians still cringe at the normalization procedures developed by physicists. <begin{rant}> These ... [math] \aligned \sum_{k=1}^{\infty}1 &= -\,\frac 1 {2} \\ \sum_{k=1}^{\infty}k &= -\,\frac 1 {12} \endaligned [/math] ... are stinking piles of something. Just because a series has an analytic continuation beyond its interval of convergence does not mean the series converges beyond its interval of convergence. <end{rant}> You have to grant that linear techniques are much more amenable to analysis than are nonlinear techniques. There are all those -ilities (controllability, stability, robustness, safety, cost) that engineers have to worry about that can be very hard to assess with non-linear techniques. There are also all those not-invented-here engineers you have to deal with. The latter problem, well, those old farts will retire someday (or maybe not; their 401Ks have shrunk by quite a bit in the last few months). The former problem is partly due to the purveyors of the non-linear techniques themselves. Some people stay in academia so they can play in sandboxes. I can understand this; I love debugging the blank piece of paper. Developing something new is much more fun and satisfying than is trying to apply those stupid -ility concepts to someone else's technique (or even to one's own technique). That said, there has been a lot of work done with non-linear controls regarding the -ilities as of late. The problem with throwing out calculus is that you throw out a lot of other things. Even when working in the realm of digital control, which inherently has some aspects of discrete maths (BTW, have you dinosaurs in the rotating machinery industry moved to digital control yet?), calculus can still play a major role. If you really want to wow your coworkers, look into the recent developments in symplectic control.

Two very good points. The argument in the web page cited in the original post is not so much against special relativity as it is against quantum mechanics. johnny rocket: The author of that web page is a philosopher, and is thus presumably versed in logic. The argument made in that page is logically invalid. Special relativity, as reaper has noted, has nothing per se to do with energy gained from Uranium 235 fission. Even if physicists do discover something wrong with the model of U235 fission, that will not falsify special relativity. Saying that it does is invoking the logical fallacy of denying the antecedent. Johnny, If you want to make a valid argument against special relativity you need to attack one of the axioms as invalid or attack the mathematics that underlies one of the steps made from those axioms. The argument made in the cited web page uses several fallacies: Poisoning the well ... and a straw man. In particular, step 4 is completely invalid. Einstein derived E = mc² in 1905 and preceded the discovery of U235 by 30 years.

No. Really I meant to say it has energy. Momentum is a separate issue. Let me be real clear: A photon has zero invariant1 mass, non-zero momentum, and non-zero energy. 1I much prefer the terms "invariant mass" and "intrinsic mass" to "rest mass". A photon always moves at c, so the term "rest mass" is a bit of an oxymoron to me. The qualifiers invariant and intrinsic when applied to mass don't ring any jumbo shrimp / death benefit alarm bells in my head.

snp.gupta, jeff was not talking about the local group. Andromeda is a only 2.5 million light years away, very small compared to the distances to which jeff was alluding. He was talking about galaxies on the other side of the Great Attractor. The Great Attractor is 250 million light years away, so he was referring to galaxies 500 million light years away. At these distances the Hubble flow overwhelms any local motion and overwhelms gravity. Thread moved to pseudoscience.

That was Fredrik, not me. I don't have any issues with either the reals or with Kolmogorov's axiomatic basis for probability theory.

Moved to pseudoscience.

With your "theory", that is. In all appearances, your "theory" differs in no regards with "theories" posed by other "purveyors of alternate explanations". Stop being so blessed sloppy. Thread moved to pseudoscience. It can be moved back to a physics forum if and when you stop being so sloppy.

You are equation inertia with mass in your first equation. In your second equation you use [math]pM=I_t[/math]. The left-hand side has units of mass2*length/time. Inertia is not a commonly used term in the physics community. In the vernacular it means mass or linear momentum (and sometimes even angular momentum), depending on context. Physicists have perfectly good words for mass, linear momentum, and angular momentum (to wit: mass, linear momentum, and angular momentum). Why use a term that has unclear meaning and doesn't add value when we have perfectly good words with clear meaning?

Again, I repeat, it does not. The invariant mass is invariant. The term mass usually refers to invariant mass, not relativistic mass. You are being sloppy with words. For a particle with non-zero invariant mass, its relativistic mass of a particle, which does vary with velocity, is [math]m_{\text{rel}} = m_0 \frac 1 {\sqrt{1+(v/c)^2}} = \frac E{c^2}[/math] Note the [math]m_0[/math] in the above. That is the invariant mass (rest mass), and that quantity does not vary with speed. Hence the name, invariant mass. In the special case that the velocity is zero, the relativistic mass is equal to the invariant mass. Hence the alternate name for invariant mass, rest mass.

You have different units, not just different values.

Relativistic mass changes with speed. Invariant mass (aka rest mass) does not. One can use relativity without invoking the concept of relativistic mass; it is merely a synonym for energy ([math]m=E/c^2[/math]) and is a completely unnecessary appendage (use energy instead). Invariant mass on the other hand is an essential characteristic of an object. When one talks about a photon having zero mass, the term obviously means invariant mass. Most physicists use the term "mass" without qualification to refer to invariant mass. When they do mean relativistic mass they generally say "relativistic mass". Note: A photon does have non-zero relativistic mass given by [math]m=h\nu/c^2[/math], which results directly from the Planck relation and the relativistic mass–energy equivalence formula.

I posted these quotes elsewhere in this forum. They are worth repeating (besides, I like them a lot). Harold Edgerton: "That's the nature of research--you don't know what in hell you're doing." Albert Einstein: "If we knew what we were doing, it wouldn't be called research, would it?" Isaac Newton: "The most exciting phrase to hear in science, the one that heralds new discoveries, is not 'Eureka!', but 'That's funny...'" Albert Einstein: "I never came upon any of my discoveries through the process of rational thinking." The creative process is neither rational nor objective. It is something else altogether. What distinguishes a scientist from a loon is that a scientist is able to (in fact, must) look at their non-rational inspirations with an objective and critical view.

Without a reference, it's a bit hard to tell to what you are alluding. I suspect it's one of these devices: Here's one in use:

Who claims science is strictly objective? The lay press (and scientists after the fact) like to present science and mathematics as completely logical as a nice, wrapped-up-with-a-pretty-bow process. I disagree. The scientific process requires creativity to advance. A frequently asked category of questions on this forum is "How did <some brilliant scientist or mathematician> come up with the idea for <something Kuhn would call a paradigm shift>?" The answer is: <some brilliant scientist or mathematician> was insanely creative. The art of connecting the dots is an art. Science is the pretty picture that results coupled with observations indicating that the pretty picture is real. A much bigger problem in my mind is the problem of theory-laden observation. Is this a bunch of pelicans or a bunch of antelopes? (from http://www.loyno.edu/~folse/Hanson.html) You might want to add "theory-laden observation" to your search list and your reading list. Just to make it a bit longer: I suggest you read up on the free lunch theorems. (The author was probably a Heinlein fan.) There's a good bibliography here: http://www.no-free-lunch.org Strictly speaking, the no free lunch theorems pertain only to machine learning techniques. However, a lot of what is said could be said of the scientific method in general. Is the the scientific method a free lunch, or is there no free lunch in science? Not necessarily. There are some things that we know we don't know. There are ways to deal with these known unknowns. A significant sub-branch of multi-criteria decision making deals with decision making under uncertainty. What these techniques can't deal with are the things that we don't know that we don't know. Donald Rumsfeld alluded to exactly this distinction between the known and unknown unknowns: "Reports that say that something hasn't happened are always interesting to me, because as we know, there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns -- the ones we don't know we don't know." This was deep -- and a sign he read something of philosophy of science (and recent literature in the field in decision analysis). The press was shallow to dismiss what he said as stupid.

Long, interesting thread. Where then to start? I'll just dive right in with the last post: That's one way to look at the reals, as the set of all Cauchy sequences of rationals. The axiomatic approach simply posits that there exists a complete Archimedean ordered field, shows that any two complete Archimedean ordered field are isomorphic to one another, and show that the sets defined by Cauchy sequences or Dedekind cuts form a complete Archimedean ordered field. Within isomorphism, there is one complete Archimedean ordered field. Another way to define the reals is the set of all decimal expansions. That this simple approach is valid is quite amazing. Even engineers can understand it! (Denigrating a group is in general unacceptable, non-politically speech. It is acceptable so long as it is self-denigration. I have been working as an engineer for a long time.) The reals between zero and one (inclusive) is even easier: [math][0,1] = \lbrace x : \exists \;\lbrace d_0,d_1,\cdots,d_n,\cdots\rbrace ,d_r\in(0,1,...,9) : x=\sum_{r=0}^{\infty}d_r10^{-r}\rbrace[/math] In short, the collection of all numbers of the form 0.d0d1...dn..., where each dr is an integer between 0 and 9 (inclusive). This is not a finite procedure, and ignores the problems of uncountable numbers. It works and is easy to grasp. (The engineer's ultimate tests.) Back to philosophy of science. Popper was on the right track regarding falsifiability as a key distinguishing factor between science and non-science. Where he went wrong was in viewing falsification as a cornerstone of science. In a very real sense, all science is inherently false in the sense that the models built by science are only approximately and provisionally correct. The rub: Just because a scientific model has been falsified in some regime (e.g., Newtonian mechanics) does not mean it is "always false". There is, in my mind, a big distinction between always false models (e.g., the caloric theory of heat, Aristotelian physics) and provisionally correct models (e.g., Newtonian mechanics, quantum mechanics, relativity). Newtonian mechanics has been falsified in the realms of the very small, very fast, and very large. Some future Einstein will find flaws in quantum mechanics and relativity. Just because Newtonian mechanics has been falsified does not mean it is not valid. We still teach it and still use it; many branches of engineering are applied Newtonian mechanics.