AP Physics Assignment Fermi Problems of Estimation The method of obtaining a quick approximation to a seemingly difficult mathematical process by using a series of "educated guesses" and rounded calculations is attributed to the nuclear physicist Enrico Fermi. Your logical imagination is important in solving the following problems. It is best to use rounded numbers with their units throughout the solution. 1). Given normal lifetime expectations, how many more seconds do you have yet to live? 2. A tuning fork sounds concert A so that the orchestra can tune. The tuning fork can still be heard 20 seconds after it is struck. How many vibrations have occurred during the 20 seconds? 3. If the area of the 48 contiguous states were divided equally among all the high school physics teachers in the country, how many square meters would each one get? 4. If five students submerge themselves at the same time in a cylinder of water 3 meters deep and 2 meters in diameter, how many centimeters would the water rise? 5. Suppose five of you could stand on each others' shoulders to form a human tower. Now imagine someone making a pile of $100 bills as tall as the human tower. How much money would it take? Facts for Fun world population (1983) 4.7 x 109 people world population (1650) 5 x 108 people domestic water use in U.S. 1.3 x 103 m3/s velocity of hair growth 3 x 10-9 m/s age of Universe 3.3 x 1017 s atomic mass unit (AMU) 1.660565 x 10-27 kg kinetic energy of earth 2.6 x 1033 J thrust of 747 jet engine 7.7 x 105 N pH of acid rain 4 x 100 sound causing ruptured eardrum 1.6 x 102 dB voltage between 747's wingtips 7.5 x 10-1 V boiling point of helium 4.2 x 100 K mass of an electron at rest 9.109534 x 10-31 kg hearing limit of teenager 2 x 104 Hz mass of delivered junk mail in U.S. 4.4 x 101 kg/s power of hurricane 2 x 1013 W == Slides from presentation on Fermi Problems: solving process problems in mathematics You are on a yacht sailing the Pacific Ocean. Your navigator announces you are over the deepest point in the ocean, the Marianas Trench. Just then, a clumsy guest accidently drops a 12- pound cannonball over the side. How long will it take for the cannonball to reach the bottom of the ocean? Did you make a completely wild guess? Did you get too bogged down in details trying to come up with the "exactly right" answer? Did you zero in on the two most important problems -- how deep is the Marianas Trench and how fast might a connonball fall through water -- then hazard a guesstimate? 99% Inspiration: Tips, Tales and Techniques for Liberating Your Business Creativity by Bryan W. Mattimore, 1994 Fermi Problems (named after Enrico Fermi, Nobel winning physicist) -- used such problems to help his students learn to think for themselves --a Fermi problem does not contain all of the information you need to solve it precisely, but you are still expected to arrive at a reasonable answer Open-Ended Problems: a number of reasonable solutions Fermi Problems: one reasonable solution, a number of possible stragegies which can lead to that answer An example of a Fermi problem: How many piano tuners are there in Chicago? --Work in your groups --Outline your method --Present your solution In math, there are "story problems" and there are "process problems" from Souviney's Learning to Teach Mathematics Fermi's suggestions to his students: 1. break the problem down into smaller, more manageable questions 2. have the courage to make some guesses/estimates and assumptions NCTM Problem Solving Standards formulate problems for everyday and mathematical situations develop and apply strategies to solve a wide variety of problems verify and interpret results with respect to the original problem acquire confidence in using mathematics meaningfully ==== University of Maryland Fermi Problems: References Publications about fermi problems. Articles John E. Carlson, "Fermi problems on gasoline consumption", The Physics Teacher, Vol. 35, No. 5, May 1997, pp. 308-309. David Chandler, "How to split hairs on Fermi questions", The Physics Teacher, Vol. 28, No. 3, March 1990, p. 170. M. St. John and Fred Reif, "Teaching physicists' thinking skills in the laboratory", American Journal of Physics, Vol. 47, 1979, p. 950. Victor F. Weisskopf, "Search for Simplicity: Mountains, waterwaves, and leaky ceilings", Am. J. Phys., Vol. 54, No. 2, February 1986, pp. 110 -111 Books Hans Christian von Baeyer, The Fermi Solution (Random House, NY, 1993). Jearl Walker, The Flying Circus of Physics with Answers (John Wiley and Sons, NW, 1977). Thanks to Rand Harrington for contributions to this site. To contribute references to this site, send them to redish@quark.umd.edu . === Naive Physics Naive physics refers to the commonsense beliefs that people hold about the way the world works, particularly with respect to classical mechanics. Being the oldest branch of physics, classical mechanics has priority because mechanical systems can be seen, whereas the motions relevant to other branches of physics are invisible. Because the motions of mechanical systems are both lawful and obvious, it is always intriguing to find instances in which people hold beliefs about mechanics that are not just underdeveloped but systematically wrong. -- Dennis Proffitt References Baillargeon, R. (1993). The object concept revisited: New directions in the investigation of infants' physical knowledge. In C. E. Granrud, Ed., Carnegie Symposium on Cognition: Visual Perception and Cognition in Infancy. Hillsdale, NJ: Erlbaum, pp. 265 - 315. Champagne, A. B., L. E. Klopher, and J. H. Anderson. (1980). Factors influencing the learning of classical mechanics. American Journal of Physics 48: 1074 - 1079. Clement, J. (1982). Students' preconceptions in introductory mechanics. American Journal of Physics 50: 66 - 71. Cooke, N. J., and S. D. Breedin. (1994). Constructing naive theories of motion on the fly. Memory and Cognition 22: 474 - 493. diSessa, A. (1982). Unlearning Aristotelian physics: A study of knowledge-based learning. Cognitive Science 6: 37 - 75. diSessa. A. (1983). Phenomenology and the evolution of intuition. In D. Gentner and A. L. Stevens, Eds., Mental Models. Hillsdale, NJ: Erlbaum, pp. 15 - 33. Howard, I. (1978). Recognition and knowledge of the water-level problem. Perception 7: 151 - 160. Kaiser, M. K., J. Jonides, and J. Alexander. (1986). Intuitive reasoning about abstract and familiar physics problems. Memory and Cognition 14: 308 - 312. Kaiser, M. K., D. E. Proffitt, and K. A. Anderson. (1985). Judgments of natural and anomalous trajectories in the presence and absence of motion. Journal of Experimental Psychology: Human Perception and Performance 11: 795 - 803. Kaiser, M. K., D. R. Proffitt, S. M. Whelan, and H. Hecht. (1992). Influence of animation on dynamical judgments. Journal of Experimental Psychology: Human Perception and Performance 18: 384 - 393. McAfee, E. A., and D. R. Proffitt. (1991). Understanding the surface orientation of liquids. Cognitive Psychology 23: 669 - 690. McCloskey, M. (1983). Intuitive physics. Scientific American 248: 122 - 130. McCloskey, M., A. Caramazza, and B. Green. (1980). Curvilinear motion in the absence of external forces: Naive beliefs about the motion of objects. Science 210: 1139 - 1141. McCloskey, M., and D. Kohl. (1983). Naive physics: The curvilinear impetus principle and its role in interactions with moving objects. Journal of Experimental Psychology: Learning, Memory, and Cognition 9: 146 - 156. Piaget, J. (1952). The Origins of Intelligence in Childhood. New York: International Universities Press. Piaget, J. (1954). The Construction of Reality in the Child. New York: Basic Books. Piaget, J., and B. Inhelder. (1956). The Child's Conception of Space. London: Routledge and Kegan Paul. Proffitt, D. R., and D. L. Gilden. (1989). Understanding natural dynamics. Journal of Experimental Psychology: Human Perception and Performance 15: 384 - 393. Proffitt, D. R., M. K. Kaiser, and S. M. Whelan. (1990). Understanding wheel dynamics. Cognitive Psychology 22: 342 - 373. Ranney, M., and P. Thagard. (1988). Explanatory coherence and belief revision in naive physics. In Proceedings of the Tenth Annual Conference of the Cognitive Science Society. Hillsdale, NJ: Erlbaum, pp. 426 - 432. Shanon, B. (1976). Aristotelianism, Newtonianism, and the physics of the layman. Perception 5: 241 - 243. Spelke, E. S., K. Breinlinger, J. Macomber, and K. Jacobson. (1992). Origins of knowledge. Psychological Review 99: 605 - 632. Further Readings Caramazza, A., M. McCloskey, and B. Green. (1981). Naive beliefs in "sophisticated" subjects: Misconceptions about trajectories of objects. Cognition 9: 117 - 123. Chi., M. T. H., and J. D. Slotta. (1993). The ontological coherence of intuitive physics. Cognition and Instruction 10: 249 - 260. Clement, J. (1983). A conceptual model discussed by Galileo and used intuitively by physics students. In D. Gentner and A. L. Stevens, Eds., Mental Models. Hillsdale, NJ: Erlbaum, pp. 325 - 339. diSessa, A. (1993). Toward an epistemology of physics. Cognition and Instruction 10: 105 - 225. Gilden, D. L. (1991). On the origins of dynamical awareness. Psychological Review 98: 554 - 568. Hubbard, T. L. (1996). Representational momentum: Centripetal force, and curvilinear impetus. Journal of Experimental Psychology: Learning, Memory, and Cognition 22: 1049 - 1060. Kaiser, M. K., D. R. Proffitt, and M. McCloskey. (1985). The development of beliefs about falling objects. Perception and Psychophysics 38: 533 - 539. Larkin, J. H. (1983). The role of problem representation in physics. In D. Gentner and A. L. Stevens, Eds., Mental Models. Hillsdale, NJ: Erlbaum, pp. 75 - 98. McCloskey, M. (1983). Naive theories of motion. In D. Gentner and A. L. Stevens, Eds., Mental Models. Hillsdale, NJ: Erlbaum, pp. 299 - 324. McCloskey, M., A. Washburn, and L. Felch. (1983). Intuitive physics: The straight-down belief and its origin. Journal of Experimental Psychology: Learning, Memory, and Cognition 9: 636 - 649. Smith, B., and R. Casati. (1994). Naive physics. Philosophical Psychology 7: 227 - 247. Spelke, L. S. (1991). Physical knowledge in infancy: Reflections on Piaget's theory. In S. Carey and R. Gelman, Eds., The Epigenesis of Mind: Essays on Biology and Cognition. Hillsdale, NJ: Erlbaum. === Free copy of SureMath for creative individuals To support your efforts in developing educational materials related to modern problem solving the publisher of SureMath has made available a limited number of copies of SureMath for individuals developing products such as those listed on the previous page. Note that SureMath is a Macintosh-only product at the present time. Those who do not have a Macintosh available can acquire a Mac Classic, SE, LC or similar Macintosh at low cost, or no cost. Several sites provide information and sources. Mac4Sale.com MacBizam ~Resources For The Older Macintosh Old Macs on the Internet At EBAY you can bid on available equipment. For example typing se/30 in the search box will lead you to an extensive list of SE/30's. Due diligence in using any auction web site is recommended. At Mac Auction you can bid on available Mac equipment. Due diligence in using any auction web site is recommended. SureMath Publishing, Inc. has obtained used computers from Computer Recylers in Cincinnati for the purposes stated above. Send e-mail to John Swartz for information. Such an acquisition provides convenient means of achieving the advantages of using SureMath. Some have set up several such Macintoshes to easily establish a simple-to-use problem solving center in their computer lab. In this way you can provide computers for 12 students for the cost of 1 machine with unused power. Those seriously concerned with education choose this 12 for the cost of one approach. For use at home you can do word processing, email etc as well as SureMath, that is school work, and relieve the demand on your expensive computer that is hogged by surfing, games and so forth. All this for $100 or so. The sensible thing to do. Of course, with the iMac now available there is no reason for not having the ease of use available with a Macintosh for your second (third, fourth) computer. Send a one-page description of the educational product you are developing, your credentials and your supporting environment to Howard C. McAllister Director of Product Development SureMath Publishing, Inc. 1737 Bent Tree Circle Fort Myers, FL 33907 A copy of SureMath will be mailed to you. There is nothing more difficult to take in hand, more perilous to conduct, or more uncertain of its success, than to take the lead in the introduction of a new order of things. Machiavelli Subject : Cooling a house with ice From: nick@vu-vlsi.ee.vill.edu (Nick Pine) Date: 15 Jul 1995 11:34:54 -0400 It's 85 F so far here today, and it's supposed to hit 100... How much ice does it take to keep a house cool in the summer? Suppose it's a completely shaded two-story house, 32' on a side, 16' tall, with R20 walls and R40 ceiling, and 20% of the walls have R2 windows, and the house leaks 1 air change per hour, and two people live in the house, and use 500 kWh/month of electricity, ie 500 kWh/30 days/month/24 h/day = 700 Watts, all of which goes into heating the house, so the house itself is heated in summertime by about (2 people x 100 W + 700 W) x 3.4 Btu/W = 3K Btu/hour. The thermal conductance of the house is the sum of each area divided by its R-value. The walls contribute 4 x 0.8 x 512 ft^2/R20 = 82 Btu/hour/F, and the windows 4 x 0.2 x 512/R2 = 205 (more than twice as much heat gain, even with no sun...) The roof only adds 32' x 32'/R40 = 26 to that number, and the volume of the house is 32' x 32' x 16' = 16384 ft^3, so air infiltration adds another 16384 ft^3/55 ft^3/Btu/F = 298 Btu/F, the biggest heat gainer of all. The total above is about 600, so when it's 100 F outside and 80 F inside, it takes (100-80) x 600 = 12K Btu/hr to keep the house cool, plus the 3K of internal heat gain in the house, ie 15K Btu/hr total, ignoring humidity. It takes about 144 Btu/pound to melt ice, and warming the water from 32 F to say, 72, requires another 40 Btu. Say 200 Btu/lb in round numbers. So each hour of summer AC requires 75 pounds of ice to begin with, ignoring the heat leaks to the ice battery itself. A month of AC requires 30 x 24 x 75 pounds of ice, 54K pounds or 27 tons, with a volume of 54K/62 = 870 ft^3, a cube 9.5 feet on a side, not counting insulation, or a 32' x 32' basement with 10" of ice in a perfectly insulated tank, under the floor, a tank in the corner, 8' high x 10.43' square, not counting insulation. I live near Philadelphia, which has average daily minimum temperatures of -5.1C, -4.0 and -2.2 in Jan, Feb and Dec of each year, according to NREL. Say it's this cold, ie 25 F, average, for 4 hours a day for 90 days, ie 2,440 cooling degree hours, with a 32 F base temperature. How large would our low-thermal mass, shallow anti-freeze pond have to be, ie how much shaded surface area do we need, with an R1 still-air film resistance, to somehow collect 54,000 pounds of ice, or 7.8 million Btu in the winter? This looks like Ohm's law for heatflow to me, ie Q = delta T x delta t x Area/R-value, with Q = 7.8 million = 2,440 Area/R1, so Area = 7,800,000/2440 = 3200 ft^2, eg a square pond, 56 feet on a side... This would work better on a night with some wind, or on a clear night with no wind. Or perhaps some snowmaking machines in a tent, over an insulated pit, with open tent flaps in the winter, or... Let's see, if an auto radiator with a fan can get rid of the heat from a gallon of gasoline in an hour, ie 100K Btu/hr, with 200 F water and 100 F air, that's 1000 Btu/hour/degree F, 1000 times better than a square foot of pond surface, so we'd need 3.2 of them. Once again, some numbers on the back of an envelope seem helpful. Nick http://forum.swarthmore.edu/workshops/sum96/interdisc/fermicollect.html http://forum.swarthmore.edu/workshops/sum96/interdisc/fermilinks.html http://www.physics.odu.edu/~weinstei/wag.html http://www.geog.buffalo.edu/ncgia/i21/papers/GISLIS96.html http://ventrella.com/Ideas/Fluid/fluid.html http://technetcast.ddj.com/tnc_program.html?program_id=50 mcharity@lcs.mit.edu