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 Electrogravitics for Advanced Propulsion Sun Sep 25,1 2016 5:14 pm
 Electrogravitics for Advanced Propulsion
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 by Thomas Valone, M.A., P.E.
 Recently, two 1956 military documents, "Electrogravitics Systems" and "The Gravitics Situation," originally published by the Gravity Research Group of London (Special Weapons Study Unit), were declassified. Outlining T. Townsend Brown's antigravity discovery (see Atlantis Rising, Number 22, p.35; AIR International, Jan., 2000; Jane’s Defence Weekly, 10 June 1995, p.34), and the subsequent Project Winterhaven, they were a vital new chapter in aviation research. For example, the documents state, "Unlike the turbine engine, electro-gravitics is not just a new propulsion system, it is a new mode of thought in aviation and communications, and it is something that may become all-embracing."
 To explain, "electrogravitics" is the science of using high voltage electricity to provide propulsive force to aircraft or spacecraft of certain geometries. Or as Jeane Manning explains, "The apparatus is pulled along by its self-generated gravity field, like a surfer riding a wave." Its discovery is often credited to Thomas Townsend Brown, a physicist who was encouraged by his professor, Dr. Paul Biefield, a former classmate of Albert Einstein. However, there are those who say that Professor Francis Nipher's experiments, electrically charging lead balls, published in the Electrical Experimenter, in 1918, predates Biefield/Brown. Unknown to many unconventional propulsion experts, T. Townsend Brown's electrogravitics work after the war involved a multinational project. American companies such as Douglas, Glenn Martin, General Electric, Bell, Convair, Lear, and Sperry-Rand participated in the
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 research effort. Countries such as Britain, France, Sweden, Canada, and Germany also had concurrent projects from 1954 through 1956.
 Furthermore, through the investigative effort of Dr. Paul LaViolette, it has become clear that electrogravitics became an integral part of the B-2 Stealth Bomber today, giving it an unlimited range. LaViolette challenges us with the question, "Could the B-2 really be the realization of one of mankind’s greatest dreams – an aircraft that has mastered the ability to control gravity?" LaViolette’s investigation is summarized in an article "The U. S. Antigravity Squadron" which has been reprinted, along with both reports mentioned above, in the book, Electrogravitics Systems, A New Propulsion Methodology. LaViolette’s book, Subquantum Kinetics: The Alchemy of Creation includes a chapter on the theory of electrogravitics and a plot of applied voltage versus disc speed from Naval Research Lab data, which starts around 40 kilovolts and 2 miles per hour.
 T. TOWNSEND BROWN
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 A curious fact revealed in T.T. Brown’s first article "How I Control Gravity" (Science and Invention, 1929) is the alignment of the "molecular gravitors". These massive dielectrics provided the most propulsive force when the "differently charged elements" were aligned (with the voltage source). This sounds like crystal plane alignment and perhaps explains the article "Gravity Nullified: Quartz Crystals Charged by High Frequency Currents Lose Their Weight" which appeared two years earlier in the same magazine in September of 1927. The editors had a change of heart however, in the following issue, and rescinded the article.
 T.T. Brown's first patent, #1,974,483 issued in 1934, "Electrostatic Motor," is a fascinating free energy machine as well as a propulsion source. Claiming an efficiency of a "million to one", Brown causes the massive dielectrics to be the workhorse of the motor, exceeding, in his words, "the well known pin wheel effect or reaction from a high voltage point discharge." Much of what we know about T.T. Brown is from his numerous patents (all of them are reprinted in the Electrogravitics Systems book), although I was fortunate enough to correspond with him in 1981 when he was at the University of Florida.
 A sample of his detailed correspondence is contained in the book, Ether-Technology: A Rational Approach to Gravity-Control by Rho Sigma (1977) which is the only other introduction to Brown's work. The important fact from that book is that the DC power supply went up to 250 kV, with a substantial force being displayed starting around 150 kV. Here we get an idea of the range of voltage necessary for successful electrogravitics that even recent military contractors mysteriously disregard. An example is R. L. Talley’s report to the Air Force concerned "with exploring the Biefield-Brown effect which allegedly converts electrostatic energy directly into a propulsive force in a vacuum environment." It was entitled, "Twenty First Century Propulsion Concept" #PL-TR-91-3009, but only tested Brown saucer designs in the range of 19 kV and predictably failed to produce results.
 Brown’s saucer tests show a propulsive force with the positive voltage leading and the negative edge trailing. The high voltage electrically charged the air around the craft with a cloud of positive ions forming in front of the craft and a cloud of negative ions behind. This has been verified with tests recently performed by researcher Larry Davenport. These tests are reprinted in the book, Electrogravitics Systems and can be seen in the commercial video, "Free Energy, The Race to Zero Point" for which I was the technical consultant.
 In March, 1952, the Townsend Brown Foundation laboratory in Hollywood, California was visited by Air Force Major General Victor E. Bertrandias. He reported to Lt. Gen. H. A. Craig that he was "frightened" by the flying demonstration because it was in private hands and he felt it was "in the stage in which the atomic development was in the early days." He expressed concern about "if it ever gets away," meaning, we presume, "into enemy hands." A confidential security investigation was thereby initiated concerning the Foundation and T.T. Brown. Shortly afterwards, an evaluation by the Office of Naval Research in September, 1952 surprisingly devalued the Brown saucers to a "well-known phenomenon of the electric wind" claiming it would perform "less at high voltage and zero in a vacuum." The Navy declassified the report in October, 1952. However, today, copies of the report are not available from the Naval Research Laboratory in Washington, DC.
 In 1956, Brown returned from a research trip in France where he verified that electrogravitics worked well in a vacuum, in other words, the environment of outer space. Interavia Magazine published an article in the same year about Brown entitled, "Towards flight without stress or strain or weight" and alluded to possible speeds of several hundred miles per hour. In 1958, Fate magazine writer Gaston Burridge described Brown’s metal discs that reached up to 30 inches in diameter. Until 1960, Brown and Agnew Bahnson worked on various designs in Bahnson’s
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 laboratory which were recorded by Bahnson’s daughter on Super-8 film. Today, a VHS converted silent video is available of those experiments entitled, "Thomas Townsend Brown: Bahnson Lab 1958-1960." In 1964 Bahnson, an experienced pilot, mysteriously flew into electric wires and died. The Bahnson heirs subsequently dissolved the laboratory project.
 ELECTROGRAVITICS PUZZLE
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 In 1985, Dr. Paul LaViolette was in the Library of Congress in Washington, DC and looked up the work "gravity" in the card catalog. Surprisingly, he found the listing for "Electrogravitics Systems," a report that was missing from the stacks. When the librarian tried to locate any other copies through interlibrary loan, she commented, "It must be an exotic document" because she could find only one in the country which was at Wright-Patterson Air Force Base. Thus, LaViolette was successful in obtaining a copy of the formerly classified document. The mystery continued: seven years later when contacting the Wright-Patterson AFB Technical Library, they surprisingly found no reference in the computer-based card catalog. They did locate the document on the shelves, however, after being asked to search for it. To summarize, the report has historic value because:
 ◦ It validates T.T. Brown's experiments; ◦ It lists the major corporations that were collaborating on electrogravitics; ◦ It includes the requirements for supersonic speed; ◦ It shows the continuity from Project Winterhaven in 1952; ◦ The report includes a list of electrostatic patents; ◦ It had been classified by the Air Force for an undetermined amount of time which underscores its importance.
 Prepared by the Aviation Studies (International) Ltd., Gravity Research Group, Special Weapons Study Unit in England in February of 1956, it defines electrogravitics as "a synthesis of electrostatic energy use for propulsion." The report historically notes that: "Electrogravitics had its birth after the War, when Townsend Brown sought to improve on the various proposals that then existed for electrostatic motors sufficiently to produce some visible manifestation of sustained motion." As mentioned in the first section of the report, both Project Winterhaven (1952) and "Electrogravitics Systems" (1956) propose "a saucer as the basis of a possible interceptor with Mach 3 capability." Another interesting detail presented is the necessity of an insulator with a exceedingly high "dielectric" constant of 30,000 for supersonic speed when the best dielectrics of that era were around 5,000. This section goes on to describe the creation of a local gravitational system by the craft which "would confer upon the fighter the sharp-edged changes of direction typical of motion in space." The January, 1955 entry states:
 "Back in 1948 and 49, the public in the U.S. had a surprisingly clear idea what a flying saucer should, or could do. There has never at any time been any realistic explanation of what propulsion agency could make it do those things, but its ability to move within its own gravitation field was presupposed from its manoeuverability. Yet all this was at least two years before electrostatic energy was shown to produce propulsion. It is curious that the public were so far ahead of the empiricists on this occasion..."
 The intriguing part of this commentary is that without any space program at the time, the report complains that the public knows how UFOs behave and refers to sharp-edged changes of direction. Later in the report, we read, "One of the difficulties in 1954 and 1955 was to get aviation to take electrogravitics seriously." However, corporations such as Douglas, Sperry, Bell, GE, Hiller, Lear, and Convair are then described with an ongoing-project perspective. For example we read that, "General Electric is working on the use of electronic rigs to make adjustments to gravity." "Glenn Martin say gravity control could be achieved in six years....Clarke Electronics state they have a rig, and add that in their view the source of gravity's force will be understood sooner than some people think." This information makes the report exciting reading and gives it an air of suspense.
 Even today, electrogravitics continues to attract public attention in the press. The latest is an article entitled "Military Power" published in a British aviation magazine, AIR International, (Jan., 2000) that includes copies of LaViolette’s drawings from the Electrogravitics Systems book. The article also cites the Aviation Week and Space Technology article from March, 1992 "Black world engineers, scientist, encourage using highly classified technology for civil applications" which caused LaViolette to investigate the B-2 Bomber connection to T.T. Brown’s electrogravitics.
 JOHN SEARLE'S ELECTROGRAVITY
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 John R.R. Searl, of England, constructed numerous craft purported to fly with high voltage (see the recent biography Antigravity: The Dream Made Reality book by John Thomas). However, one correction to the "Antigravity" article from Atlantis Rising Number 22 issue is that the positive pole was traditionally at the periphery of Searl’s crafts. This is important because as Searl describes his control of the imbalance of positive voltage on the edges, to steer the craft, he found that the saucers would travel toward the more positive side, exactly like T.T. Brown’s saucers behave!
 Throughout the sixties and the seventies, J.R.R. Searl produced many newsletters detailing the work he was doing. Since I corresponded with him in 1981, I also received some of these reports. The importance of his experiments lies in the electrogravitics phenomena associated with them. In the 6/1/68 issue of the "Searl National Space Research Consortium" newsletter, Barrett reports that the ionization of the air and permanent electric polarity of dielectrics were common along with the antigravity effects. In the 6/14/71 issue of the newsletter, Bernhard Vaegs reports that "a pink halo surrounded the craft" and describes the effect of the millions of volts that were generated. This type of description is found throughout the reports and probably was measured by the length of the spark discharge considering the approximate voltage breakdown of air. Barrett describes in the 6/1/68 issue a vacuum layer that surrounds the craft preventing ionizing breakdown of the air. The similarities between Searl's high voltage propulsion and T.T. Brown's high voltage propulsion that both are based upon the principles of electrogravitics as theoretically predicted by Dr. Paul LaViolette in his previously mentioned book, SubQuantum Kinetics.
 THE HUTCHISON EFFECT
 In 1980, George Hathaway, a professional engineer licensed in Canada, along with entrepreneur, Alex Pizzaro, formed a small company to develop and promote what is referred to as "The Hutchison Effect." It is named after its inventor, John Hutchison, who liked to experiment with combinations of Tesla coils and Van de Graaff generators at the same time. Much of the information about the "lift and disruption" effects has been reported at various conferences (such as in the Third International Symposium on Non-Conventional Energy
 Technology held in Hull, Quebec in 1986). Videotapes of much of the phenomena have been shown on Japanese TV as well. Hathaway also assembled a three-hour videotape that documents the TV interviews, reports, and actual events. To summarize, the experiments were conducted with 250 KV of DC power on the Van de Graaff and about the same voltage of AC power on the Tesla coil. The total real power was about 1.5 KW continuously, according to Hathaway. Besides the disruptive effects, which were numerous, the lifting of various heavy objects by the field was most impressive. These events can be seen at the end of the commercial video, "Free Energy, The Race to Zero Point." In regards to the AC contribution to the field, Hathaway reports that he measured a small voltage of 2 millivolts per meter in the active region (besides the DC offset). This is a small AC signal but on top of the high voltage DC signal, it performs amazing feats.
 The importance of the Hutchison Effect to Brown electrogravitics is the AC "ripple" on the high DC voltage. A reference to this may be found in a military report by Dr. Dennis Cravens who gave T.T. Brown a high rating of "practicality." Cravens reported in his evaluation of Brown that older, high voltage supplies always had some AC ripple to the regulated signal, and wonders if this had any effect on Brown’s phenomena (Cravens, T.L. "Electric Propulsion Study", AL-TR-89-040, #ADA 227121, Science Applic. Inter. Corp., Torrance, CA 90501). Dr. LaViolette has also found this factor to have particular electrogravitic significance.
 THE B-2 STEALTH BOMBER CONNECTION
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 Thanks to Dr. Paul LaViolette reporting in his article, "The U.S. Antigravity Squadron", there is substantial evidence that the electrogravitics research of the 1950's actually resulted in the B-2 Stealth Bomber "auxiliary propulsion system." Summarizing Dr. LaViolette's article, with references cited therein, the following facts are the most convincing:
 ◦ The B-2 charges the leading edges of its wing-like body, with high voltage; ◦ The B-2 is shaped just like T.T. Brown suggested an electrogravitic craft should look, for maximum charge separation; ◦ Northrup tested leading-edge charging in 1968; ◦ B-2 electrically charges the exhaust similar to the suggestion by T.T. Brown that the craft should be powered by needle-point flame-jet generators which electrically charge the exhaust; ◦ Aviation Week admits the existence of "dramatic, classified technologies" applicable to "aircraft control and propulsion" on the B-2; ◦ Aviation Week also disclosed that a high-density dielectric ceramic RAM made of powdered depleted uranium happens to have a mass density of 3 times that of the high-K dielectrics tested in the 1950's; ◦ The B-2's Emergency Power Units (EPU) can drive an electrical generator at high altitudes or even in space, since the fuel can be made to decompose rapidly even without oxygen; ◦ Edward Aldridge, the Secretary of the Air Force, admits that the B-2 creates no vapor trail at high altitudes. ◦ The decomposed gases from the EPU's can function as the ion-carrying medium, in much the same way as the hot exhaust gases from the air breathing flame jet generators.
 These details create a sense of excitement about the world's foremost aircraft. Dr. LaViolette argues that the electrogravitic drive will function better at higher speeds due to the better flow of the ions. Therefore, it is likely, he says, that the B-2 actually is a supersonic aircraft, especially since the 1968 leading-edge charging experiments were for supersonic softening of the shock wave.
 House Representative Robert Walker was quoted recently in Popular Science ("Secrets of Groom Lake") as promoting the idea of declassifying military secrets that will help commercial development. We hope that this trend will continue so that advanced Shuttle designs may also acquire an electrogravitic drive.
 Through a proposal he submitted in 1990, Dr. LaViolette made NASA aware that an electrogravitic drive would be a feasible propulsion method for the Mars journey because calculations show that the transit time can be less than one month, instead of half a year to a year. It is especially attractive since it uses so little power when it is operational and verified by Brown to work well in a vacuum. More information about electrogravitics and Brown’s patents are available in my book, Electrogravitics Systems: Reports on a New Propulsion Methodology. Many of the books and videos mentioned in this article are available, as a public education service, from our non-profit organization, Integrity Research Institute, www.integrity-research.org
 (c) 2000 Thomas Valone Integrity Research Institute www.integrity-research.org Thomas Valone, a licensed professional engineer, college teacher, and physicist, is the author of numerous articles and books on non-conventional energy and propulsion. His first book, The
 Homopolar Handbook: A Guide to Faraday Disk and N-Machine Technologies, is now available in paperback.
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 From http://users.erols.com/iri/TTBROWN2.htm
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 Electrogravitics & UFO propulsion - PAUL A. LaVIOLETTE, PH.D
 For more information about electrogravitic propulsion see [url=http:// nexusilluminati.blogspot.com/search/label/electrogravitic propulsion]http:// nexusilluminati.blogspot.com/search/label/electrogravitic%20propulsion[/url]
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 rexresearch.com
 Thomas Townsend BROWN
 Electro-Gravity Device
 Office of Naval Research File 24-185 (15 September 1952)
 An Investigation Relative to T. T. Brown
 by Willoughby M. Cady (ONR, Pasadena)
 1 ) Introduction
 This report summarizes a technical investigation of certain claims of Thomas Townsend Brown concerning a proposed method of propulsion for aerial vehicles. The investigation, which began May 3, 1952 and which is concluded hereby, is mainly analytical, no allotment having been made for an experimental program.
 2 ) Brief Description of the Alleged Phenomena
 Mr. Brown claims to have discovered several new physical effects, and considers them to be interrelated. We will be concerned here with those relating to the force upon and between electrically charged bodies in air or under oil.
 2.1 ) It is asserted that the force exerted upon a negatively charged body by a positively charged body is greater than the force exerted upon the latter by the former; that the force differential is proportional to the mass of the dielectric inserted between the bodies; and that the phenomenon is evidence of a new "electrogravitic" effect. Paragraph 4 contains a fuller description of these phenomena.
 2.2 ) A corollary effect is the force tending to move a metallic disc which by means of a plastic disc supports a fine wire, AA tangentially disposed in its plane, as shown in Figure 1. When the wire is positively charged (e.g., 30 kilovolts) and the disc negatively charged, a propulsion force of a fraction of an ounce appears which moves the disc (and with it the wire) in the
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 direction of the arrow C. These effects have been witnessed by the writer. Further discussion is presented below in paragraph 6.
 Figure 1: Demonstration Flying Saucer
 2.3 ) An apparatus has been examined superficially, which shows an unexpected torque on an oil-immersed disc when charged. This, according to Mr. Brown, is arranged to record automatically and frequently the magnitude of the angular deflection of the disc. These records have been tabulated and partially analyzed by Mr. Brown. Greater detail regarding this device is given in paragraph 5 below.
 2.4 ) There has been briefly inspected a third electrical device, purporting to be a communication equipment operating on waves related to the electrogravitic effect. It appears that the transmitter is simply a relaxation oscillator consisting of a high tension DC power supply, a high resistor, and a condenser which periodically discharges through a short gap. The receiver, shown in Figure 2, consists of two Pyranol capacitors 0-5 mf, two titanium oxide capacitors 500 mf, a Brush recorder, and a bypass capacitor 0.1 mf to prevent the recorder from oscillating. Two terminals in the circuit are charged as indicated before reception and left floating during reception. The bridge of four capacitors is housed in a cabinet, indicated by a dashed line, which forms an obviously imperfect shield against electromagnetic radiation. When the transmitter is operated in an adjoining room, the Brush recorder jumps with each spark. Mr. Brown states that the effect is present, but reduced, if the 25 kv potentials are not applied. If the Pyranols are short-circuited and removed, the effect is still reported present. No attempt
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 has been made at conscious electromagnetic shielding of either the transmitter or the receiver. In the writer's opinion the response may be the result of a chance rectifying contact.
 The relation between the communication device and the mechanical effects which form the topic of this report appears to remote that no further mention will be made of the former.
 Figure 2: Electro-Gravitic Communication Device
 3 ) The Investigative Method
 This study is limited to the technical claims of Mr. Brown and is not concerned with matters of personality or personnel, or with the Townsend Brown Foundation. Although propulsion is the immediate center of interest, the claimed basic effect and its several manifestations are of the essence also. Therefore the analysis is not restricted to the propulsion apparatus.
 Rough quantitative data have been taken by the writer on the force between charged bodies, and on the force on the disc shown in Figure 1. He has also carried out a brief examination of the recorded data on the oilimmersed disc. Whenever possible, word-of-mouth data have been ignored. It is regrettable that during the time of the investigation the greater part of the experimental data gathered over the years by Mr. Brown are unavailable, being in storage elsewhere.
 Full reliance is placed on the sincerity of Mr. Brown in his written reports of the data, but no reliance is placed on his interpretations. His orally reported data have been partially discounted as being less reliable than his written records. There is no evidence of conscious deceit.
 Most of the interviews have taken place at the Townsend Brown Foundation,
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 306 North Vermont Avenue, Hollywood, California.
 4 ) The Basic Effect
 At Zanesville, Ohio in 1927, Mr. Brown at this then Laboratory of Research Devices, performed experiments on force acting upon charged conductors. The apparatus took two forms, as described in a manuscript "Tapping Cosmic Energy", dated 28 October 1927, which was submitted at that time to the Physical Review for publication but which was rejected. Apparently no basic experiments have been performed since 1928.
 4.1 ) In one form the conductors were two independent spheres suspended close but not touching by means of fine wires which also served as electrical leads. The spheres were of lead, and weighed 14,740 gr each. The wires supporting them hung from the ceiling 45 cm apart. When the spheres were oppositely charged to a potential difference of 125 kv they moved toward each other slightly in response to their electrostatic attraction. The negative sphere was displaced allegedly about twice as far as the positive sphere. The inequality of the displacement is the essence of the phenomenon.
 4.2 ) In the other form of the basic experiments the condenser was formed by a block of insulating material 3-1/3 c 2-1/2 x 15", to the ends of which electrodes were attached. Whether the condenser was suspended or whether it was mounted at the end of a centrally pivoted horizontal beam, it showed a force causing the arm to rotate, usually in such a direction that the positive electrode moved forward. The force is claimed to have been proportional to the mass of the block, although no data are given in the manuscript and none are available. The name "gravitor" is used for this apparatus.
 The condenser dielectric was either homogenous or laminated. In one laminated version, 10,000 sheets of 0.001" lead foil were separated by an equal number of 3 x 3" sheets of 0.0008" cellulose acetate in a formica box, to make 10,000 condensers in a series arrangement weighing 10 kg. The acceleration exhibited by this device when suitably charged is claimed to have been 2.5 cm per sec2 when the electrodes were charged, evidencing a force of 25,000 dynes or about 25 gm; the homogenous type is stated by Mr. Brown to have had comparable thrust. The force was usually toward the positive electrode, although in some units it was toward the negative; this effect was attributed to a "polarization". The force is stated to have been stronger at some time that others, correlating with the position of the sun and moon. Data are displayed which relate to the angular speed of a pivoted gravitor, and which show half a dozen large fluctuation in a day; some of these can perhaps be construed a being dips simultaneous with the rise or setting of the sun or moon.
 On 17 June 1952 the undersigned accompanied (censored) of the Townsend Brown Foundation to the Glendale plant of the Bendix Aviation Corporation. An experiment of the type described in paragraph 4.1 had been set up by Bendix personnel. (Sentence censored). The observations in Table 1 were
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 recorded by the undersigned.
 Table 1: Observations by the Writer on 17 June 1952 Experiment at Bendix Aviation Corp., Glendale, CA
 From a wooden scaffold a cylindrical brass weight 8" long and 4" in diameter was hung with axis vertical by means of about 5' of #14 Cu wire (0.06408" diameter). A similar wire about 6" away suspended a similar weight level with the first. By means of the wires the weights were electrified to a measured voltage; the corona current was not measured. An optical projection system provided on the wall an image on a scale and insulated pointers attached to the weights. The weights were either hollow (5 lb) or solid (30 lb). In certain of the observations the weights were linked by an insulating bar.
 4.3 ) An examination of the data makes it clear that the inequality in deflections reported by Brown is still present here, though only to the extent of 1-30% (tests 2-4) and less than that under conditions where the corona discharge is reduced (of tests 1 & 5).
 When cigarette smoke was drifted into the vicinity of the wires it showed the existence of a wind. Just above the weights the wind blew from negative top positive, while higher on the wires the wind was opposite.
 5 ) The Brown Electrometer
 In a small thermostated room in the second basement of the Banks-Huntley Building (address censored), Los Angeles, there is a continuous experiment which shows an extension of the basic phenomenon. This apparatus is the Brown Electrometer. The device is housed in an oil-filled tank of half-inch steel, perhaps two feet in diameter and a foot deep, with a half-inch steel
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 cover. In the tank there is suspended a horizontal disc 12" in diameter and one inch thick, made of six segments of pine wood alternating with six of marble around an 8" hub of Bakelite. The sections are separated by 1 x 2 x 1/16-inch copper strips. Alternate strips are grounded to a central metal stem supporting the disc which is attached to a music-wire suspension 0.018" in diameter and 6-3/8" long; the remaining copper strips connect to a central iron tit at the center of the hub, which dips into a cup of mercury. The disc is surrounded by a grounded metal shield which is attached to the tank and which stands at a clearance of 1/2-inch above and below the disc and 1-1/2 inch radially; this shield is centrally perforated at the top and bottom for the suspension and the tit.
 A sequencing device in a cabinet below the tank causes a half-wave smoothed DC power supply to impress upon the mercury cup a wellregulated negative potential of 11.1 kv resulting in a leakage current of 0.45 ma. The sequencer leaves the voltage on for 30 seconds, causing the disc to rotate several degrees. After 30 seconds, a short spark is made to jump from a point projecting about 9 inches horizontally in air from the stem of the disc, and perforates a rice-paper strip wound on a drum which rotates once each day. The voltage is then turned off and the disc returns to its relaxed position, which is recorded by a separate spark. The cycle repeats every three minutes. The apparatus appears to be well engineered and extremely reliable.
 5.1 ) Mr. Brown has given attention for several years to the variations in the deflections of this electrometer. On the spark record the rest position is 50 units and the deflected reading is about 80. Until about a year and a half ago the deflected position was about 20, but suffered a gradual change to its present value at that time. Aside from the reversal of sign, the deflection undergoes fluctuations whose character is less dramatic but still large, e.g., 10-50%. The cause of these fluctuations has been sought by Mr. Brown by seeking for correlations between them and various outside influences, such as magnetic storms, temperature, pressure, the time of the year, civil time, lunar time, sidereal time, and the Dow-Jones average. Only the last five correlations are claimed. Tabulated data are available only for the year 1937. Hourly recordings throughout that year are tabulated, both for civil and lunar time. The writer has seen no convincing evidence that the stock market and the automatic recorder are correlated, although it has been claimed that a chi-square analysis has been performed, with positive results.
 5.2 ) The undersigned has conducted a partial statistical analysis of the data from the automatic recorder for 1937 as follows:
 5.2.1 ) Use was made of a table furnished by Mr. Brown entitled "Hourly Readings of Instrument, 1937" and particular attention was directed toward the readings of 0100, one hour after midnight, on each day of the first half of 1937. During the first half of January, for instance, the readings on days at this hour were 26.20, 25.90, 26.20, 26.10, 26.80, 25.60, 25.40, 26.00, 26.00, 25.60, 26.60 and 25.90. The mean is 26.05, the standard deviation
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 0.55 and the probable error of the mean 0.15. Table 2 gives the halfmonthly means and probable errors for time 0100 and for time 1300. This table shows that there was a strong and real variation during the spring of 1937; but the great similarity between the readings at 0100 and 1300 implies that there is no conspicuous diurnal effect. (See also paragraph 5.2.2)
 5.2.2 ) A direct search was undertaken by the writer to see whether the data for the first half of 1937 reveal any solar diurnal period. Attention was confined to the first day in each half-month. On these twelve selected days the readings at time 0200, for example, were 26.20, 26.00, 27.90, 29.00, 28.10, 29.80, 30.40, 29.80, 28.90, 27.50, 27.40, and 28.10. The mean was 28.29 and the probable error of the mean 0.46. The probable errors, as far as calculated, were all in this neighborhood. Table 3 gives the means calculated as in the above example. It is seen that there is no strong diurnal periodicity, with the dubious exception of a peak at 1100 hours; this result confirms that of paragraph 5.2.1.
 5.2.3 ) The undersigned has scrutinized a table of data captioned "1937 Readings of Instrument with Lunar Hour Angle" in a search for evidence of a variation of the readings with the position of the moon, as rising, setting, etc. The form taken by this search was as follows. For the calendar month January the average deflection for each lunar hour was noted; e.g., at lunar hour angle 1300, when the moon was one hour past the meridian, the readings for January averaged 26.74. At the same lunar angle in the succeeding calendar months the average readings were 28,60, 30.18, 29.50, 28.02, and 28.52. The average of those six averages gives 28.59 as the mean of all observations taken at lunar hour angle 1300 during the first half of 1937. This process was repeated for each of the 24 hours. The means are listed in table 4. Their constance is most striking; they show no dependence upon lunar hour.
 Table 2: Half-month mean deflections at two times a day, during the first half of 1937
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 Table 3: Hourly deflections averaged over the first day of the halfmonth in January-June 1937. The Probable errors are above 0.46
 Table 4: The deflection of the instrument as averaged over the first half of 1937 for each lunar hour angle
 5.3 ) By the foregoing sketchy analysis of the data from the Brown Electrometer, the following conclusion are implied:
 (a) Through the weeks and months the data drift significantly; (b) There is no evidence of a solar diurnal period synchronizing with the sun, as claimed by Brown; (c) There is no evidence of a lunar diurnal period synchronizing with the moon, as claimed by Brown. On the contrary, the data appear most emphatically to deny such a period.
 5.4 ) Two related experiments have been described by Mr. Brown which throw light on the reason for the deflections if not the reason for their fluctuations.
 5.4.1 ) In one, the sector disc was held stationary while a grounded metal disc was suspended horizontally over it in the oil, by a torsion fiber. When the sector disc was charged, a motion of the adjacent oil was noted by Mr. Brown. The suspended disc deflected, but not when a glass plate was
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 interposed between the sector disc and the suspended disc.
 5.4.2. ) The other related experiment concerns an oil-immersed annulus described by Mr. Brown as 2 feet in diameter and 6 inches thick, made of lead oxide (PbO) and paraffin in eight segments separated by 8 copper strips alternately charged positive and negative. By means of a probe the potential distribution on the PbO and paraffin surfaces was explored and it was noted that whereas the gradient was uniform across the edges of the lead oxide sectors, the gradient on the paraffin was steepest near the negative electrodes. This observation means that the steepest gradients are all in the same direction, e.g., clockwise. It was also noted that the application of high voltage to the electrodes of this disc caused a lively motion of the oil.
 6 ) The Model Flying Saucers
 At the quarters of the Townsend Brown Foundation a room is devoted to demonstrating a new method of propulsion (para. 2.2) which is claimed to depend upon the electrogravitic effect.
 6.1 ) At the center of the room there stands a tall iron post carrying at its top a transverse Plexiglass beam about 6 feet long, which is freely pivoted at its center to rotate in a horizontal plane. From each end of the beam hangs a model flying saucer, Figure 1, of which the metal part is composed of two spun aluminum discs 12" in diameter, between which is bolted an 18" Plexiglass disc. The periphery of this disc is indented with nearly semicircular scallops. The fingers remaining between the scallops constitute standoff insulators supporting a bare 0.005-inch stainless steel wire AA at a distance of 3 inches from the aluminum spinnings and insulate it from the aluminum. This will be called the outboard wire in the following discussions. It embraces perhaps one-quarter of the circumference. A similar wire, BB, is supported at the opposite quadrant, but is connected to the aluminum spinnings; it appears to be without effect. Each of these two identical assemblies is suspended by an insulating string from one end of the Plexiglass beam. By rubber-covered leads, shown in Figure 3, from the beam spinnings can be given one polarity while the outboard wires are given the opposite polarity. The leads are energized through rotating contacts at the pivot. If the beam rotates counter-clockwise as seen from above, the discs revolve about the post, in such a way that the outboard wire is at the leading edge.
 A full-wave rectified unsmoothed high-tension supply in an adjacent room provides voltage for the demonstration. The voltage is controlled by a Variac. An examination showed no hidden connections.
 6.2 ) When the power was turned on, the discs became negative and the outboard wires positive. Corona discharge could be heard as a gentle hiss or crackling. It could be seen in the dark, particularly around the outboard wire. As soon as the potential was applied, the discs and the beam began to revolve, the leading edge being at the outboard wires.
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 6.2.1 ) The thrust was roughly measured by a method in which the rotation was prevented, as shown in Figure 3. The weight on the pan was adjusted until the indicated angle was 45 degrees, at which time the thrust was known to be equal to the force in the vertical thread. This method of thrust measurement is valid if the pivot is frictionless; actually, the friction seems to introduce an uncertainty of about 0.5 gm in the thrust data. The procedure described provides a value of the thrust of the two discs combined.
 Figure 3: The method involving three threads A, B & C as used to measure the thrust of a demonstration flying saucer
 6.2.2 ) The data shown in Table 5 were gathered by the writer on 6 June with the collaboration of Mr. Brown and (censored), an assistant. Column 1 gives the setting of the Variac in the supply circuit for the high-tension transformers. The corresponding voltages, with respect to ground at the positive and negative outpost cables from the rectifier are given in columns 2 & 3; at each polarity the voltage was measured with the aid of a 0-2 milliameter and a high series resistor. The current in each lead, recorded in Columns 4 & 5, was measured by the same milliameter when appropriately reconnected; it may be resumed that most of the current represented corona discharge between the discs and the outboard wires. Column 6 gives the measured thrust. In Column 7 is recorded the time taken for the discs to revolve once completely when free to do so, after they had had time to reach their full speed. The entires in Column 8 give the thrust to be expected on the basis that it is caused by the electric wind (see para. 7 below). The last Column shows the efficiency of the propulsion, calculated (with due heed to the conversion of units) from the ratio of the electrical power input (Col. 2 x Col. 4 plus Col. 3 x Col. 5) to the mechanical power output (Col. 6 x Velocity). The data in Columns 2 & 3 were gathered first, then those in Columns 4, 5 & 6, and finally in Columns 2 & 3 again. The repeat measurements of the voltages nearly always agreed with the first
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 measurements to within 0.75 kv or less. The average is recorded in Columns 2 & 3.
 Table 5: The results of measurements on the demonstration apparatus for flying saucers
 6.2.3 ) Three additional observations were made. In one, slips of thin cardboard were cut, folded double and so hung on the outboard wire of each disc as to cover practically its entire length. Their purpose was to confine the positive ions to the region of the wire and so reduce the corona. It was observed that the thrust, originally about 7 gr as shown in Table 5, was reduced to about 1 gr.
 6.2.4 ) In the second additional observation the polarity of the discs was made positive while the wire was negative. The thrust was then 3 / 1 gr (in the same direction as before) instead of about 7, the Variac setting being 40, 45 and 50.
 6.2.5 ) The third additional observation was that when the disc was charged and immobilized as in Figure 2, a brisk wind was developed. The origin of the wind was apparently near the outboard wire. Its direction was from the wire toward and past the aluminum spinning as shown by cigarette smoke.
 7 ) Possible Theories of the Townsend Brown Phenomena
 7.1 ) Mr. Brown has no theory for the effects that he has demonstrated or claimed. He has advanced an empirical explanation for the magnitude of the force on a suspended condenser, but has not advanced supporting evidence. While granting the existence of the electric wind, Mr. Brown seems nevertheless to believe that some new force is operating which he describes as electrogravitic. This force is variable with time as shown by the Brown Electrometer. The origin of the force is of smaller interest to Mr. Brown than is the cause of its variations. No cause for the variations has been proposed, other than the alleged correlations with astronomical bodies.
 7.2 ) It might be suggested that electrostriction is significant in Mr. Brown's effects. This suggestion is untenable because (1) the deflection of a suspended condenser (Table A [sic], Tests 6 & 7) is far greater than any conceivable electrostrictive expansion or contraction; (2) electrostriction
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 would cause no acceleration of the center of the center (sic) of gravity; and (3) the steady rotation of the flying saucer models cannot be caused by electrostriction, which is a dynamic effect only while the voltage is changing, and not in the steady state.
 7.3 ) The force exerted upon a wire carrying a current in a magnetic field could conceivably contribute to the phenomenon. This hypothesis appears tenable at first sight, since the earth always provides the field, and with the corona discharge there is always associated a current. It is easily disproved, however, by (1) the sign of the deflections shown in Table 1 was unchanged in test 3 when the polarity, and thereby the direction of any magnetic effect, was reversed; (2) the direction of the thrust by the model flying saucer was unchanged by changed polarity; (3) the equal and opposite corona currents in the leads to the saucers should result in equal and opposite magnetic force, and zero net thrust; and (4) there is no report that the Brown Electrometer shows normal deflections during magnetic storms.
 7.4 ) The writer proposes that the origin of the observed phenomena is the electric wind. The effects described above all relate to the forces upon electrically charged bodies or upon adjacent insulating material. Such forces have long been the subject of observation and mathematical theory, and are discussed accordingly in such works as W.R. Smythe's "Static and Dynamic Electricity". It is commonly held by physicists that no practical discrepancies exist between theory and experiment in static electricity, if the experiments are conducted in the manner contemplated in the theory. It must be emphasized in this connection that the theory of electrostatics is written around the assumption that the insulators in the apparatus are perfect non-conductors, and that the fluid (e.g., air or oil) in which the apparatus is immersed is also a perfect non-conductor. For an experiment performed in a non-conducting medium the verdict of electrostatics is quite plain: the Townsend Brown effects should not exist.
 The experiments by Mr. Brown, including all those described in paras. 4, 5 and 6 were carried out in a partially conducting medium, i.e., air or oil. In such a medium the simple laws of electrostatics are not applicable. It is true that in the air or oil the conduction of electricity is usually so slight that the departures from the laws of electrostatics are quite negligible. When, however, large voltage gradients are involved, the density of ions is very greatly increased for well-known reasons, and the conductivity of the medium increases commensurately. If the voltage is high enough the passage of the ions through the air causes the crackling sound and bluish glow characteristic of the corona discharge. In the presence of such a discharge, the laws of electrostatics are not expected to hold. We will show that, so far from being forbidden, the Townsend Brown effects are actually predicted in the presence of a corona discharge.
 7.5 ) We will adapt the treatment cited in (name censored) ("Conduction of Electricity Through Gases, 3rd ed., Vol. II, 1933, page 544) in considering the force on a charged conductor brought about by the ionic conductivity in a gas. Consider a positively charged fine wire in air, parallel to a negative
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 conductor. The well known concentration of the electric field near the fine wire will then produce a corona discharge; positive ions are generated at the wire. The wire being positively charged repels the ions. The force on the wire is roughly equal and opposite to that on the ions. If E is the field intensity and u the space charge density, the force on the ion is:
 Where the integration covers the region occupied by the space charge. Here if we ignore the non-parallelism of the E vectors, we may write roughly
 It is known that the velocity W of an ion in a fluid is related to the electric field in which it moves through the relation
 W = kE
 Where K is ionic mobility.
 For a positive ion in air at atmospheric pressure its value will be taken as 1.32 (cm/sec)/(volt/cm), while for the negative ion the value will be taken as 2.51 (cm/sec)/(volt/cm).
 since
 is the current,
 Where d is the distance between the conductors. This force on the ions is communicated to the air by molecular collisions, and creates what is known as the electric wind; it is observed in the neighborhood of sharp points as well as fine wires. The thrust may be regarded as the force of reaction which results from the generation of the wind; the thrust is in the direction opposite to the wind, as in a jet airplane.
 7.5.1 ) It is known that the mobility of a negative ion is greater than that of a positive, the ratio being about 1.9 at atmospheric pressure. The force on a
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 positive conductor should therefore be, and according to (censored) and (censored) is observed to be, greater than that on a negative conductor.
 7.5.2 ) Further confirmation of the above theory is found in the variation of F with pressure. Theory predicts that since k is well known to vary as the inverse square of the pressure the force should vary directly as the square root of the pressure; this prediction agrees with the early results of Arrhenius, Wied. Ann. 63: 305 (1897).
 7.5.3 ) The efficiency of the electric wind as a propulsive agent is easily calculated. The power output is the product ViD/k of the thrust and the vehicle velocity V; while the power input is iv is the potential difference. The efficiency is then given by
 At a given vehicle velocity the efficiency can be improved by increasing the electrode spacing D or by decreasing the voltage necessary to sustain the corona.
 7.6 ) Physically the theory of the electric wind is a shown in Figure 4, which shows a section view of two parallel charged wires in air. Positive ions are formed at the positive wire and surround it, but the attraction of the negative wire draws the positive cloud so that it lies primarily to the left of the positive wire in Figure 4. The slow mobility of the positive leads to a considerable positive charge in this region, which is repelled by the positive wire and which repels it. Negative ions are generated at the negative wire, and form an unsymmetrical cloud about it. Since their mobility is fairly fast, they are less crowded than the positives. The force on the positive wire is made of two parts; attraction to the left by the negative cloud (e.g., 20 gr) and repulsion to the left by the ion cloud (e.g., 10 gr). the force on the negative wire is: attraction to the right by the positive wire (e.g., 20 gr) and repulsion to the left by the ion cloud (e.g., 5 gr). The net force on the positive wire would in this hypothetical case be 10 gr to the left, while that on the negative wire would be 15 gr to the right. The difference, 5 gr to the right would be directly observed if the weights hanging from the wires were connected by a rigid insulating link.
 Figure 4: The ion clouds near two parallel oppositely charged wires in air
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 8 ) Test of the Electronic Wind Hypothesis
 It is believed by the writer that the electric wind in air and in oil leads to all the well-documented results described herein. These results will be summarized and compared with the theoretical predictions.
 8.1 ) (See para. 4.1) The positively charged sphere in the Zanesville experiments moved less that the negatively charged weight. This is as expected because the electrostatic force on the positive weight was partly balanced by the kinetic reaction from the electric wind which presumable existed (para. 7.5 & 7.6) because of the corona discharge at the wires.
 8.2 ) (See para. 4.3) The inequality in the motion of the weights in the Bendix experiment was less noticeable, thus discouraging corona and reducing the current. The observed electric wind was caused by corona at the wires, which was more or less strong according as the wires were rough or dusty in different areas.
 8.3 ) (See para. 4.2) the deflection of a suspended condenser when charged was claimed. This is as expected if the leads or terminals sustain a corona. The "polarization" effect would seem to be evidence that one or the other lead or terminal had rough areas, sharp corners, or dust resulting in corona at one end to the other, and a thrust in one or the other direction. If the terminals have the same geometry, the force will, because of the lesser mobility of the positive ions, be directed toward the positive terminal; if the negative terminal is much rougher or sharper, it will be the seat of a corona of which the force might outweigh that at the positive terminal. The observed fluctuations in the effect are to be expected as the result of changing dust on the wires or terminals, which changes the corona current; the deflections in the Bendix experiment were somewhat irregular.
 8.4 ) (See para. 5) The disc of the Brown Electrometer shown in Figure 5 develops a torque in response to an applied voltage. The field in the oil is strongest in the regions A.B.V.D. where the oil is in contact with the electrodes separating the pine sectors from the marble sectors M. Positive ions formed at the electrodes as in a corona discharge will produce positive space charges at the regions A. & D. while regions B. & C. will have negative space charges. In the oil these will cause effects analogous to the electric wind in air. If the fields A.B.C.D. are equally strong the ion densities will be such as to give no net torque to the disc. We will assume, on the other hand, that the surface of the pine sectors allows a leakage current; this will have the effect of reducing the field in the regions A. & B. and will
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 leave the regions C. & D. dominant in the production of force. The positive ions at D. will tend to rotate the disc clockwise while the negatives will tend to rotate is counterclockwise. Since their mobilities differ there will be a net torque in one or the other direction, as observed. If the chemistry of the oil or the insulating sectors changes with time the surface conductivity of the marble could become larger than that of the pine, causing a reversal of the torque. Aside from surface conductivity, other attributes may distinguish the marble from the pine. For example, a net torque might be caused by the influence of different dielectric constants from the field distribution, or if alternate sectors all project slightly from the electrodes.
 Figure 5: Electrometer
 A net torque might result from or be influenced by the surface conductivity of the pine and marble. If one is more conductive than the other, the field pattern will be made unsymmetrical (See para. 5.4.2) and the torque will be altered. It is possible that the slow fluctuations in the Brown Electrometer are caused by changes in the surfaces. The faster fluctuations may be caused by particles suspended in the oil, which may temporarily rest at such points as A. or B., changing the corona there temporarily.
 8.5 ) The experiment reported in para. 5.4.1 would seem to corroborate the idea that a swirling of the oil is essential to the operation of the electrometer. This idea is in accord with the electric wind theory, whether or not the discussion in para. 8.4 is valid in detail.
 8.6 ) It is claimed that the deflections of the electrometer show certain periodicities. If these are real they show the inadequacy of the electric wind hypothesis. Their reality is assessed in the crude and preliminary analysis contained in para. 5.2 and summarized in para. 5.3. There is no manifest solar diurnal period and no manifest lunar diurnal period. No test has been applied for the claimed sidereal diurnal period or for the correlation with
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 the value of financial securities. Neither has it been possible to determine whether the slow changes in deflection (Table 2) are repeated annually as claimed; however, the fact that the deflections once annually permanently reversed direction would seem to discourage this claim. This reversal is attributable to changed surface conductivity of the sectors of the disc. It is concluded that the short study, para. 5 of the reading of the Brown Electrometer does not weaken the hypothesis that all the (censored: probably reads, "phenomena reported by Mr. Brown") are the result of electric wind.
 8.7 ) The thrust observed in the experiment on the model flying saucers is given in Table 5, para 6.2.2. The thrust to be expected on the electric wind hypothesis is tabulated as calculated from the equations, para. 7.5. This agreement is satisfactory in order of magnitude. Actually the calculated thrust is about twice the observed. The discrepancy may be due to the approximations made in the mathematics of para. 7.5.
 8.8 ) (See para 6.2.3) A partial shielding of the insulated wire on a model flying disc partially eliminated the thrust, presumable because the current was probably thereby reduced. The shielding also had the effect of making the approximations in para.7.5 still worse. Qualitatively, then, the electric wind hypothesis is again verified.
 8.8 ) (See para. 6.2.4) With polarity reversed so that the insulating wire was negative, the thrust fell form about 7.5 gr to about 3 gr, the ratio being 2.5. Unfortunately the current was not measured. Assuming it to have been unchanged in magnitude by the reversal, we would predict, from the result, that the mobility of the negative ions is 2.5 times greater than that of the positives. This is in fair agreement with the known factor of 1.9.
 8.10 ) (See para. 6.2.5) A wind was observed to be generated near the outboard wire of the model flying saucers. This is as expected on the hypothesis under evaluation. The thrust experienced by the disc may be regarded as the jet reaction caused by the formation of this wind.
 9 ) Conclusion
 9.1 ) All the well documented effects disclosed by Mr. Brown appear to be explained as caused by ionization of the air or oil in which the apparatus is immersed.
 9.2 ) Mr. Brown claims that a gravitational anomaly exists in the neighborhood of a charged condenser. This effect has not been well documented by Mr. Brown, nor has the writer undertaken to refute it.
 9.3 ) The thrust developed by the model flying saucers has its origin in the well-known phenomenon of the electric wind.
 9.4 ) The efficiency of propulsion by the electric wind, as exemplified by the model flying saucers is of the order of 1.4 percent. At speeds higher than 2.3 ft/sec attained by the model flying saucer the efficiency might be
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 substantially improved. It is presumable less at high altitudes and zero in a vacuum.* If the efficiency of conversion of fuel energy into electrical energy is 21 percent the overall efficiency of propulsion of the model flying saucers by the electric wind is 0.3 percent. This compares with about 25 percent for a propeller-driven airplane and about 15 percent for a jet airplane.
 End of Report
 Classification of this report cancelled by authority of Office of Naval Research Ltr. 2340, by Capt. Dewey Brown, CI Div., Dir., Special Investigations, HQ USAF, Wash. DC., 1 October 1952.
 * N.B.: The "presumably less at high altitudes and zero in vacuum" hypothesis was later disproven by Brown during experiments in France, where it was shown that in fact the efficiency of his model flying saucers actually increased when flown in high vacuum. ~ Ed.
 N.B.: The remainder of the report is comprised of miscellaneous military letters that do not contain any pertinent information. ~ Ed.
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 ESSENTIAL
 MODERN PHYSICS
 Study Guide Workbook
 (with Full Solutions)
 Chris McMullen, Ph.D.
 Essential Modern Physics Study Guide Workbook Chris McMullen, Ph.D.
 monkeyphysicsblog.wordpress.com improveyourmathfluency.com chrismcmullen.com
 Copyright © 2019 Chris McMullen, Ph.D. All rights reserved.
 Zishka Publishing ISBN: 978-1-941691-28-1 Textbooks > Science > Physics Study Guides > Workbooks> Science
 CONTENTS
 Introduction 1 Special Relativity Concepts 2 Time Dilation and Length Contraction 3 The Lorentz Transformation 4 Relative Velocity 5 Relativistic Momentum and Energy 6 Blackbody Radiation 7 The Photoelectric Effect 8 The Compton Effect 9 Bohr’s Model 10 The de Broglie Relation 11 Heisenberg’s Uncertainty Principle 12 Differential Equations 13 Schrödinger’s Equation 14 Probabilities and Expectation Values About the Author
 INTRODUCTION
 The goal of this study guide workbook is to provide practice and help carrying out essential problemsolving strategies that are standard in modern physics. The aim here is not to overwhelm the student with comprehensive coverage of every type of problem, but to focus on the main strategies and techniques with which most physics students struggle. This workbook is not intended to serve as a substitute for lectures or for a textbook, but is rather intended to serve as a valuable supplement. Each chapter includes a concise review of the essential information, a handy outline of the problem-solving strategies, and examples which show step-bystep how to carry out the procedure. This is not intended to teach the material, but is designed to serve as a time-saving review for students who have already been exposed to the material in class or in a textbook. Students who would like more examples or a more thorough introduction to the material should review their lecture notes or read their textbooks. Every problem in this study guide workbook applies the same strategy which is solved step-bystep in at least one example within the chapter. Study the examples and then follow them closely in order to complete the exercises. Many of the exercises are broken down into parts to help guide the student through the exercises. Each exercise tabulates the corresponding answers on the same page. Students can find full solutions at the end of each chapter. The prerequisites for using this workbook include first-year physics (including energy, waves, and electricity and magnetism) and calculus (including derivatives and integrals). Although Schrödinger’s equation in quantum mechanics is a differential equation, students do not need previous exposure to differential equations: This workbook provides a concise introduction to basic differential equations in Chapter 12, and shows how to apply these methods in the remaining chapters.
 1 SPECIAL RELATIVITY CONCEPTS
 Relevant Terminology
 Galilean relativity – our experience with the relative motion of objects traveling at speeds much slower than the speed of light.
 Special relativity – the physics of the relative motion of objects where at least one object is traveling at a very high speed (compared to the speed of light).
 Ether – a hypothetical substance once believed to fill space; it was believed to serve as a medium for the transmission of light waves.
 Photon – a single particle in a beam of light.
 Interferometer – a device involving the interference of two beams of light, which was used by Michelson and Morley to measure the speed of light relative to the earth.
 Time dilation – the phenomenon whereby time appears to travel more slowly for objects moving fast (close to light speed) relative to other observers.
 Length contraction – the phenomenon whereby objects moving fast (close to light speed) appear shorter relative to other observers.
 Simultaneity – when two events occur at the exact same moment relative to an observer, the events are said to be simultaneous for that observer.
 Momentum – mass times velocity.
 Inertia – the natural tendency of any object to maintain constant momentum.
 Mass – a measure of inertia.
 Vacuum – a region of space completely devoid of matter (it doesn’t even contain air).
 Inertial reference frame – a frame that travels with constant velocity.
 Galilean Relativity
 In our everyday experience with objects that travel much slower than the speed of light, relative velocities appear to obey the formula for vector addition. Suppose that one observer (designated R) is at rest while a second observer (designated M) is moving with speed v relative to the first observer. Suppose also that each observer sets up a coordinate system with the x-axis along the relative velocity
 If each observer measures the velocity of an object, the x-components of the velocities that they measure (uR and uM) are related by the following vector addition formula, provided that neither
 observer nor the object are traveling close to light speed.
 uM = uR – v
 This equation is actually pretty simple: It’s just subtraction. The challenge is to remember what the notation means (M stands for moving, while R stands for rest) so that you can apply the equation correctly. We will explore this equation further in the examples that follow.
 The Mysterious Ether
 Physicists once believed in a hypothetical substance called the ether, which was believed to permeate all of space. At the time, all other waves besides light were known to travel in a medium. You can see ripples travel along the surface of water. Sound waves create alternating regions of compression (high pressure) and rarefaction (low pressure) in a medium such as air, water, wood, or metal. Light was also known to be a wave, yet sunlight can travel through space (a near-perfect vacuum). Since all other waves required a medium in which to propagate, the concept of the ether could explain the transmission of light through space.
 It turns out that the ether hypothesis is incorrect, as demonstrated by the Michelson-Morley experiment. Light can travel through a perfect vacuum (without an ether).
 The Earth, Light, and the Hypothetical Ether
 The result of the Michelson-Morley experiment—that the ether doesn’t exist—came as a big shock to the physics community at the time. To understand why, we must explore the ether as it had been believed to exist. The ether was believed to permeate all of space. The reference frame of the ether was believed to serve as an absolute reference frame. That is, the speed of light was only believed to
 travel c = 2.9979 × 108 m/s in a reference frame that was at rest relative to the ether. Furthermore, it was believed that the vector addition equation of Galilean relativity applied to objects moving any speed, including light itself. (Like the ether, this also proved to be incorrect.)
 The earth orbits the sun and therefore must be moving relative to the hypothetical ether (as the direction of earth’s velocity is constantly changing). From the point of view of the ether, the earth is moving relative to the ether with instantaneous speed v. From the point of view of earthlings, the earth seems to be stationary and we would instead interpret the ether to be moving with speed v. (You should have experience with this. If you are sitting in a bus that is moving, objects outside of the bus appear to be moving relative to you.) From the perspective of “stationary” earthlings, the ether is seen as an “ether wind” (when you run through air that is originally still, the air seems to pass by you like a sort of wind).
 Imagine that you get in a motorboat and travel along the surface of a river. The boat would travel with a speed of 30 m/s on a still pond, but there is a river current of 10 m/s. When the boat is headed downstream, it would be traveling 30 + 10 = 40 m/s relative to the land. When the boat is headed upstream, it would be traveling 30 – 10 = 20 m/s relative to the land. When the boat is headed crossstream, apply the Pythagorean theorem to determine that the boat travels
 relative to the land.
 The same principle as the motorboat and river was believed to apply to the earth traveling through the hypothetical ether. Imagine that you shine a beam of light from earth and wish to measure the speed of
 light. According to the ether hypothesis, the light would travel with speed c = 2.9979 × 108 m/s relative to the ether. The earth travels with speed v = 30,000 m/s relative to the sun. As the earth orbits the sun, the direction of its velocity constantly changes. The speed of light would equal c + v when the light is heading “downwind” (when the earth happens to be traveling along the ether wind), the speed of light would equal c – v when the light is heading “upwind,” and the speed of light would
 equal
 when the light is heading “across wind.” However, when Michelson and Morley investigated this, no such changes in the speed of light were detected. The laws of Galilean relativity do not apply to light or to objects moving fast (close to light speed). The ether does not exist. The speed of light turned out to be a universal constant (independent of the motion of the observer or source).
 Why c + v, c – v, and
 Seemed to Make Sense
 Imagine a father and son playing catch on a boat that is at rest. They throw the ball with a speed of 20 m/s east and west. They continue to play catch the same way even when the boat starts traveling 50 m/s to the east. When the ball travels 20 m/s east relative to the boat, it is traveling 50 + 20 = 70 m/s to the east relative to the land. When the ball travels 20 m/s to the west relative to the boat, it is traveling 50 – 20 = 30 m/s to the east relative to the land. This is the way relative velocities work at speeds that are small compared to the speed of light. It agrees with everyday experience.
 Now imagine a spaceship traveling 0.7c (or 70% the speed of light) relative to earth. The spaceship suddenly turns on its headlights. How fast is the beam of light traveling relative to earth? Based on our experience with low speeds (Galilean relativity), it’s intuitive to expect the beam of light to be traveling 0.7c + c = 1.7c relative to the earth. But that’s not what happens. It turns out that the beam of
 light travels c = 2.9979 × 108 m/s relative to the ship and that the beam of light also travels c =
 2.9979 × 108 m/s relative to earth, with no contradiction! Although c + v seemed to make sense, it turned out to contradict experiments.
 The Michelson-Morley Experiment
 Michelson and Morley used an interferometer that was designed to detect small changes in the speed of light. When an incident beam of light reached a glass slab, it split into two: One beam reflected from the surface of the glass towards mirror M1, and a second beam refracted through the
 glass towards mirror M2. After reflecting off mirrors M1 and M2, the beams met back up at the glass,
 and light from each beam reached a telescope, where an interference pattern was viewed. The entire apparatus could be rotated.
 The two beams would travel in different directions relative to the ether, and thus the two beams would have different speeds relative to the earth if the ether hypothesis were true. For example, if the beam heading toward M1 were traveling upwind with speed c – v (in which case it would then travel
 downwind with speed c + v after reflection), then the beam heading toward M1 would be traveling
 crosswind with speed
 Thus, the two beams would return to the glass at different times, creating an interference pattern when viewed through the telescope (since the beams would be slightly out of phase due to the time lag).
 However, the Michelson-Morley experiment failed to detect any change in the speed of light as the apparatus was rotated. It turns out that light (which is an electromagnetic wave) does not require a medium (such as the ether) in order to propagate. Light can travel through a perfect vacuum. There is no preferred or absolute reference frame for measuring the speed of light. It turns out that the speed of light is a universal constant, independent of the motion of the source or the observer.
 The Problem
 The Michelson-Morley experiment contradicted the expectations of the ether hypothesis. How could the speed of light be the same in each beam of the interferometer, regardless of the orientation of the apparatus?
 If a boat is traveling north, it can launch a cannonball farther to the north (relative to the land) than it could if the boat were at rest. This is the principle of vector addition applied in Galilean relativity, which agrees with human experience (with objects traveling much slower than the speed of light).
 Imagine a spaceship traveling one-half the speed of light (0.5c) relative to earth. Also imagine a beam of light traveling parallel to the spaceship. An observer on earth measures the speed of the beam of
 light to be c = 2.9979 × 108 m/s. What will an observer on the spaceship measure the speed of the beam of light to be? According to the Michelson-Morley experiment, the answer is the same: c =
 2.9979 × 108 m/s. (Note that the answer isn’t 0.5c.)
 Your experience with relative motion at low speeds is much different. If you’re sitting in a bus traveling 40 m/s and a car passes you traveling 50 m/s, each second the car gets 10 m further ahead of the bus, so the car seems to be traveling 10 m/s relative to you. However, if you’re in a spaceship and proceed to measure the speed of light, the Michelson-Morley experiment shows that you will get c =
 2.9979 × 108 m/s regardless of how fast the ship is traveling. Even if the spaceship travels 0.99c
 relative to earth, an observer on the ship still measures the speed of light to be c = 2.9979 × 108 m/s (and not 0.01c).
 Albert Einstein introduced his theory of special relativity to resolve this seeming paradox, but it came with some interesting consequences: time dilation and length contraction. The underlying issue is that time, space, and light behave much differently than our everyday experience with low-speed motion suggests. The two different observers (in the spaceship and on earth) actually measure distance and time differently due to time dilation and length contraction. If there is a meterstick on the spaceship and an observer on earth proceeds to measure the length of that meterstick as the spaceship travels very fast (close to light speed), the observer on earth measures the meterstick to be significantly less than one meter long. Similarly, if there is a pendulum on the spaceship that oscillates with a period of exactly one second relative to an observer on the spaceship, an observer on earth measures the period of that same pendulum to last significantly longer than one second. When the relative speed between two observers is in the neighborhood of the speed of light, the observers significantly disagree on such basic notions as what meters and seconds are! As bizarre as this may seem, Einstein’s theory of special relativity not only explains the Michelson-Morley experiment, it also agrees with countless other scientific tests.
 Einstein’s Theory of Special Relativity
 The theory of special relativity applies to objects that move with constant velocity (meaning that they move with constant speed and also travel in a straight line). Einstein developed his theory of special relativity from two fundamental postulates:
 1. The laws of physics are the same in any inertial reference frame. (An inertial reference frame is any coordinate system that has constant velocity. Recall that constant velocity means both constant speed and traveling in a straight line.) This means that any physics experiment will yield the same results in any laboratory that travels with constant velocity, whether the laboratory is at rest on earth or moving in a spaceship with a velocity of 0.8c relative to the earth.
 2. Any observer in an inertial reference frame would measure the speed of light to be c = 2.9979
 × 108 m/s, regardless of the velocity of the observer and also regardless of the velocity of the light source.
 One consequence of special relativity is that there isn’t any preferred (or absolute) reference frame (such as an ether). Any inertial reference frame is equally as good as any other.
 What does it mean to be at rest? You can be at rest relative to earth, but earth is revolving around the sun. Even if an object is at rest relative to the sun, the sun is traveling through space relative to other stars. Since the laws of physics are the same in all inertial reference frames, it would be impossible to find a particular reference frame that you could say is truly at “rest.” As long as you’re moving with constant velocity, you’re entitled to consider yourself to be at “rest” and to consider everything else moving relative to you. Even if you’re in a bus that is traveling 30 m/s west past a station, you may consider yourself to be at rest and may consider the station as traveling 30 m/s to the east (opposite to you). A woman at the station may consider herself to be at rest and consider you to be traveling 30 m/s to the west. You are both entitled to be correct when working out physics with your own inertial reference frames. According to Einstein, it’s all relative.
 As we will explore mathematically in Chapters 2-5, the theory of special relativity comes with a few seemingly strange consequences. In particular, even seemingly fundamental concepts like length and time are relative, and different observers in different inertial reference frames measure length and time differently:
 • Two events that appear to occur simultaneously in one inertial reference frame may appear to occur at different times in another inertial reference frame.
 • Length contraction: When an object is moving relative to an inertial reference frame, the object appears shorter (along the direction of motion) than it does relative to an inertial reference frame that is at rest relative to the object.
 • Time dilation: Time passes more slowly on a clock in a moving inertial reference frame than it does for an inertial reference frame that is at rest relative to the clock.
 Simultaneity Is Relative
 One consequence of the postulates of special relativity is that whether or not two events appear to occur simultaneously (at the same time) depends on the inertial reference frame from which the events are observed. If two events are observed to occur simultaneously in one inertial reference frame, they may not be observed to occur simultaneously in another inertial reference frame.
 For example, consider the spaceship traveling close to the speed of light in the illustration above. The spaceship passes very close to a space station that is orbiting the earth. The space station is traveling so much slower than the speed of light that it is practically at rest relative to the very fast spaceship. One astronaut is stationed at point Q on the space station, while another astronaut is stationed at point P on the spaceship. There are lights on the space station at the points labeled 1 and 2. These two lights remain off almost all of the time. The lights are programmed to flash quickly at the exact instant that the two ends of the spaceship happen to be (momentarily) positioned directly above the lights. At this exact instant, point A is directly above point 1, point B is directly above point 2, and points P and Q are exactly midway between points 1 and 2 (and are thus midway between points A and B).
 Observer Q is at rest relative to the space station and is exactly midway between points 1 and 2, such that the light emitted by each point during the flash travels the same distance to reach observer Q. Thus, observer Q on the space station sees the two lights flash simultaneously. In contrast, observer P is traveling close to the speed of light relative to points 1 and 2. Light from point 2 reaches observer P before light from point 1. Observer P doesn’t see the two lights flash simultaneously. Which observer is correct? According to special relativity, both are correct. Whether you are practically at rest (like the space station) or traveling close to the speed of light relative to the earth (like the spaceship), the laws of physics are the same. There isn’t a preferred inertial reference frame that makes observations more “correct.”
 Time Dilation
 Time passes more slowly on a clock in a moving inertial reference frame than it does for an inertial reference frame that is at rest relative to the clock. This is known as time dilation.
 You can see how time dilation is a direct consequence of the postulates of special relativity by considering the illustration above. The spaceship is traveling to the right with a speed that is close to the speed of light. The spaceship passes by a space station that is orbiting the earth. The space station is traveling so much slower than the speed of light that it is practically at rest relative to the very fast spaceship. As the spaceship is passing the space station, an astronaut inside of the spaceship turns on a flashlight, shining a beam of light straight upward in the diagram (perpendicular to the direction that the spaceship is traveling). The spaceship has transparent walls such that an observer inside of the space station can see the beam of the flashlight shining inside of the spaceship as it passes the space station.
 Relative to the astronaut inside of the spaceship (observer P), the flashlight beam appears to travel straight upward, as shown in the left diagram above. Relative to an observer inside of the space station (observer Q), the flashlight beam appears to travel diagonally up and to the right, as shown in the right diagram above. (Of course, the photons in the beam of light have inertia, which is the natural tendency of all objects to travel with constant momentum. If you ride in an airplane traveling 500 mph and throw a ball straight upwards, you will catch the ball in your hand because of inertia. The ball certainly won’t smack the back of the airplane mid-flight. If you’ve forgotten about inertia, it may help to review an introductory physics textbook.)
 Observer P sees the beam of light take a shorter path (distance LP, which is straight upward), whereas
 observer Q sees the beam of light take a longer path (distance LQ, which is diagonal). According to
 the second postulate of special relativity, both observers must measure the speed of light to be the same value. Either observer takes the distance (L) traveled and divides by the corresponding time (t) measured to determine the speed of light. The subscripts P and Q indicate which observer makes the measurement.
 Since LQ > LP, in order for both observers to measure the same value for the speed of light, it follows
 that tQ > tP. Since observer P is at rest relative to the flashlight, while observer Q is moving relative
 to the flashlight, the inequality tQ > tP means that time travels more slowly for clocks (and observers
 and objects) that are moving relative to an event. This is called time dilation. We will explore the mathematics of time dilation further in Chapter 2.
 The effect is mutual. Note that observer Q is moving relative to P, but also that observer P is moving relative to Q. Therefore, time dilation depends on your perspective:
 • Relative to observer P on the spaceship, time passes more slowly for observer Q who is on the space station. This is what we found in our example since we let observer P shine the flashlight inside of the spaceship.
 • Relative to observer Q on the space station, time passes more slowly for observer P who is on the spaceship. If we had let observer Q shine the flashlight inside of the space station (instead of letting observer P shine the flashlight inside of the spaceship), we would have found that the time measured by observer P would have been longer.
 Length Contraction
 When an object is moving relative to an inertial reference frame, the object appears shorter (along the direction of motion) than it does relative to an inertial reference frame that is at rest relative to the object. This is known as length contraction.
 You can see how length contraction comes about by considering the illustration above. The spaceship is traveling from earth’s sun to Alpha Centauri with a speed that is close to the speed of light. Observer S is an astronaut aboard the spaceship, while observer E is stationed on earth. Each observer proceeds to measure the distance from the sun to Alpha Centauri (L) and the time of the trip (t).
 Observer E on earth measures the distance to be LE and the time to be tE, such that the speed of the
 spaceship is v = LE/tE. Since the earth is “moving” relative to the spaceship, time is dilated for
 observer E, meaning that tE > tS. Observer S is at “rest” relative to the spaceship, and sees Alpha
 Centauri getting closer to the spaceship with the same speed v. (The perspective is differentwhether earth is at rest and the spaceship is moving, or whether the spaceship is at rest and the stars are moving—but the speed is the same.) Observer S measures the distance to be LS and the time to be
 tS such that v = LS/tS.
 Since observer S is at “rest” relative to the spaceship, time passes “normally” for observer S, such that tS < tE. If tS < tE, how can v = LS/tS and v = LE/tE both result in the same speed? The answer is
 that LS < LE. This means that the distance between the stars appears shorter for observer S than it
 does for observer E. Note that observer S is moving relative to the distance between the stars. Therefore, distance is shorter (along the direction of motion) relative to an inertial reference frame that it is moving relative to the distance than it is for a reference frame that is at rest relative to the distance. This is known as length contraction. We will explore the mathematics of length contraction further in Chapter 2.
 The effect is mutual. Note that observer S is moving relative to E, but also that observer E is moving relative to S. Therefore, length contraction depends on your perspective:
 • Relative to observer S on the spaceship, distances measured along the direction of the spaceship’s motion (such as the distance between the sun and Alpha Centauri) appear shorter than they do for observer E on earth. This is what we found in our example since observer S is moving relative to the distance between the two stars.
 • Relative to observer E on earth, the length of the spaceship is shorter than it is for observer S on the spaceship. If we had considered measurements of the length of the spaceship (instead of the distance between the stars) in our example, this is what we would have found.
 • Relative to observer S on the spaceship traveling from the sun to Alpha Centauri, a second spaceship parked on earth would appear shorter (along the line connecting the two stars) than it is for observer E on earth.
 Note that the effects of time dilation and length contraction that we discussed in this example involve two different perspectives:
 • For observer E on earth, v = LE/tE, we noted that time was dilated (tE > tS) for observer E
 because the earth is “moving” relative to the spaceship.
 • For observer S on the spaceship, v = LS/tS, we noted that length was contracted (LS < LE) for
 observer S because the spaceship is “moving” relative to the two stars.
 Relativistic Mass
 As the relative speed between two inertial reference frames gets closer to the speed of light, the effects of special relativity—including time dilation and length contraction—become more pronounced. In the limit that the relative speed approaches the speed of light, time slows down to a complete stop and length contracts to zero. However, you can’t actually reach this limit. An object that has mass can be accelerated to nearly light speed (like 0.99c or 0.999c), but can never reach the speed of light exactly.
 In Galilean relativity, it would be very easy to accelerate an object faster than the speed of light. According to Newton’s second law of motion, the net force acting on an object equals the object’s mass times its acceleration:
 (This equation applies to objects that have constant mass, but that’s not a problem for Galilean relativity, where ordinarily an object’s mass isn’t expected to change while it accelerates.) The mass
 of an electron is 9.1 × 10–31 kg. If you applied a force of just 1 N (one Newton) to an electron, according to Galilean relativity, the electron would experience an acceleration of
 What does this acceleration mean? It means that starting from rest, after a just 1 s (one second), an
 electron would have a speed of v = 1.1 × 1030 m/s (since acceleration describes the rate at which velocity increases). That’s way, way faster than the speed of light in vacuum, which is c = 2.9979 ×
 108 m/s.
 In the laboratory, it doesn’t happen that way. Although it is very easy to accelerate electrons to very high speeds, once the speed of an electron reaches the neighborhood of the speed of light in vacuum, it becomes increasingly harder to accelerate the electron. We can accelerate electrons up to 0.9c or even 0.99c, but trying to reach 0.99999c is extremely difficult. Why? According to Einstein’s theory of relativity, mass isn’t constant: The faster an electron travels, the more mass the electron has (relative to the laboratory). We call this relativistic mass.
 Recall that mass is a measure of inertia in the following sense: The more mass an object has, the more difficult it is to overcome the object’s inertia in order to accelerate the object. The relativistic mass of an object describes the object’s inertia. As the object travels closer to the speed of light (relative to an inertial reference frame), the greater its relativistic mass (and relativistic inertia): It becomes harder and harder to accelerate the object.
 There are two types of mass:
 • An object’s rest mass tells you how difficult it is to accelerate the object relative to an observer for which the observer and object are both at rest.
 • An object’s relativistic mass tells you how difficult it is to accelerate the object relative to an observer for which the object is traveling close to the speed of light.
 Proper Time, Proper Length, and Rest Mass
 When you compare measurements of time, distance, or mass made by observers in different inertial reference frames, it’s important to be able to determine which of the measurements will be larger and which will be smaller. If you can identify the proper time, the proper length, and the rest mass properly, this will help with your comparisons.
 • The proper time corresponds to a time interval measured by a clock that is at rest relative to the events. Observers who are moving relative to the events measure a greater time interval due to time dilation.
 • The proper length corresponds a distance measured by a tape measure (or other device used for measuring distance) that is at rest. Observers who are moving relative to the distance measure a shorter distance due to length contraction.
 • The rest mass corresponds to the mass of an object measured by an observer who is at rest relative to the object. Observers who are moving relative to the object measure a larger mass called relativistic mass.
 Note: In some relativity questions, the proper time and proper length are not measured by the same observer: Proper time and proper length may come from different perspectives (that is, two different inertial reference frames).
 Symbols and SI Units
 Constants
 Strategy for Solving Galilean Relativity Problems
 To solve a problem involving Galilean relativity (which applies to problems with relative speeds that are small compared to the speed of light in vacuum), follow these steps:
 • Setup a coordinate system with the x-axis along the direction of the relative velocity,
 • Galilean relativity involves vector subtraction. For one-dimensional problems, simply subtract according to the following equation:
 uM = uR – v
 •• uR is the velocity of an object as measured by an observer at rest (called R).
 •• uM is the velocity of an object as measured by an observer (called M) that is moving relative
 to the observer that is at rest.
 •• v is the relative speed between the two observers.
 Strategy for Solving Conceptual Special Relativity Problems
 To solve a conceptual problem involving time dilation, length contraction, or relativistic mass, follow these steps. (For mathematical problems, see Chapters 2-5.)
 • It may help to draw a diagram and label both objects and observers.
 • The laws of special relativity apply to inertial observers—observers who travel with (approximately) constant velocity (which means that speed and direction are both constant). Any inertial observer is free to claim to be at rest, and can make equally valid arguments based on this claim.
 • When applying time dilation, identify the proper time: The proper time is measured by an observer whose clock is at rest relative to the events. Any observer who is moving relative to the events will measure a longer time due to time dilation.
 • When applying length contraction, identify the proper length: The proper length is measured by an observer who is at rest relative to the distance being measured. Any observer who is moving relative to the distance will measure a shorter distance due to length contraction.
 • When working with relativistic mass, identify the rest mass: The rest mass is measured by an observer who is at rest relative to the object. Any observer who is moving relative to the object will measure a greater mass called relativistic mass.
 Example: A monkey is riding on a boat that is traveling 12 m/s to the east along a river. As the boat passes a boy who is standing on the bank of the river, the monkey throws a banana 18 m/s to the east relative to the monkey. How fast is the banana moving relative to the boy?
 Setup a coordinate system with +x directed to the east. Identify the given information:
 • The relative speed between the monkey and the boy is v = 12 m/s.
 • The monkey is the moving observer. The velocity of the banana relative to the monkey is uM =
 18 m/s to the east.
 • The boy is at rest. The velocity of the banana relative to the boy is uR.
 Since these speeds are small compared to the speed of light, we may apply the equation for Galilean relativity:
 The banana is moving 30 m/s relative to the boy.
 Note: It is standard in physics to neglect air resistance unless stated otherwise in a problem.
 Example: A monkey is riding on a boat that is traveling 12 m/s to the east along a river. As the boat passes a boy who is standing on the bank of the river, the monkey throws a banana 18 m/s to the west relative to the monkey. How fast is the banana moving relative to the boy?
 Compare these two examples carefully. What is different? This time the monkey throws the banana to the west, which is opposite to the boat’s motion. Setup a coordinate system with +x directed to the east. Identify the given information:
 • The relative speed between the monkey and the boy is v = 12 m/s.
 • The monkey is the moving observer. The velocity of the banana relative to the monkey is uM =
 –18 m/s. Velocity includes direction: For one-dimensional problems, minus signs distinguish between forward and backward.
 • The boy is at rest. The velocity of the banana relative to the boy is uR.
 Since these speeds are small compared to the speed of light, we may apply the equation for Galilean relativity:
 The banana is moving 6 m/s relative to the boy. Since uR is negative, the banana is heading west
 (along –x) relative to the boy.
 Example: A monkey drives a blue car 30 m/s to the north. On the same street, another monkey drives a red car 20 m/s to the south. What is the velocity of the red car relative to the monkey in the blue car?
 Don’t overthink it. This problem is simpler, since there isn’t a banana (or other object) moving relative to both observers. We’re just trying to determine the relative velocity
 between the two monkeys. You can reason this out as follows:
 • In one second, the blue car will travel 30 m north relative to the ground.
 • In one second, the red car will travel 20 m south relative to the ground.
 • Thus, in one second, the red car will be 20 + 30 = 50 m south of the blue car.
 • Since the red car will be 50 m further south of the blue car each second, the velocity of the red car is –50 m/s relative to the blue car, meaning 50 m/s to the south.
 Example: One astronaut is in a space station that is orbiting the earth. Another astronaut is in a spaceship. The space station is practically at rest relative to the very fast spaceship, and the spaceship is traveling close to the speed of light relative to the earth. The spaceship and space station both have transparent walls such that either astronaut can make observations (with the aid of a telescope) of what’s going inside of the other space craft.
 (A) How does the astronaut inside of the spaceship appear to age relative to the astronaut inside of the space station?
 Identify the proper time. The astronaut on the spaceship measures the proper time because this astronaut is at rest relative to himself (or herself). An observer who is moving relative to the spaceship will measure a longer time due to time dilation. Therefore, relative to the astronaut on the space station, the astronaut on the spaceship appears to age more slowly than the astronaut on the space station. (Let’s not worry about make-up, genetic differences in aging, etc. Let’s treat the astronauts equally, as if they are identical twins.)
 (B) Both astronauts measure the length of the spaceship. Compare their measurements.
 Identify the proper length. The astronaut on the spaceship is at “rest” relative to the distance being measured. An observer who is moving relative to the spaceship will measure a shorter distance due to length contraction. Therefore, the spaceship appears to be shorter relative to the astronaut inside of the space station than it does relative to the astronaut inside of the spaceship.
 (C) Both astronauts measure the mass of the spaceship. Compare their measurements.
 The astronaut in the spaceship measures the spaceship’s rest mass, whereas the astronaut in the space station measures the spaceship’s relativistic mass, which is larger.
 Chapter 1 Problems
 1. A monkey is standing on the top of a train that is traveling 36 m/s to the south. A girl is standing on the ground beside the railroad tracks. As the train passes the girl, the monkey throws an apple with a speed of 12 m/s relative to the train.
 (A) If the monkey throws the apple to the south, what is the speed of the apple relative to the girl?
 (B) If the monkey throws the apple to the north, what is the speed of the apple relative to the girl?
 Want help? Check the solution at the end of the chapter.
 Answers: 1. (A) 48 m/s (B) 24 m/s
 2. A monkey is riding inside of a train car that is traveling to the west with constant velocity. A girl is standing on the ground beside the railroad tracks. The monkey paints a red X on the floor of the train car (that is, the X is on the bottom of the train car, not on the ground). As the train car is passing the girl, the monkey holds a banana directly over the X and releases the banana. The train car has large windows (with no curtains), such that the girl is able to watch the banana fall and see where it lands.
 (A) Where does the banana land? Explain your answer.
 (B) What path does the banana take relative to the monkey?
 (C) What path does the banana take relative to the girl?
 Want help? Check the solution at the end of the chapter.
 Answers: 2. (A) on the X
 (B) straight line (C) parabola
 3. On a still pond, a monkey rides a boat 32 m/s to the east. This monkey throws an orange 20 m/s to the east relative to himself. The monkey’s father rides a boat 24 m/s to the east, the monkey’s mother is standing still on a pier, and the monkey’s uncle rides a boat 10 m/s to the west.
 (A) Find the velocity of the monkey relative to his father.
 (B) Find the velocity of the monkey relative to his mother.
 (C) Find the velocity of the monkey relative to his uncle.
 (D) Find the velocity of the monkey’s father relative to the monkey.
 (E) Find the velocity of the monkey’s mother relative to the monkey.
 (F) Find the velocity of the monkey’s uncle relative to the monkey.
 (G) Find the velocity of the monkey’s father relative to the monkey’s mother.
 (H) Find the velocity of the monkey’s mother relative to the monkey’s father.
 (I) Find the velocity of the monkey’s father relative to the monkey’s uncle.
 (J) Find the velocity of the monkey’s uncle relative to the monkey’s father.
 (K) Find the velocity of the monkey’s mother relative to the monkey’s uncle.
 (L) Find the velocity of the monkey’s uncle relative to the monkey’s mother.
 (M) Find the velocity of the orange relative to the monkey.
 (N) Find the velocity of the orange relative to the monkey’s father.
 (O) Find the velocity of the orange relative to the monkey’s mother.
 (P) Find the velocity of the orange relative to the monkey’s uncle.
 Want help? Check the solution at the end of the chapter.
 Answers: 3. (A) 8 m/s E (B) 32 m/s E
 (C) 42 m/s E (D) 8 m/s W (E) 32 m/s W
 (F) 42 m/s W (G) 24 m/s E (H) 24 m/s W
 (I) 34 m/s E (J) 34 m/s W (K) 10 m/s E
 (L) 10 m/s W (M) 20 m/s E (N) 28 m/s E
 (O) 52 m/s E (P) 62 m/s E
 4. Two identical spaceships travel with a relative speed close to the speed of light. As shown below, the spaceships travel in opposite directions. At the exact moment that the spaceships pass one another, scientists aboard each ship create identical clones of a chimpanzee. In this way, the two chimpanzees are effectively identical twins. Their names are Marco and Polo.
 (A) Scientists aboard Marco’s ship have one photograph of Marco on his 30th birthday, and compare it to a photograph taken with a telescope of exactly how Polo looked after 30 years according to calendars kept by Marco’s scientists (after accounting for the time it took for light to reach Marco’s ship from Polo’s ship). How do the photographs compare?
 (B) Scientists aboard Polo’s ship have one photograph of Polo on his 30th birthday, and compare it to a photograph taken with a telescope of exactly how Marco looked after 30 years according to calendars kept by Polo’s scientists (after accounting for the time it took for light to reach Polo’s ship from Marco’s ship). How do the photographs compare?
 (C) Which team of scientists is correct? Explain your answers to parts (A) and (B).
 (D) Which ship appears longer according to measurements made by Marco’s scientists?
 (E) Which ship appears longer according to measurements made by Polo’s scientists?
 (F) Which ship has more mass according to measurements made by Marco’s scientists?
 (G) Which ship has more mass according to measurements made by Polo’s scientists?
 (H) Which ship appears taller according to measurements made by Marco’s scientists?
 Want help? Check the solution at the end of the chapter.
 Answers: 4. (A) Polo looks younger than 30
 (B) Marco looks younger than 30
 (C) both! (D) Marco’s (E) Polo’s
 (F) Polo’s (G) Marco’s (H) same
 5. As illustrated below, when a spaceship is parked at rest inside of a space garage, the ship is too long to fit inside of the garage.
 As illustrated below, when the same spaceship is traveling close to the speed of light, it can momentarily fit inside of the same space garage according to observers stationed in the space garage. (This space garage is special: There are front and back doors which sense the presence of the spaceship, and which open or close almost instantly. Of course, the spaceship will only be inside of the space garage with both doors shut for a tiny fraction of a second relative to observers stationed in the space garage.)
 (A) Explain how the spaceship is able to fit inside of the space garage with both doors closed simultaneously (for a tiny fraction of a second) from the point of view of observers stationed in the space garage.
 (B) From the point of view of astronauts inside of the spaceship, is the spaceship able to fit inside of the space garage with both doors closed simultaneously? Explain.
 Want help? Check the solution at the end of the chapter.
 Answers: 5. (A) length contraction
 (B) no; the doors do not appear to be closed simultaneously relative to observers in the spaceship
 6. When cosmic rays interact with atoms high in earth’s atmosphere, particles called muons can be produced. Muons are particles similar to electrons, except that they have about 200 times as much mass and are unstable. Muons decay very quickly: A muon produced at rest only lasts a couple of microseconds before decaying into other particles. Based on the short average lifetime of a muon (as measured in the muon’s rest frame), almost none of the muons produced in the upper atmosphere should reach earth’s surface, yet a very large number of these muons are detected at earth’s surface.
 (A) From the point of view of scientists stationed on the ground, explain how a large number of muons are able to reach earth’s surface even though their average lifetime (as measured in the muon’s rest frame) is too short for them to survive that long.
 (B) Now explain this from the reference frame of the muons (instead of the reference frame of scientists stationed on the ground).
 Want help? Check the solution at the end of the chapter.
 Answers: 6. (A) time dilation
 (B) length contraction
 7. For each question below, state whether it is theoretically possible or impossible according to special relativity. Explain your answers.
 (A) A chimpanzee could travel to a star that is 10,000 light-years away in her natural lifetime. Note: A light-year is the distance that light travels in one year.
 (B) A chimpanzee could go on a space trip and appear younger than her own daughter when she returns to earth.
 (C) A chimpanzee could go on a space trip and appear younger than she was when she left the earth.
 Want help? Check the solution at the end of the chapter.
 Answers: 7. (A) possible
 (B) possible (C) impossible
 Solutions to Chapter 1
 1. Setup a coordinate system with +x directed to the south. Identify the given information:
 • The relative speed between the monkey and the girl is v = 36 m/s.
 • The monkey is the moving observer. The velocity of the apple relative to the monkey is uM.
 This value will be different in parts (A) and (B) since the apple is thrown in a different direction in each part.
 • The girl is at rest. The velocity of the apple relative to the girl is uR.
 Since these speeds are small compared to the speed of light, we may apply the equation for Galilean relativity. As usual, we neglect air resistance unless stated otherwise.
 uM = uR – v
 (A) Since the apple is thrown south and we chose +x to point south, uM = 12 m/s.
 The apple is moving 48 m/s relative to the girl.
 (B) Since the apple is thrown north and we chose +x to point south, uM = –12 m/s.
 The apple is moving 24 m/s relative to the girl.
 2. (A) The banana lands directly on the X. Why? Because the banana has inertia. Recall from Newton’s laws of motion (which are taught in first-year physics) that inertia is the natural tendency of all objects to maintain constant velocity. According to Newton’s second law of motion, a net external force is needed to accelerate an object (and thus change the object’s velocity), since
 When the monkey releases the banana, a net gravitational force acts downward, causing the banana to accelerate downward. However, there are no forces acting horizontally, so the banana doesn’t accelerate horizontally. The banana maintains a constant horizontal component of velocity (vx), while
 the vertical component of velocity (vy) changes. That’s why, in projectile motion, horizontally we
 have
 and vertically we have
 Horizontally, vx is constant (because ax = 0), whereas vertically there is uniform acceleration (ay =
 –g = –9.81 m/s2 is constant).
 You can verify this by riding in an airplane. If you throw an eraser straight upward relative to you while sitting inside of the airplane, you will catch the eraser because it has inertia. The eraser surely won’t land behind you, even if the airplane is traveling 300 m/s and if the eraser is in the air for half a second (in which case the airplane travels 150 m horizontally).
 (B) Relative to the monkey, the banana appears to fall in a straight line downward, no different than if the train had been parked when the monkey released the banana. The laws of physics are the same in any inertial reference frame, meaning that the result of dropping a banana from rest will be the same whether the train is at rest or moving with constant velocity.
 (C) Relative to the girl, the banana follows the arc of a parabola, beginning with a horizontal tangent. The same path would result if the girl had the banana and threw it horizontally. The banana follows the path of a projectile, which is parabolic.
 From the monkey’s point of view, the monkey claims that the train is at “rest” (but that the girl and ground are moving). According to the monkey, vx = 0 and the banana falls straight downward.
 From the girl’s point of view, the girl claims that she is at “rest” (but that the train is moving). According to the girl, vx isn’t 0. Relative to the girl,
 and
 which can be combined to make
 which is the equation of a parabola. To derive this equation, you need to combine the following equations:
 3. Setup a coordinate system with +x directed to the east.
 4. (A) Marco appears to age normally relative to the scientists on Marco’s ship. Because the proper time for Polo’s aging process is measured by Polo’s team, Marco’s team will measure Polo’s aging process to occur more slowly due to time dilation. Therefore, when Marco’s team compares their photographs, Polo will appear younger.
 (B) Polo appears to age normally relative to the scientists on Polo’s ship. Because the proper time for Marco’s aging process is measured by Marco’s team, Polo’s team will measure Marco’s aging process to occur more slowly due to time dilation. Therefore, when Polo’s team compares their photographs, Marco will appear younger.
 (C) Both teams are correct. Marco’s team believes that Polo appears younger when Marco celebrates
 his 30th birthday, and Polo’s team believes that Marco appears younger when Polo celebrates his 30th birthday, and both teams are correct because both teams are inertial observers (since both spaceships travel with constant velocity). According to special relativity, there is no preferred reference frame; and the laws of physics are the same for all inertial observers. (For the chimpanzees to actually meet up, note that one would have to accelerate.)
 (D) Marco’s ship appears normal relative to the scientists on Marco’s ship. Because the proper length for Polo’s ship is measured by Polo’s team, Marco’s team will measure the length of Polo’s ship to be shorter than normal due to length contraction; Marco’s appears longer.
 (E) Polo’s ship appears normal relative to the scientists on Polo’s ship. Because the proper length for Marco’s ship is measured by Marco’s team, Polo’s team will measure the length of Marco’s ship to be shorter than normal due to length contraction; Polo’s appears longer.
 (F) Marco’s team measures Marco’s ship to have its rest mass. When Marco’s team measures the mass of Polo’s ship, Polo’s ship’s relativistic mass appears greater than its rest mass. (Let’s assume that both teams of scientists have the same combined rest mass.)
 (G) Polo’s team measures Polo’s ship to have its rest mass. When Polo’s team measures the mass of Marco’s ship, Marco’s ship’s relativistic mass appears greater than its rest mass.
 (H) The height is the same for each. (Length contraction occurs only along the direction of the velocity. The height, which is perpendicular to the velocity, is unaffected.)
 5. (A) Since the proper length of the spaceship is measured by astronauts inside of the ship, astronauts stationed in the space garage measure the length of the spaceship to be shorter than normal due to length contraction. The concept of length contraction explains how the ship is short enough to fit in the garage while traveling close to the speed of light.
 (B) The spaceship has its usual length relative to astronauts inside of the spaceship. You can’t explain this in terms of length contraction because length contraction would make the space garage appear shorter than usual relative to astronauts inside of the spaceship. The answer has to do with the fact that different inertial observers often disagree on whether or not two events occur simultaneously; this is one of those times. Whereas astronauts stationed in the garage momentarily see the spaceship fit inside of the garage with both doors temporarily closed, astronauts aboard the spaceship don’t see the two doors closed at the same instant. Rather, astronauts aboard the spaceship see the door on the right side of our diagram close first (while the back of the spaceship sticks out on the left side of our diagram). They then see the front door open. The spaceship continues a short ways until the back of the spaceship is safely inside of the garage. At this point, the door on the left side of our diagram quickly closes (while the front of the spaceship sticks out on the right side of our diagram).
 6. (A) Since the proper time would be measured by a reference frame traveling with the muons themselves, scientists on earth’s surface measure a longer lifetime due to time dilation. That is, because the muons travel close to the speed of light relative to the earth, their average lifetime is much longer than it would be if they were at rest, which allows them to travel a longer distance before they decay.
 (B) From the reference frame of the muons, their lifetime is normal (they measure the proper time). Instead, the muons “see” length contraction: Since scientists on earth’s surface measure the proper length for the muons’ trip, a reference frame attached to the muons would measure a shorter distance. That is, because the muons travel close to the speed of light (although from their perspective, the muons are at rest and the earth is traveling close to the speed of light towards the muons), the distance between earth’s surface and where they are produced in earth’s atmosphere is much shorter than would be if they were at rest, which allows them to reach earth’s surface in less time. It’s instructive to compare how these different perspectives lead to equivalent conclusions through two quite different effects.
 7. (A) Theoretically, it is possible, provided that the chimpanzee travels in a spaceship with a speed that is close enough to the speed of light (that’s the hard part). Relative to observers on earth, the chimpanzees would age much more slowly than normal through time dilation. (However, the initial part of the trip requires acceleration, and the end of the trip involves deceleration, both of which involve general relativity, which is a step beyond special relativity.)
 (B) Theoretically, it is possible, for the same reason as part A, provided that the chimpanzee travels in a spaceship that is close enough to the speed of light. If the chimpanzee ages slowly enough due to time dilation, she could appear younger than her daughter when she returns to earth. However, whatever the age difference is between the chimpanzee and her daughter, at least that number of years must pass on earth for this to be possible. For example, if the chimpanzee is 30 years old and her daughter is 12 years old, at least 18 years must pass on earth during the trip (plus additional years depending on how fast the ship travels; we’ll explore the mathematics involved in time dilation in Chapter 2).
 (C) This is theoretically impossible (without plastic surgery or age defying medicine). While time can slow down due to time dilation, in special relativity time can’t go backwards.
 2 TIME DILATION AND LENGTH CONTRACTION
 Relevant Terminology
 Time dilation – the phenomenon whereby time appears to travel more slowly for objects moving fast (close to light speed) relative to other observers.
 Length contraction – the phenomenon whereby objects moving fast (close to light speed) appear shorter relative to other observers.
 Proper time – a time interval measured by a clock that is at rest relative to the events. An observer who is moving relative to the events measures a greater time interval due to time dilation.
 Proper length – a distance measured by an observer who is at rest relative to the distance. An observer who is moving relative to the distance measures a shorter distance due to length contraction.
 Inertial reference frame – a frame that travels with constant velocity.
 Time Dilation
 Time passes more slowly on a clock in a moving inertial reference frame than it does for an inertial reference frame that is at rest relative to the clock. This is known as time dilation.
 In the diagram above, the spaceship is traveling to the right with a speed that is close to the speed of light. The spaceship passes by a space station that is practically at rest relative to the very fast spaceship. As the spaceship is passing the space station, an astronaut inside of the spaceship turns on a flashlight, shining a beam of light straight upward in the diagram (perpendicular to the direction that the spaceship is traveling).
 Relative to the astronaut inside of the spaceship (observer P), the flashlight beam appears to travel straight upward (left diagram). Relative to an observer inside of the space station (observer Q), the flashlight beam appears to travel diagonally (right diagram).
 In the illustration above:
 • LP is the distance that the light travels according to observer P.
 • LQ is the distance that the light travels according to observer Q.
 • LS is the distance that the spaceship travels horizontally during this time.
 These three distances are related by the Pythagorean theorem:
 According to the second postulate of special relativity, both observers must measure the speed of light to be the same value. Either observer takes the distance (L) traveled and divides by the corresponding time (t) measured to determine the speed of light (c). The subscripts P and Q indicate
 which observer makes the measurement.
 Observer Q can also measure the speed (v) of the spaceship by dividing the horizontal distance (LS)
 traveled by the corresponding time (tQ).
 Multiply each equation by the corresponding time.
 LP = c tP
 LQ = c tQ
 LS = v tQ
 Substitute these expressions into the equation from the Pythagorean theorem.
 Solve for tQ.
 Divide the numerator and denominator each by c2.
 Length Contraction
 When an object is moving relative to an inertial reference frame, the object appears shorter (along the direction of motion) than it does relative to an inertial reference frame that is at rest relative to the object. This is known as length contraction.
 In the diagram above, a spaceship is traveling from earth’s sun to Alpha Centauri with a speed that is close to the speed of light. Observer S is an astronaut aboard the spaceship, while observer E is stationed on earth.
 Observer E on earth measures the distance to be LE and the time to be tE, such that the speed of the
 spaceship is
 v = LE / tE
 Observer S on the spaceship measures the distance to be LS and the time to be tS such that
 v = LS / tS
 According to observer E, the ship is “moving” while the stars are at rest, whereas according to observer S, the earth and stars are “moving” while the spaceship is at “rest,” but either way the speed of the spaceship is the same:
 Solve for LS.
 Since the spaceship’s clock is at “rest” relative to the journey, observer S measures the proper time and the passage of time aboard the spaceship appears dilated relative to observer E. Use the time dilation equation with tS as the proper time:
 Note: When applying the above equation to other problems, the proper length will often not be measured by an observer on earth. That just happened to be the case in this example. In a given problem, you must apply the concept of proper length to determine which observer measures the proper length.
 Note that the effects of time dilation and length contraction that we discussed in this example involve
 two different perspectives:
 • For observer E on earth, v = LE / tE, we noted that time was dilated (tE > tS) for observer E
 because the earth is “moving” relative to the spaceship.
 • For observer S on the spaceship, v = LS / tS, we noted that length was contracted (LS < LE) for
 observer S because the spaceship is “moving” relative to the two stars.
 Time Dilation and Length Contraction Equations
 In the equations below, t0 represents the proper time (measured by an observer who is at rest relative
 to the events) and L0 represents the proper length (measured by an observer who is at rest relative to
 the distance). Note that t0 and L0 are not necessarily measured by the same observer in a problem (in
 fact, these were measured by different observers when we derived the equation for length contraction in the previous section).
 Symbols and SI Units
 Constants
 Strategy for Solving Time Dilation and Length Contraction
 Problems
 To solve a problem involving time dilation or length contraction, follow these steps:
 • It may help to draw a diagram and label both objects and observers.
 • When applying time dilation, identify the proper time (t0), which is measured by an observer
 whose clock is at rest relative to the events.
 • When applying length contraction, identify the proper length (L0), which is measured by an
 observer who is at rest relative to the distance being measured.
 Note that t0 and L0 are not necessarily measured by the same observer in a problem.
 Example: A chimpanzee astronaut sleeps for 8 hours according to a spaceship’s clock while traveling at 0.5c (half the speed of light) relative to the earth. For how much time does the chimpanzee appear to be sleeping relative to an observer on earth?
 Which observer measures the proper time? The spaceship’s clock is at rest relative to the chimpanzee. Therefore, the chimpanzee measures the proper time: t0 = 8 hr. The observer on earth
 measures a greater time, td, due to time dilation. The relative speed is v = 0.5c. Use the time dilation
 equation.
 Example: A spaceship has a length of 20 m when it is parked near the surface of the earth. When a chimpanzee astronaut in the spaceship travels at 0.8c relative to the earth, what length does the spaceship appear to have relative to an observer on earth?
 Which observer measures the proper length? The chimpanzee is at rest relative to the length of the spaceship. Therefore, the chimpanzee measures the proper length: L0 = 20 m. The observer on earth
 measures a shorter distance, Lc, due to length contraction. The relative speed is v = 0.8c. Use the
 length contraction equation.
 Chapter 2 Problems
 1. A chimpanzee astronaut travels in a spaceship at 0.6c relative to the earth. According to the chimpanzee, the spaceship is 30 m long and the trip takes 12 years.
 (A) How long is the spaceship relative to an observer on earth?
 (B) How long does the trip take relative to an observer on earth?
 (C) How far does the spaceship travel according to an observer on earth?
 Note: One light-year (ly) is the distance that light travels in one year.
 (D) How far does the spaceship travel according to the chimpanzee?
 (E) Explain your answers to parts C and D.
 Want help? Check the solution at the end of the chapter.
 Answers: 1. (A) 24 m (B) 15 years
 (C) 9.0 ly (D) 7.2 ly
 (E) time dilation vs. length contraction
 2. How fast must a spaceship travel relative to another observer in order to appear half as long as it really is?
 Want help? Check the solution at the end of the chapter.
 Answer:
 3. Muons that are produced at rest have an average lifetime of
 A beam of muons is produced that travels 0.99c relative to the earth.
 (A) How far does classical physics expect the muons to travel on average? (For this question, pretend that the muons don’t follow the laws of special relativity.)
 (B) Relative to observers on earth, how far will the muons actually travel on average?
 (C) Relative to a reference frame attached to the muons, how far do the muons travel on average?
 (D) Explain your answers to parts B and C.
 Want help? Check the solution at the end of the chapter.
 Answers: 3. (A) 0.65 km
 (B) 4.6 km (C) 0.65 km
 (D) time dilation vs. length contraction
 4. A chimpanzee astronaut travels in a spaceship at (12/13)c relative to the earth. According to observers stationed on earth, the trip takes 26 years.
 (A) How far does the spaceship travel relative to the earth?
 (B) How far does the spaceship travel relative to the chimpanzee?
 (C) How long does the trip take relative to the chimpanzee?
 Want help? Check the solution at the end of the chapter.
 Answers: 4. (A) 24 ly (B) (120/13) ly = 9.2 ly (C) 10 yr
 Solutions to Chapter 2
 1. The relative speed is v = 0.6c.
 (A) Regarding the length of the spaceship, the chimpanzee measures the proper length since the chimpanzee is at rest relative to the spaceship: L0 = 30 m. Use the equation for length contraction to
 determine what an observer on earth measures.
 Recall that the way to divide by a fraction is to multiply by its reciprocal.
 (B) Regarding the time of the trip, the chimpanzee measures the proper time since the ship’s clock is at “rest” relative to the journey: t0 = 12 yr. Use the equation for time dilation to determine what an
 observer on earth measures.
 (C) Multiply the speed of the ship by the time of the trip as measured by an earth observer.
 Note that one lightyear (ly) equals c times 1 yr. That is, the speed of light times one year equals the
 distance that light travels in one year. (If you prefer meters, use c = 2.9979 × 108 m/s and convert 1 yr
 to 31,536,000 seconds to get 8.5 × 1016 m.) Unlike part A, the observer on earth measures the proper length for this distance (which is at rest relative to earth).
 (D) Multiply the speed of the ship by the time of the trip as measured by the chimpanzee.
 Since the observer on earth measures the proper length of the trip (since the starting and ending points of the journey—which are likely the sun and a nearby star—aren’t moving relative to the earth), the chimpanzee measures a shorter length due to length contraction. We could have obtained the same answer from the length contraction equation (using d for the distance of the trip, so as not to confuse it with the L that we used for the length of the ship in part A). It’s instructive to compare part D with part A, since in part A the chimpanzee measured the proper length, whereas in part D the observer on earth measures the proper length.
 (A) Multiply the speed of the muons by the average lifetime.
 (B) The average lifetime will be dilated relative to observers on earth.
 (C) In the muons’ reference frame, the lifetime is normal, but the distance that they travel (on average) is contracted.
 (A) Multiply the speed of the ship by the time of the trip as measured by an earth observer.
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 How did German radio beams reach distant English cities during WWII? [closed]
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 Improve this question
 During WWII, the Germans were using radio beacons (the system) to guide their bombers into English territory. They set up two beacons, one in Kleve, a city in West Germany, and one at Stollberg Hill (North Frisia). The two radio beams intersected over Derby.
 "Knickebein"
 The Knickebein system supposedly relied on line-of-sight propagation, but the distance between these locations is simply too great to allow for LOS. According to William CY Lee (page 350, equation 11.3.1),
 −−−
 √ −−−
 √
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 the radio horizon (i.e. the service range of the beacons) can be calculated as where is the distance in miles, is the aircraft altitude and is the ground-station antenna height in feet. If we plug in the numbers (say, the height of the beacon in Kleve (239ft) and the flight height of the German bombers (19,200ft)), we get a maximum beam range of 217,8 miles. However, Derby and Kleve are far away from each other, about 330 miles. So how could the beams reach Derby?
 R = 2ha +
 √ 2hb
 √
 R ha hb
 Knickebein
 electromagnetic-radiation history geometry antennas radio-frequency
 Share Cite Improve this question
 Follow
 edited Aug 18, 2019 at 17:35
 Qmechanic ♦ 208k 48 570 2.3k
 asked Aug 18, 2019 at 15:29
 Hepper 51 4
 Check the frequency and propagation characteristics of that frequency. Remember you receive radio broadcasts when you can’t see the antenna. – Jon Custer Aug 18, 2019 at 16:23
 The frequency was 30-33MHz. – Hepper Aug 18, 2019 at 16:37
 1 Answer Sorted by: Highest score (default)
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 As pointed out by Hepper and Jon Custer, at the frequencies of operation of the system, significant bending of the radio waves will occur along with reflection off the ionosphere- making over-thehorizon transmission easy.
 Share Cite Improve this answer Follow answered Aug 18, 2019 at 17:11
 niels nielsen 96.6k 18 135 254
 Using the atmosphere itself as a waveguide – DakkVader Aug 18, 2019 at 17:51
 not exactly- 30MHz has a wavelength of 10 meters, and the atmosphere is way thicker than that. – niels nielsen Aug 18, 2019 at 18:36
 I always thought that (V)HF radio waves do not bend. – Hepper Aug 18, 2019 at 18:53
 30 megahertz is not vhf, and its waves do bend and bounce. the ability to bend gradually goes away above that frequency, and by the time you are at ~50 megahertz, you are in the line-of-sight regime. 100MHz is line-ofsight only. – niels nielsen Aug 18, 2019 at 19:49
 The German Knickebein radar stations emitted guide beams. The bombers had to follow this beam in order to reach their target. Wouldn't ionospheric refraction/reflection obscure the path of the beam? – Hepper Aug 18, 2019 at 20:26
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 Physics World
 FEATURE
 The invisibility of length contraction
 To cite this article: David Appell 2019 Phys. World 32 (8) 41
 View the article online for updates and enhancements.
 This content was downloaded from IP address 129.105.215.146 on 24/10/2019 at 00:47
 physicsworld.com
 Physics World Augus t 2019 41
 Feature: Theoretical physics
 If the Starship Enterprise dipped into the Earth’s atmosphere at a sub-warp speed, would we see it? And if the craft were visible, would it look like the object we’re familiar with from TV, with its saucer section and two nacelles? Well, if the Enterprise were travelling fast enough, then – bright physicists that we are – we’d expect the craft to experience the length contraction dictated by special relativity. According to this famous principle, a body moving relative to an observer will appear slightly shorter in the direction the body’s travelling in. Specifically, its observed length will have been reduced by the Lorentz factor (1–v2/c2)1/2, where v is the relative velocity
 of the moving object and c is the speed of light in a vacuum. However, the Enterprise won’t be seen as shorter despite zipping along so fast. In fact, it will appear to be the same length, but rotated. You might not have heard of this phenomenon before, but it’s often called the “Terrell effect” or “Terrell rotation”. It’s named after James Terrell – a physicist at the Los Alamos National Laboratory in the US, who first came up with the idea in 1957. The apparent rotation of an object moving near the speed of light is, in essence, a consequence of the time it takes light rays to travel from various points on the moving body to an observer’s eyes.
 The invisibility of
 length contraction
 The idea that objects contract in length when they travel near the speed of light is a widely accepted consequence of Einstein’s special relativity. But if you could observe such an object, it wouldn’t look shorter at all – bizarrely, it would seem to have been rotated, as David Appell explains
 David Appell is a freelance science writer living in Oregon, US, www. davidappell.com
 Paramount Television/Kobal/Shutterstock
 physicsworld.com
 Physics World August 2019
 42
 Feature: Theoretical physics
 Terrell’s insight was that even if those rays leave different parts of the moving body at the same time, they won’t reach the observer’s eyes at the same time. The net result is a distorted view. Amazingly, for an object that subtends a small solid angle, the Terrell effect will cancel out the Lorentz contraction, making the object appear to have been rotated. Indeed, it allows you to see partly around the object and observe its reverse side. Of course, the rapidly moving Enterprise would also be subject to the Doppler effect, which would render it a different colour. Indeed, the frequency shift might even make the craft totally invisible to human eyes. The Enterprise would also appear brighter or dimmer due to relativity’s “headlight effect” (which I’ll come back to later). All in all, these effects will make a rapidly moving object, such as the Enterprise, appear nothing like it does at rest. Now, if you think that’s confusing – don’t worry, you’re not alone.
 Length contraction: the short story
 The idea of length contraction was postulated by the Irish theoretical physicist George FitzGerald in 1889 and by the Dutch theorist Hendrik Lorentz in 1892. Also known as Lorentz–Fitzgerald contraction or just Lorentz contraction, it was invoked to account for the negative outcome of Albert Michelson and Edward Morley’s memorable experiment of 1887. The two Americans had famously tried – and failed – to detect the “aether”, a hypothetical stationary medium that was supposed to carry electromagnetic waves, much as the surface of water carries water waves. Keen to rescue the hypothesis of a stationary aether, Lorentz and Fitzgerald’s solution to Michelson and Morley’s null result was to suggest that objects travelling at a substantial fraction of the speed of light will – when viewed by another observer
 – appear shortened in the direction of their travel. Based on a calculation by Oliver Heaviside of how the magnetic vector potential in Maxwell’s equation transformed between reference frames, their idea was, in truth, just one of a few proposed “solutions”. None was completely satisfactory, however, and for many years, length contraction remained an ad hoc hypothesis. It was only in 1905 that Albert Einstein cut through the confusion when he published his special theory of relativity. It did away with the aether altogether and proposed two postulates. The first is that the laws of nature are the same in all “inertial” reference frames – i.e. those frames moving at a constant velocity with respect to one another, in which any object with no net forces on it will appear at rest. The second postulate states that an observer in any inertial reference frame will measure the speed of light in a vacuum to be the same. From his theory, Einstein derived length contraction as well as time dilation, the equivalence of mass and energy, and more. In particular, he realized that an observer at rest on a high-speed train will measure the length of a passing high-speed train as shortened in the direction of its motion, by the factor (1–v2/c2)1/2 where v is the relative velocity between the observer and the other train. (Paradoxically, an observer on that train will measure the first as shortened by the same factor. Isn’t special relativity fun?) Length contraction has never been directly measured. But its effects show up in the magnetic force that acts between parallel, current-carrying wires. Bizarrely, this force, which is purely magnetostatic, appears in one wire due to length contraction as experienced by the charge carriers in the other wire’s frame. (It’s complicated. You can find more information in a paper by Paul van Kampen from Trinity College Dublin – Eur. J. Phys 29 879). Current-carrying wires aside, there is little doubt that Lorentz contraction occurs if length is measured – that is, when different points on a moving object are all measured at the same instant of time in the observer’s stationary frame of reference. But when you view an object with your eyes or a camera, it’s a different story. You’re then recording the photons the object emits – and these photons arrive at your eyeball or camera lens at the same time. What both Einstein and Lorentz overlooked, however, is the fact that the photons may have been emitted by the object at a different time, especially if the object is large. In fact, in 1922 Lorentz erroneously claimed, in the Dutch version of his Lectures of Theoretical Physics, that the contraction could be photographed.
 Lost in history
 Over the years, few people paid much attention to observing Lorentz contraction. Anton Lampa, an Austrian physicist who had once helped Einstein get his first university professorship, did publish a paper on this topic in 1924 (Zeitschrift für Physik 27 138). Concerning the appearance of a moving rod to an observer, his work was unfortunately largely overlooked. Indeed, in his famous 1940 children’s book
 Curiouser and curiouser A row of stationary dice (bottom), with other dice moving from left to right (top) at 90% of the speed of light. All cubes, whether moving or at rest, have the same orientation. However, we cannot see the Lorentz contraction of the upper cubes, which instead are rotated. Indeed, due to the fact the speed of light is finite, we can actually see the “rear” sides of the upper cubes.
 CC BY-SA 2.0-de/Ute Kraus
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 1 Why Lorentz contraction disappears for a cube moving at near the speed of light
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 (a) To understand why an object appears rotated, not contracted, when moving at relativistic speeds, imagine you’re looking at a cube of length L moving from left to right. You, the observer, are standing far enough away, compared to the cube’s dimensions, that the solid angle subtended by the cube is small and light rays from it are essentially parallel.
 (c) Now consider a cube at rest or moving slowly, but rotated around an axis through the top and bottom by some angle relative to the line of sight. For an observer looking head-on, two sides would now be visible but not seen to be of equal area.
 (b) If the cube is moving much more slowly than the speed of light, all you’d see – when the object’s directly in front of you – would be the cube’s “front” side.
 (d) If an unrotated cube is moving at a substantial fraction of the speed of light, but without considering special relativity, there is no length contraction in any direction. Shown here is the top of the cube at successive times, with the dashed square showing the position at an initial time and the solid square its position after travelling from B1 to B2. As the speed of light is finite, photons emitted from E2 will arrive after those emitted from B2; the eye will not see them as simultaneous. But “retarded” photons from E1 will be seen at the same time as those from 2B and C2 when the time for a photon to travel from 1Eto B1 is the same as the time for the cube to travel from 1Bto B2 – that is, when L/c = b/v, where b is the distance from B1 to B2. (Again, we’re assuming the observer is far enough away so that the light rays are essentially parallel.)
 (e) Continuing from above, but still not considering special relativity, the observer will not only see the “front” of the cube but, due to the “retarded” photons, also its “left” side, with its length L reduced toL(v/c). The cube will appear distorted though not rotated, since the “front” is not foreshortened and still has a length L.
 (f) Finally, let’s consider both photon retardation and relativistic effects, as we must in the real world. Moving at near the speed of light, the “front” of the cube is foreshortened and appears to have a length L(1–v2/c2)1/2. Compared to part b, the cube appears the same as one of length L but rotated by an angle θ, where (as in part c) cosθ =L(1–v2/c2)1/2 /L i.e. θ = arcsin(v/c). The moving cube will thus look the same as an undistorted, rotated, non-relativistic cube of length L. The Lorentz contraction has amazingly “disappeared”.
 Figures redrawn from: Peter Signell, Project Physnet, Michigan State University. Used with permission
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 Mr Tompkins in Wonderland, the Russian-born physicist George Gamow got length contraction all wrong. He showed bicycles as simply shortened in the direction they are travelling, instead of being distorted and elongated when approaching an observer, or contracting as they receded into the distance. Researchers only properly started to take notice of the practicalities of observing Lorentz contraction when Terrell published an internal Los Alamos article in 1957 about the effect, followed two years later by a paper in the Physical Review (116 4). His work was spotted by the theorist Victor Weisskopf, who, while serving as president of the American Physical Society, wrote an article for Physics Today presenting Terrell’s findings in a simpler form (September 1960 p24). Like Terrell, Weisskopf also noted an earlier 1959 article by the British mathematical physicist Roger Penrose, in which he analysed the appearance of a relativistically moving sphere. In his 1905 paper on special relativity, Einstein had said that such a sphere would look like an ellipsoid, contracted in the direction of motion. But Penrose reckoned that the object would still be spherical, albeit rotated. Indeed, thanks to Penrose’s insights, the Terrell effect is sometimes referred to as the “Penrose–Terrell effect” (though the two worked independently). Terrell’s and Penrose’s simple insight – overlooked by nearly every physicist before them – was that light rays that simultaneously leave a moving object do not necessarily strike the eye or camera film at the same time. The eye or lens sees images from photons that strike it simultaneously, but these photons – especially for a rapidly moving object – do not leave the object simultaneously. Surprisingly enough, in certain circumstances this difference exactly cancels the
 Lorentz length contraction and the moving object appears rotated.
 Rotation, not contraction
 To appreciate why Lorentz contraction disappears for an object moving at relativistic speeds, Peter Signell from Michigan State University has considered the simple case of a cube moving left to right when viewed head-on (figure 1). Amazingly, the “front” of the cube nearly disappears – and the observer can see almost the entire “left” side. Indeed, it’s easy to extend this thinking to other shapes (such as Penrose did with spheres) and to other viewing angles. Of course, to fully understand the appearance of a rapidly moving object, you have to calculate the geometry not only for arbitrary viewing angles but also for arbitrary distances, velocities and sizes (i.e. subtended angles). You also have to include two purely relativistic effects: the relativistic Doppler effect and relativistic aberration – the “headlight” effect mentioned earlier whereby light emitted isotropically by a moving object is seen by a stationary viewer as bunched in the direction of motion. Do all that and a “small” object moving from left to right will appear to change colour from highfrequency blues to low-frequency reds – and may even be invisible if its apparent colours are Doppler-shifted outside the visual range of your eye or recording medium. The object will also become dimmer, while its apparent angle of rotation will change from near zero far away to nearly 90°. And because the left-hand side of the object becomes visible, the object will appear to be receding before it has even reached the observer. As for a sphere, it will – as Penrose showed retain a circular outline no matter what its speed
 2 Length contraction
 (a) A row of dice at rest moving from left to right in a single file at 95% of the speed of light. (b) The moving dice are length contracted, so that one might (wrongly) expect them to look as here. (c) If you actually observe the dice, however, they will appear rotated.d() But when some perception in depth is provided, you’d see them as sheared rather than rotated.e)( Shown here is the predicted “classical” appearance of the dice, with no length contraction. You can view a short film of part c online at bit.ly/2xDc55j.
 U Kraus 2008 Eur. J. Phys. 29 1
 To fully understand the appearance of a rapidly moving object, you have to calculate the geometry not only for arbitrary viewing angles but also for arbitrary distances, velocities and sizes
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 and subtended solid angle, albeit rotated and partly distorted. It will also undergo all the colour and intensity changes noted above. The bottom line is that the observed shape of any object will not depend on its Lorentz transformation. Yes, length contraction occurs – and can be detected by careful measurement. But if you try to observe length contraction, you can’t because your view of it is compensated for by the finite speed of light.
 Cool findings
 Over the years, only a few hardy researchers have studied the geometrical appearance of large objects moving at relativistic speeds. In 1961 Mary Boas of DePaul University in Chicago showed that straight lines will appear curved (Am. J. Phys. 29 283), while four years later David Scott and M R Viner of the University of Toronto showed that extended objects – those not meeting Terrell’s assumption of a small subtended angle – are distorted in shape too (Am. J. Phys. 33 534).
 “Their grossly altered appearance,” Scott and Viner wrote, “might be described as a non-uniform shear deformation in each of the two perpendicular planes through the line of motion.” Weirdly, the noncentral parts of such bigger objects will, at larger subtended angles, appear rotated and behind the centre along a hyperbola. Strange things also happen to the apparent speed of a relativistically moving object. In 2005 Robert Deissler, who is now at Case Western Reserve University in Ohio, showed that an object with a measured speed v will appear to be travelling faster than it actually is (Am. J. Phys. 73 663). That’s because the object’s apparent position is always behind the actual position as light takes a finite amount of time to travel to the eye or lens. According to Deissler’s calculations, the apparent velocity of a distant object approaching an observer is v/(1–v/c), which means that if v ≥ c/2 the object will appear to be moving faster than the speed of light (you can do the sum to check). And if the object’s speed starts getting close to c, it’ll seem to be travelling infinitely fast. Indeed, this effect, which is purely geometrical, has been observed in some stars and galaxies that seem to be moving faster than the speed of light. Considering all these effects – geometrical plus relativistic – is a challenge, but one rendered easier by computer graphics, as described by Zachary Sherin, Ryan Cheu and Philip Tan from the Game Lab at the Massachusetts Institute of Technology and Gerd Kortemeyer from Michigan State University in 2016 (Am. J. Phys. 84 369). Many such graphics and videos have also been collected at the website spacetimetravel.org, a translation of a German website created by Ute Kraus and Corvin Zahn from the University of Hildesheim in Germany (figure 2). (Kraus has written more about her work in “First-person visualizations of the special and general theories of relativity” Eur. J. Phys. 29 1).
 Some six decades after Penrose and Terrell’s publications, the Terrell effect is still not widely known – Terrell died in 2009 after a lifetime at Los Alamos
 – and many textbooks and science presenters still get length contraction wrong. But when you realize what Penrose and Terrell had to say, you’ll surely wonder why you never thought about Lorentz contraction in this way before. As the German philosopher Arthur Schopenhauer once wrote, in a remark that’s often erroneously attributed to Erwin Schrödinger: “The task is...not so much to see what no-one has yet seen; but to think what nobody has yet thought, about that which everybody sees.”
 High-speed vision
 What’s the fastest speed we can see with the human eye? One way to answer this question has been suggested by Jack Singal, a physicist at the University of Richmond in Virginia. He imagined an object – a Borg spaceship from Star Trek if you like (shown above) – moving at speed v, passing linearly across one’s field of vision, visible if it were at rest, and directly overhead. For simplicity, he ignored any potential distortion from the atmosphere and assumed the ship is bright enough to see. Now if A is the angle (in radians) over which the human eye can focus, T is the neurological “refresh rate” of human visual perception, and D is the closest distance between you and the object, then according to simple geometry, and ignoring any unfocused peripheral vision, v =DA/T. Given that A ≈ 1° (0.017 radians) and T ≈ 0.001 s, then if the Borg ship is 100 km away, skimming the very top of the Earth’s atmosphere, the fastest visible speed will be 1750 km/s or roughly 0.6% of the speed of light. (Such an object would take only four minutes to get from Earth to the Moon, and its purely relativistic length contraction would be, for a 1000 m-long ship, only 2 cm.) However, that limiting observational speed may be too high. Jordan DeLong, a psychologist who is a data-science director of the Los Angeles market-research firm Research Narrative, thinks the situation is more complex. “The eye isn’t uniformly sensitive like a digital camera,” says DeLong, who has carried out research on vision especially as it relates to film. “The periphery is better at detecting motion, but not as sensitive to detail and colour.” And brighter visual stimuli are transmitted more quickly to the brain. DeLong points to a classic study from 1985, in which a team led by Craig Meyer, now of the University of Virginia, immobilized the heads of five male subjects, who were asked to track a moving spot as closely as possible. Using a device to monitor their eye movements, Meyer found that – in this near-perfect set-up – their eyes could track a spot to an upper limit of about 90° per second (Vision Research 25 561). “That could be a small object really close to the eye that’s moving slowly, or a large thing in the distance moving more quickly,” says DeLong. The equation above yields a maximum detectable velocity at a distance of 100 km of only about 160 000 km/s, or 0.05% of the speed of light.
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