There were two Murderers' Rows in 1927

There's the one we know best....

1927 New York Yankees

...and the one which actually changed the world forever.

Solvay Conference, 1927

Why I feel that time travel is necessarily impossible

When considering time travel, I feel it is necessary to weigh in with as logically complete an argument as I can muster. I do not know about the progress that may have been done with subatomic or massless particles. I do not claim that time travel of these objects may be found to be achievable by human technology. I do claim that these events are necessarily infinitesimal: if there is found to be a wall of progress, beyond which certain diminutive particles become too massive to transport, or beyond which the duration of the time displacement becomes too great to affect, then my points below will be just as valid.

My argument here is against travel through time of macroscopic objects, up to and including human beings. The time travel allowable must be of a non-negligible quantity. Let us say 1 second, but for pedantic readers, many important things have happened in less than 1 second, so you can go as low as you like, say 1 ms or 1 us. I do not believe that time travel (into the past) of massive objects, for non-negligible periods of time, is physically or logically possible.

1) What limitations do I propose of the term massive?

The limitations of “massive” are not hard and fast. If it is technically possible to transport 1 g of mass, then it will likely become possible to transport 100 kg, or 1000 kg or more. But this development is not necessary for my point to remain valid. If it is not possible to transport 100 kg, 1 g is enough mass for a suitably advanced microcomputer to influence past events. Even if 1 g is not possible, 1 mg of VX nerve gas could be sent back in time to the exact spot in which Hitler was standing in 1923, and kill him. As we drift downwards to other small quantities, there approaches a point in which the displaced object is too small to make any perceivable difference; that is what I would refer to as the cutoff point of “massive.” Something that is the mass of an electron naturally has nonzero mass but I would not refer to it as “massive.”

2) Is there just one timeline?

If there are multiple universes, and each event divides the present universe into parallel universes, this is in accordance with the “multiverse” theory. This means that a “timeline” doesn’t really exist; at best, going back in time would be to travel backwards from the present universe (the “leaf” of an infinite and constantly expanding binary tree of universes) along any of an infinitely long path of parent universes. If this theory is accurate, then time travel to the past is not logically impossible. Even if one were to go back in time and kill him or herself as an infant, the ramifications to the “present day” would be felt in a parallel “present day” universe in which the traveler did not exist. Since he must have lived in a divergent timeline in order to become an adult, no paradox exists.

The multiverse theory cannot be disproved at this point in time, but I do not feel that it is a valid model.  It is fairly magical and whimsical. If all events led to another set of universes based on the possible outcomes, then there would be an infinite number of universes. The observable universe that we know at the present moment is finite, and so it is possible that universes could communicate with each other. My argument here is not an appeal to ridicule, since the notion of communicating with other universes is strange but not inherently impossible, but that if communication could possibly exist, the infinite sophistication of technology that would come about through time travel would render it inevitable that in the present universe in which we currently exist, evidence of this communication would be manifest. Since such evidence is not manifest, I am inclined to disagree with the multiverse theory.

3) What constitutes physical or logical impossibility?

Something can only be definitively stated to be impossible if it causes logical error. This is to say,  if it defies logic for such a condition to exist. 2+2 = 5 is impossible. Using mathematical rigor that is beyond my ability to competently provide, it can be proven that 2+2 = 4. For something to be impossible, it must result in a paradox. Paradoxes are by their very definition impossible and unachievable. The Novikov self-consistency principle essentially states the same. It is basically a tautology, and of course like all things scientific it can be denied, but if you contend that paradoxes ARE possible, then you defy all existing human logic and you will have a very hard time creating a new one to suit your beliefs.

4) What is the grandfather paradox?

This is a very common theoretical paradox. An individual travels back in time and kills his own grandfather, thus making his own existence impossible. This constitutes a paradox in a one-timeline universe because he was never alive to kill his grandfather, but this event definitely happened in the past. There are lots of paradoxes, but I will use this one.

Having qualified these terms, I have a few premises (bulleted) that I want the reader to accept.

•  If time travel caused paradoxes to exist, then time travel must necessarily be impossible, since paradoxes are by definition impossible.

This should not be controversial. If you are a multiverse fan or you have some sort of alternate theory, then you are probably shaking your head since nothing near and dear to you is under assault yet. But let me go onward.

• A one-timeline universe in which time travel to past periods with macroscopic objects for non-negligible periods of time is possible, results in intentional attempts to create paradox.

This is a direct assault. I think it is crucial that this premise is accepted and I will go to all lengths to ensure that it is. I’ll start with an example in the form of one of my favorite middle-school novels, the very brilliant Timeline by Michael Crichton, published 1999. Robert Doniger, the ruthless founder of a technology company that invents time travel, gives us a monologue on why the grandfather paradox is invalid. He says that the traveler might have the most tenacious will to commit the act but the mere fact that history and his own memory records differently means that the act must not have happened. Something must have stopped the traveler and preserved the timeline. It could be as minor as getting second thoughts, or it could be as extreme as the grandfather shooting first and killing the traveler. Whatever the means, paradoxes do not occur.

Doniger’s explanation is not very convincing because it comes from the perspective of a writer of science fiction. The writer of science fiction often depicts the origins of time travel, gives us a good story to read, and never has to contemplate what would happen if, instead of one time travel journey, there were an infinite number of time travel journeys.

Maybe the perpetrator of grandfather paradox simply fails en route almost all the time. Maybe even the best-laid plans of geniuses to create paradoxes are somehow foiled. Maybe there are 1 million time travel events over 1000 years, trying to cause paradoxes, and somehow each one of them was defeated before it changed the timeline in any impossible way.

But we are talking about an INFINITE number of time travel events, which makes 1 million the same as zero. If there is any possibility of causing a paradox, if somehow even the most foolproof method of just going back in time to change history is always foiled, we should examine the preponderance of probability, as I refer to it. What is more probable? 1) Time travel is possible and an infinite number of time travel events somehow do not lead to a single activity that causes a time paradox; 2) Time travel is not possible, making it unnecessary to prevent every conscious effort to create a paradox.

I’m going to keep hammering on about infinity. Of course a time paradox isn’t “supposed to happen”, but how much probability can stack against it? If a time traveler came to the past with a million bombs, each of which is 99% likely to work, and every single one is a dud, is that more unlikely than time travel being impossible? But that’s not even fanciful. Let’s propose some other things that very definitely would happen in an infinite number of time travel events.

1.       A scientist and his colleague sit together in a sealed room for 10 minutes motionless. They have a time machine which can take one of them back exactly 5 minutes and can only be engaged from a remote console; the time travel machine will self-destruct in 15 minutes. At the 10 minute mark, one of the scientists goes back in time 5 minutes with a katana and decapitates the other scientist (sitting motionless), making the journey impossible.

2.       A pregnant woman who knows that her child will be born with a defect from the father goes back in time 6 months and persuades her earlier self not to have sex with that particular man.

3.       A very dedicated Japanese Imperial sympathizer goes back in time to 1945 and just minutes before the Trinity blast in Alamogordo, he arrives and sabotages the device on the spot. The test is declared a failure and Fat Man is not dropped on Nagasaki, which means that WWII does not end in August 1945, in contravention to known history.

4.       A Nazi sympathizer goes back in time and demonstrates his technology to the Germans. They bring their armies of 1942 and 1943 back to 1941, more than tripling the size and power of the force for Operation Barbarossa. The Soviet Union capitulates in 1941. Thereafter, the Germans conquer the world, in contravention to known history. They ostensibly preserve the "new" timeline by remembering to send exactly the same armies back in 1942 and 1943.

5.       The original inventor of time travel is identified by a hater of time travel, who goes back in time and kills the inventor.

All of these and millions more paradox attempts are guaranteed because time travel will perpetuate itself. Once time travel is invented, it will be used to go back in time, and then exist at a previous time period. Eventually the time travel events will encompass every moment. The leads me to the next main premise of my argument.

• Time travel’s existence will lead to an infinite number of time travel events, and an infinite number of attempts to create paradox.

As a prelude, let me address why each and every single time travel event can’t be managed so as to not cause paradoxes. I need a new example: Time Squad, a Cartoon Network show from the early 2000s.

This cartoon is actually pretty entertaining because it shows revered historical figures to be totally different from the way posterity records them. It shows Abraham Lincoln as a hooligan who has to be “reformed” by the Time Squad. If we suspend our disbelief concerning the fact that Lincoln’s second life was kept secret from the press of the 1860s, then their activities make total sense. We know that Abe Lincoln was president 1861-1865, and that he fought and won the Civil War, and that he signed the Emancipation Proclamation. Time Squad did not alter the past to change history. They went back into time to ensure that history was preserved in its current form. This causes no paradoxes, and it’s completely logical in a one-timeline universe. Of course they went back in time! Because somebody had to have gone back in time to help these historical figures- it obviously happened in the past, even if nobody knew about it. This is totally plausible even in the one-timeline model.

The problem is that Time Squad is made up of responsible persons who have not intervened in ways that are contrary to recorded history. They have not done anything that flagrantly violates what we acknowledge as fact. But they are one team working in a very regimented way with a very limited number of time travel events. If time travel were a real human technology, the infinite amount of time would tend to make it widespread.

I propose the following set of assertions that support my contention that an infinite number of time travel events follows from the mere invention of time travel.

1.       Human beings may endure catastrophes that set them back hundreds or thousands of years in technology, but on an overall trend, they must inevitably get higher in technology.

2.       If a technology exists once, there must exist a means, however complicated and expensive, to replicate it. Nobody has yet made a device so miraculous that its purpose cannot be met by a reverse-engineered device. We are not talking about what is “practical” to replicate. The pyramids could not be practically made again unless hundreds of billions of dollars were spent, and the workforce simply wouldn’t stand for it, but it is theoretically possible.

3.       Technology tends to become more widespread over time, inexorably and despite the best efforts and intentions of government and organizations and communities. We are at a point in the 21^st century where nuclear weapons could potentially be developed by most of the world’s countries. Almost all of them may choose not to do so, but as far as time travel is concerned, it would only take one exception to cause a time paradox. If North Korea developed time travel technology, it would spell the end.

4.       Free will does exist, and there are people who do want to see the world destroyed. Even if it takes millennia to happen upon such an evil person who would use time travel to contradict history, the passage of time will generate so many time travel events that the presence of nefarious intentions is assured in the long run.

5.       There is no human extinction event that will wipe out people before time travel technology can spread. This may be the hardest for some people to swallow. But if you consider that humans would have a desire to prevent the extinction of their species, if most of the world were rendered uninhabitable for a long time, one would reasonably predict that the nation or entity that controlled time travel would save itself from the extinction event by going into the past before the event. If you believe that extinction will come before time travel is even discovered, then you’re dodging the question of whether it’s possible or not. We have to assume that time travel WOULD be discovered before extinction, or else there is no point in discussing it.

6.       Going onward from Premise 5, if a nation or group of people would choose to avoid an extinction event by going backwards into the past or forwards into the future, then the timeline becomes basically circular. At any point on the timeline there are an infinite number of people who have chosen to avoid death by going into the past, because these people who went into the past inflated the population of previous populations, who later grew into the future population that also went back into the past to live there as well. Perhaps this doesn’t cause any visible paradoxes on its own;  maybe overcrowding is prevented by colonizing other planets. Perhaps censuses are no longer taken. Maybe the extinction-avoiders implement a government that kills off most of their population and rigidly limits the number who can be saved, so as to avoid interfering with the past timeline. However, the presence of this circularity does still make the number of time travel events virtually infinite, and see Premise 4 for a treatment of the ramifications of that condition.

The hardest part of my argument is yet to be made. We are at the point now, I hope, where you agree that time travel will create a “circular” timeline, if only one timeline exists.

D.      An infinite number of time travel events means that in the long run, a paradox is guaranteed to occur.

I know, you’re thinking “How can he say that?” If paradoxes are impossible, of course there must be some way to make sure that they do not happen. I propose that the actual method of not creating paradoxes is by time travel not existing in the first place, since if it did exist, there would be a million trillion billion quadrillion people at every second in the entire universe trying to go back in time and screw up the timeline. Whether they used reliable tools and obedience of solid mechanical rules, or half-hearted and poorly planned attempts, every action has to be counteracted by increasingly improbable counteractions. The timeline could only be preserved if there was an equally infinite number of “time preservers”, like the Time Squad, preemptively preventing paradoxes. However, their job too would become impossible, since the presence of infinite numbers of future travelers in the past would make the witness of such events unavoidable. Right now, if time travel existed, somebody from the year 2145 would unquestionably be trying to blow up a bomb in your neighborhood, and even if it took a million tries for such a person to successfully get to the year 2012 with a bomb, and even if it was defused at the last second, you would still bear witness with your own eyes to time travelers. A billion time travelers might somehow keep a secret, but an infinite number of time travelers would with certainty have success in making time travel known to previous generations.

I have met with some resistance on the topic of assuming time travel would be developed before extinction, if time travel were possible. Let us assume that extinction occurs; the argument becomes easy:

Premise: Time travel is not discovered before extinction of intelligent life.

Premise: Extinction causes the end of future technological developments.

Premise: Without future technological developments, nothing new can be discovered.

Premise: If time travel is never discovered, it remains impossible.

Conclusion: Time travel remains impossible.

If you have no further objections (please add them in the comments), then I’d like to condense my argument to conclude.

Premise: Time travel is discovered before extinction of intelligent life.

Premise: If time travel caused paradoxes to exist, then it must necessarily be impossible, since paradoxes are impossible.

Premise: A one-timeline universe in which time travel to past periods with macroscopic objects for non-negligible periods of time is possible, results in intentional attempts to create paradox.

Premise: Time travel’s existence will lead to an infinite number of time travel events.

Premise: An infinite number of time travel events means that in the long run, a paradox is guaranteed to occur.

Conclusion 1: Time travel in one timeline will generate paradox.

Conclusion 2: Time travel within one timeline is impossible.

Hope I’ve made you think. For a more entertaining story, let’s change gears to a most patently incorrect treatment of time travel, along with my most hated phrase: “ripple effect.” Of course I am talking about Back to the Future.

The most agonizing part for me about Back to the Future and the other two sequels is the notion that although there is one timeline, the past can be changed to affect it, even as a future observer traveling into the past.

Where does the Doc make it clear that there is one timeline? When he suggests that a paradox could arise. If there were more than one timeline, then paradoxes couldn’t occur. If Marty actually failed to get his mother to fall in love with his dad in 1955, then there would exist an alternate 1985 in which Marty McFly never existed, but that is obviously not the 1985 from which he came and of which he had memory. There would be no paradox here, just diverging timelines. If Marty stayed in 1955 for 30 years, he would witness an alternate 1985 that was distinct from his own memory, because the point of divergence occurred when he traveled back to 1955 and started a new timeline.

If there is only one timeline, then nothing like the “ripple effect” can occur. Marty being born around 1967 is an established fact. He knows when he was born. There does not exist any doubt about the fact that he was born, because he is a living, breathing teenager. If Marty could go back in time to make himself not exist, then he couldn’t have existed to go back in time to do it, because there is one timeline. A paradox is by definition impossible, so it’s imaginary to even consider what a paradox might be like. Let’s say that Marty failed spectacularly in getting his mom and dad to hook up in 1955. Let’s say that he actually just abandoned Lorraine without success and returned to 1985. He didn’t de-materialize. The timeline would have to remain intact. It would come out later that Marty’s parents were lying about kissing for the first time at the Enchantment under the Sea dance, and they got together later. Marty’s world could not be changed when he returned to 1985. George would still have to be bullied by Biff, because that is what happened in the one timeline. If Marty’s old 1985 showed his dad as a weak wimp, then unless he was in The Matrix or being fed some kind of virtual reality simulation, that was the only real 1985, and when he gets back his dad must still be the same weak wimp.

This happily precludes paradoxes. But it means that the ripple effect is garbage. If they were going back in time to prevent Biff from becoming rich, then their deeds done in 1955 predated Biff becoming rich in 1958-1985, and so it could never have happened in the first place. In our single timeline, they would have arrived in 1955 and burned the sports almanac three years before Biff ever won his first horse race, and so Biff would never have won all that money, and he never would have taken over Hill Valley, and Biff’s matchbook would either have to say that he was a casino owner or a car detailer, but it couldn’t possibly be blurred from one to the other based on meddling with the past. The past predates the present; the past has already happened relative to the present. The ripple effect is itself a paradox, because it allows the time traveler to change events in the past that would have affected the motivation to do the travel in the future, even when removing the problem.

The problem is not that paradoxes are possible, but that BTTF’s creators have taken liberties with the possible effects of time travel. Everything that is done by a time traveler in the past must have already influenced the future. It would create no paradoxes whatsoever if a man was his own grandfather. Perhaps he knew that his grandmother was a single parent but did not know who her lover was, and when he went back in time he had sex with her as a young woman, causing the creation of his family in the first place. His memory is intact, everything happened in a way consistent with the one timeline. Consequently, the very fact that Biff is rich in 1985 must be part of the timeline, and if they tried to go back to 1955 to stop it, they could not stop it from happening. Something would have to have intervened to stop it. The proof is in 1985. How could Biff be rich if he never got the almanac 30 ears ago? Moviegoers don’t really ponder the fact that the decision to stop Biff becoming rich came 30 years after it had to have happened or not, and consequently that bit of history was already written.

How can you get an average of numbers which are secret?

This is one of the more "blue sky" questions posed in the Gamesman section of IEEE Potentials, Vol. 31, No. 4 for August 2012. I like this little magazine very much and it comes free with membership in the IEEE, at least for students. All EE students should join IEEE as soon as possible.

Seven employees are in a meeting room and the topic of conversation turns to salaries. Each employee would like to know how his/her salary compares to the average but, at the same time, does not want to violate the company policy that prohibits disclosing salaries. What strategy can the employees use to get the information they want?

I did not think at length although it seemed interesting to me. My initial theory was that it could be done by some form of obfuscation with multipliers. Each employee could pick a different random number, say 65 or 213, by which to multiply their salary and then write it down as an unmanageable fractional representation that could not be estimated off the top of the head by anyone present. Then one of them could feed the sum of the fractional quantities inline into his or her TI-89 and without reducing the fractions (since this is the data to be obscured), calculate the average of them and record it. The original fractional values on a piece of paper can be shredded. The security of this comes in the fact that the calculator is the only thing that will consider the un-obscured salary figures, since none of the employees will see the numbers long enough to reduce it in their heads.

The magazine's solution might be more secure than mine: The first employee can add his salary to an arbitrary number, then whisper the sum to the second employee. The second can then add her salary and whisper it to the third employee, and so on. At the end, the seventh employee can whisper the total to the first employee, who can then subtract the arbitrary number he chose initially and divide by seven to obtain an average. 

WWII: The Scientists' War

Exodus of the European Scientists

In the previous post, Fermi's flight from Fascist Italy to the United States in 1938 was briefly mentioned. In fact, this emigration was far from atypical; the United States was increasingly called home by a number of European Jewish scientists (or in Fermi's case, European scientists with Jewish family members or spouses).

Fermi's defection completed a powerful assemblage of theoretical nuclear physics talent in America. Some of the names are all-time legends in physics: Albert Einstein had come to the US in 1933, Edward Teller in 1935, and Leo Szilard earlier in 1938. Between Germany, Hungary, and Italy, the major Axis or Axis-aligned European powers had hemorrhaged some of their best brains for the sake of their insane national liberations.

Albert Einstein (1921)

Einstein seemed amazingly prescient about the future of Nazism. Einstein was vacationing in America in 1933 when Hitler came to power in Germany. Even though fascist governments were generally seen as benign by the world at large (Mussolini was praised for "getting the trains to run on time"), Einstein had lost faith in German democracy permanently, and he already knew of Hitler's stance on Jews. Einstein never returned to Germany, and he became a naturalized American citizen in 1940. Leo Szilard's letter to President Roosevelt in 1939 was the genesis of the atomic bomb project. Einstein did not draft the letter, but it is invariably called "the Einstein letter" because Einstein signed it himself. The professor must have succumbed to pressures from fellow scientists, who knew how revered he was all over the world, and insisted that only the magical name "Einstein" would be able to command the attention of the US president.

Einstein was largely a pacifist, and his participation in the Manhattan Project was nil, but his theories made the detonation of an atomic device possible. He later regretted his role in starting the program, but waffled on whether or not it was the right idea at the time, sometimes saying it was necessary to do so before the Germans did, and sometimes falling back on his pacifism. Einstein's last years were marked by political support for the newly-created state of Israel, founded in 1948.

Other prominent scientists rendered refugees included Otto Stern (German, emigrated 1933) Stanislaw Ulam (Polish, 1941), Ugo Fano (Italian, 1939), and Felix Bloch (Swiss, 1939).

Niels Bohr (1922)

The legendary Niels Bohr was a Danish Nobel laureate from 1922, and he initially resisted any affiliation with any government's nuclear program. Clearly the rope would tighten around him at some point; Denmark was occupied by the Nazis in 1940. Bohr was tipped off that the Germans planned to arrest him and force him to work on their nuclear bomb project in 1943, and the Danish Resistance herded Bohr and his family out of the country to neutral Sweden, which provided him with passage to Britain, from which he continued on to the United States.

Bohr was a minor advisor to the Manhattan Project, although he made note of the fact that he was among many distinguished peers and had fallen somewhat behind on the times: "This is why I went to America. They didn't need my help in making the atom bomb." Despite this, Bohr's work on atomic structure and quantum mechanics are essential contributions to modern physics.

It would be somewhat unkind to say that America was weak in physics prior to this WWII influx, but the public was largely unaware of the names of the people whose theories would be so critical just a few years later. More to the point, few of the greatest American folk heroes were scientific men. Thomas Edison was certainly a gifted, hardworking inventor, but his work was entirely empirical and he was uninterested in theory or mathematical proofs. Consequently, the work of Edison was always done with what was already understood and measurable physically, and could never be thought of as revolutionary or theoretical. Henry Ford likewise "went with the gut" when it came to knowing the right way to do it. 

Philo Farnsworth (1939)

I'd like to take this opportunity to nominate Philo Farnsworth as the most unsung American physics prodigy of all time. As a 14-year-old high school freshman in 1921, he conceived of how to transmit television signals electronically. He was persuaded by his high school teacher to keep pursuing the idea until eventually it became the world standard. The young man never received a college degree and only took a few classes at Brigham Young University in Utah before financial issues forced him to take care of his family. Farnsworth's idea eventually became the only standard for television broadcast and reception for the rest of the 20th century. Astonishingly, Farnsworth was virtually unknown for his pioneering theories, memorably stumping the panel on the TV show I've Got A Secret in 1957 regarding his secret: "I invented electronic television." Farnsworth went on to design a small device (Farnsworth-Hirsch Fusor) capable of generating nuclear fusion, but frustratingly, it was unable to produce power from this reaction. Farnsworth was never a household name and it never will be, but he was endlessly influential and held 165 patents, all without a college degree. I tend to think of Farnsworth as a sort of 20th century version of Michael Faraday, who was even weaker in terms of formal education, but had extraordinary intuition and a gift for effective experimentation. The difference is that Faraday's name is still widely known, but Farnsworth is as obscure now as he was 60 years ago.

Going back into the 19th century and earlier, the Americans always remained an inventive people, but they did not have the same hard science background as was flourishing in Europe, Germany in particular. From 1901 to 1933, Germans held 11 Nobel Prizes in Physics, more than any other nation, while the Americans held just 3. Over the 1930s and through WWII, American scientists gained 4 more, while Germany gained none. Since WWII, over 75 Nobel Prizes in Physics have been awarded to Americans, and the American people (either native- or foreign-born) increasingly became the dominant nationality in physics. As of 2012, this trend seems to show little slowdown.

Vannevar Bush during WWII

The Proximity Fuze

The most widely-known American-born physicist in the prewar era was probably Vannevar Bush, who did pioneering work in analog computing and later was a trusted confidante to FDR in terms of cooperation between government demands and the body of scientists. Bush's reputation is stellar in the 21st century, but he is sometimes remembered most for his pessimism regarding innovation, saying in 1944 that "I don't see how a serious scientist or engineer can play around with rockets." Bush's stewardship of the Allied effort to develop a proximity fuze ranks as one of the most difficult and complicated engineering challenges of the war, even compared to the Manhattan Project. The VT (variable time) fuze was the first-generation proximity fuze and it had the following things in common with the contemporaneous nuclear bomb:

1. The Germans had a concurrent development program that failed to invent the same thing, either because of lack of scientific knowledge or Hitler's insistence on six-month timetables for results, killing most of the more ambitious engineering goals.
2. However, the American project was late in coming, and the UK had independently started its project earlier. The success of the proximity fuze was most primarily due to the raw materials, quantity of engineering manpower, and funding that America had, which Great Britain did not. In the case of the nuclear bomb, the British program "Tube Alloys" predated the Manhattan Project, but it never came close to success during the war because it simply did not have the funding or the quantity of scientific genius behind it. Even though the American English word for this device would be "proximity fuse", the standard term is by consensus "proximity fuze", using the British variant, probably because the British first started the research.
3. The American effort was led by a native-born American under US military control, and the contracts were provided to American companies, notwithstanding the British influence which was subsumed into the American project, which was thereafter treated as a joint Allied project and shared fully between the UK and USA.
4. The success of the project was of extreme importance for the Allies. Bush estimated that proximity fuzes increased the effectiveness of anti-aircraft artillery sevenfold. For their effectiveness in land-based artillery, Patton claimed that the invention of the fuze "required a full revision of the tactics of land warfare," which seems almost great enough praise to be used to describe the atomic bomb.
5. Usage was limited because of the enormous fear that the Allies had that the technology would fall into enemy hands. Of course, the Americans took great care in delivering the first two atomic warheads. Although individual proximity fuze-equipped munitions were infinitely cheaper and more replaceable than nuclear weapons, the American War Department decided that these fuzes should only be used in places where they could not be captured (especially shooting down incoming V-1s over British soil). Eisenhower protested the stupidity of this decision, eventually getting proximity fuzes on the front line in time for the Battle of the Bulge in 1944-1945, resulting in terrifyingly successful bombardments of unprepared German positions. A similar request by MacArthur in 1950 to allow the use of nuclear weapons in the Korean theater was a complete failure, since Truman feared the Soviets would retaliate in kind. No US commander has ever seriously requested nuclear weapons in any war since the Korean War. It's possible that only MacArthur's legendary status made him arrogant enough to make such a suggestion to the US President, and that any US general who made such a suggestion in the subsequent political climate would have been relieved of duty hastily, with an immediate psychiatric evaluation afterwards.
6. The scale of the research and development amounted to hundreds of millions of dollars. The total spent by the US military on 22 million VT fuzes during WWII amounted to over $1 billion. Per-unit cost fell from $732 at the outset, to just $18. The Manhattan Project was only somewhat more than twice the price, and it produced a total of four nuclear weapons by the end of the war (one test, two dropped, one ready by the end of August 1945) making it a pretty good deal, at least to modern sensibilities.

Nuclear Weapon Types

Of course there were many other American WWII scientists of great importance: Nobel Prize winners Arthur Compton (discoverer of the Compton Effect) and Ernest Lawrence (inventor of the cyclotron) were two American physicists who offered a great deal to the Manhattan Project. Compton, like Fermi, was affiliated with the University of Chicago, which was where the earliest efforts at nuclear fission reactors were undertaken. Lawrence invented the calutron, a device for separating the isotopes of uranium. Scaled up to massive proportions at the Oak Ridge Site, Lawrence's calutrons used obscene quantities of electric power and had a dismal yield at first, but ultimately they produced enough uranium-235 for Little Boy, the bomb dropped on Hiroshima. This single bomb accounted for roughly half of the existing U-235 in the world. It was so much more expensive to build a gun-type uranium weapon that it was only done because it was basically guaranteed to work. The design was as simple and solid as conceptually possible, and a test was neither prepared for, nor carried out.

Plutonium as an alternative to uranium has become the standard of nuclear weapons. Plutonium gun-types were envisioned ("Thin Man") but they would never have worked due to the higher spontaneous fission of plutonium. Trinity and Fat Man were implosion-type plutonium weapons. These were more complicated, but once the concept was proven in the test, they were the obvious choice for mass production. Difficult though it was to produce plutonium in reactors, it paled in comparison to getting enough U-235 for a gun-type weapon.

The principle of operation of a gun-type weapon is to fire a sub-critical quantity of fissile material into another sub-critical quantity of fissile material. Together, when the materials are brought together, they form a critical mass, and a nuclear chain reaction occurs. The gun-type can be thought of as a stepping stone to more sophisticated nuclear weapons, although it was roughly as powerful as the earliest implosion-type weapons. The US developed this method not because it was cheaper or more effective, but because the scientists felt absolutely certain that it would work, and it was designed in such a simple manner that it was practically foolproof. Because it was designed to always work, the system is extremely dangerous for long-term storage, since many failure modes will result in an unintended nuclear explosion. Although this is much simpler, most early nuclear programs have not tried it because of lack of sufficient uranium, since so much is required. Only the US and South Africa have developed gun-type weapons. South Africa has completely given up on nuclear weapons since 1989, and there have been no gun-types in the US arsenal for decades.

The implosion-type weapon uses a single sub-critical mass of plutonium, achieving criticality by the simultaneous detonation of multiple explosive charges that compress it. Unlike with the gun-type, this is not foolproof, and it is relatively safe because a combination of failures that would cause an inadvertant detonation is virtually impossible. Making it happen in the first place is difficult, and so this was why the Manhattan Project scientists chose to test the device (Trinity) before dropping an identical device on Nagasaki (Fat Man). As production quality improved, implosion-type nuclear weapons developed to the point of utter reliability and multiple layers of safety. All other national nuclear weapons programs have only used the the implosion-type, including the UK, France, the Soviet Union (later Russia), the People's Republic of China, India, Pakistan, and North Korea.

What is a Fermi problem?

Nobel Laureate Enrico Fermi, c. 1943

Dr. Enrico Fermi was one of the most brilliant physicists of the 20th century, winning the Nobel Prize in 1938, and producing the first self-sustaining power-generating fission reaction in 1942. His theoretical and practical contributions to atomic physics can scarcely be overstated. In 1938, Fermi and his family left their hometown of Rome as a consequence of Mussolini's anti-Semitic laws. Fermi himself was not a Jew, but his wife was. They settled in Chicago and Fermi became a naturalized American citizen in 1944.

Fermi was as close as modern scientists have come to a completely apolitical technocrat. Neither did he help draft the 1939 Einstein letter, nor did he provide an active role in the postwar order as his confederates Einstein, Szilard, and Teller did. Fermi's role in the Manhattan Project was crucial, but he was basically anti-war. He was also an intensely humble scientist who could be found machining his own parts or helping graduate students move conference tables around the university. Tragically, Fermi died in 1954 as a result of stomach cancer.

One of the stories about Fermi is that he did the first back-of-the-envelope guess about the yield of the first atomic explosion (Trinity test in 1945) by standing at a known distance from the explosion holding strips of cut paper. He threw the pile into the air periodically. When the shock wave hit, the airborne paper was displaced a certain distance that he estimated. Then, in his head very quickly, Fermi estimated the yield of the bomb at 10 kilotons of TNT. The actual yield as determined by measurement instruments was 19 kT. Granted, he wasn't precisely accurate, but it's remarkable how "in the ballpark" a scientist can be simply by using estimation.

To posit a question which requires estimation and layers of assumption is known as creating a "Fermi problem". The "solution" to a Fermi problem is sometimes indeterminate and should be carefully quantified in terms of the uncertainty of each assumption, which can be substantial.

The original question that Dr. Enrico Fermi asked while teaching at the University of Chicago was this: How many piano tuners are there in the city of Chicago?

According to Wikipedia, Fermi's assumptions are given as follows:

1. There are approximately 5 million people in Chicago (c. 1940).

2. The average household size in Chicago is roughly 2 persons.

3. Roughly one household in 20 has a piano that is kept in a good state of repair.

4. A piano in good state of repair requires tuning once per year.

5. A piano tuner can perform a complete tuning, plus travel time, in 2 hours.

6. Piano tuners work standard 8-hour days, 5 days per week, and take off 2 weeks per year.

Using the above logic, we estimate 125,000 piano-owning households in Chicago, and the 125,000 piano tunings per year can be done by a total of 125 piano tuners.

When a guest lecturer in my extremely large freshman physics class posed this seemingly insipid question, some of the very large student body promptly went to sleep, and others took it as some kind of joke being wound up. Only a moderate portion of the students actually sought to do a dimensional analysis and helped walk the professor through our class-consensus estimates. I do not believe that many of us present had been familiar with the concept of Fermi problems before. Perhaps in recent years it has become (and will continue to become) more important to teach physicists, engineers, mathematicians, and students of all hard sciences to learn the art of estimation in research.

Actually, our guest lecturer planned to devote as much time as he felt fruitful to the development of the point of this Fermi problem. To that end, the students and he hammered out a list of not merely guesses, but upper and lower bounds for those guesses. Wow, sophisticated! But don't give us too much credit, since the lecturer did get us to answer the problem using more or less the same approach.

1. Chicago's population was estimated at between 3 and 8 million by the students, some of whom had some personal knowledge of the city, and the uncertainty largely rested on whether the city proper or the metro area was being considered. Fermi's estimate of 5 million was probably more consistent with a metro area. Calculations were done with both 3 and 8 million.

2. We were not aware of Fermi's guess of 2, but we estimated that single workers would be pretty commonplace in a highly urbanized area like Chicago, and that households of 1 would be equal in number to childless couples, which together would outnumber those with kids two to one. Of those with more than 2 residents, one of the students proposed doing some kind of summation up to 12 (10 kids and 2 parents being very uncommon) with the total probability from 3 to 12 being 0.33. We arrived at an average household size of about 2.5.

3. Pianos seem to be less common, since they remain expensive while families have found other ways to encourage gathering such as television and home computers. We reckoned that 2-4% of households had pianos, of which most (75%) were regularly tuned, since a piano way out of tune is of little musical benefit. This amounts to 1.5% to 3% of Chicago households.

4. My old friend Dan plays the piano as well as the pipe organ and violin, so he dutifully told the class in boring detail about how often tuning can become necessary, depending on how often the instrument was used. Eventually the lecturer simply suggested once per year might be accurate, and we accepted it.

5. Dan pointed out that the job of tuning a piano could be done in 1 hour. When we were asked to think of how the schedule of the piano tuner might be padded with real-life constraints such as driving his vehicle to the owner's home, and waiting for the phone to ring in the first place, we reasoned that only maybe half of the time on the job was spent doing tuning. Therefore, in a roundabout way, we also reckoned 2 hours per job to be fairly accurate. However, we took "piano tuner" in the most literal sense, and only included those who actually performed that kind of work, and not support staff for the businesses that do this work.

6. We did agree that piano tuning is neither a common nor extreme vocation, and that the practitioners are likely to set themselves very reasonable 8-hour days. The stereotypical 2-week vacation and 5-day workweek were thrown in as well. It was assumed that tuners would not have to work weekends. Here we agreed completely with Fermi again.

Using our estimates, taking just Chicago proper resulted in a range of piano tuners from 18 to 36, or if the whole metropolitan area was considered, from 48 to 96.

The real advantage of an estimation analysis like a Fermi problem is that it teaches a number of fundamental skills that help in the life of a scientist:

1. Ability to get a "reality check" on figures to within an order of magnitude, if not higher precision. If we check Fermi's guess, and try to do research on exactly how many piano tuners exist, is it reasonably accurate? If an estimator feels it is exceedingly likely that there are more than 12 piano tuners but fewer than 1,250, then he or she has a certain amount of confidence that Fermi's guess is within an order of magnitude of the "real" value.

2. Proper dimensional analysis skills. Multiplying jobs per day by days per year equals jobs per year.

3. Comfort in dealing with ranges of numbers, or in uncertainty values. We might have said that Chicago's population was 5.5 million, plus or minus 2.5 million.

4. Familiarity with maintaining proper significant figures. For example, if Chicago had 8 million or 8.001 million people, what difference is implied? If you add a 100 kg man to a 40-tonne boat, is the weight now exactly 40,100 kg?

5. Crossover fields of knowledge and "common sense". If asked for the population of Chicago, a common-sense adult should know that it is in the millions (certainly not 100,000) but not greater than the population of the whole region (certainly not 30 million). Asking engineers and scientists to familiarize themselves with pianos seems silly, but it's merely an example of how arcane knowledge could be called upon. Here is some more motivation (not that any was needed by geeky undergrads in the first place) to have a very diverse base of knowledge.

6. Intellectual flexibility in quite an intangible way. The scientist who is comfortable with relying on estimates while carefully examining the limitations of those estimates, is the one who will actually get a job. A scientist who is obsessed with Ivory-Tower exactness and does not like the real world, will have a hard time finding a job, and will also find it difficult to make substantial contributions to his or her field.

Brig. Gen. Leslie Groves

Order-of-magnitude estimates are occasionally called for when approaching theoretical physics topics, but they tend to infuriate the uninitiated! Perhaps no man was so plagued by imprecision as General Leslie Groves, who was responsible for managing the Manhattan Project. Although he had almost unlimited resources (given the time period), Groves was initially unsure if the atomic bomb could even exist at all.

General Leslie Groves, the military officer in charge of the Manhattan Project, viewed the Los Alamos scientists as in need of constant reminders of reality. Their initial estimates for a critical mass of fission material were only precise to within an order of magnitude, which would have made it either easy (10 times less than the estimate), possible (close to the estimate) or impossible given the uranium-refining capability of the day (10 times greater than the estimate). He likened order-of-magnitude accuracy in the following anecdote: "The wedding party is planned to have 100 people. However, maybe 10 people will show up, and maybe 1000 people will show up." Clearly effective planning is difficult in such a situation, and his anger is in fact clearly understandable when explaining the awesome responsibility of planning the atomic bomb project.

But an estimation problem is either a thought experiment for learning purposes, or the start of a more solid investigation with greater accuracy. It should be taken for what it is. The Fermi problem continues to be an important lesson in physics and related disciplines.

To build a pyramid

This was written by me as a freshman undergrad at Case Western Reserve University in 2006 for PHYS 123, Physics I Honors. It is reprinted in full, original form, with no attempt made to make the conclusion more realistic. Some of my assumptions were silly and made the problem trivial, but these were never graded for strict accuracy. 

The Great Pyramid was the tallest structure in the world from about 2570 BC to AD 1300 (when it was surpassed by the Lincoln Cathedral in England).  Its specifications are given below:

Length of one side of base (base is square) = 230.4 m

Height (original, estimated) = 146.6 m

Number of stones = 2.4 million

Total mass = 5.9 million tonnes

Average density = 2300 kg/m³

Egyptologists, from the Ancient Greeks who subjugated the old empire of the Pharaohs, to the British archaeologists who had such an interest in the Egyptian colony, to the present day scholars, have consistently marveled at the investment of labor and planning that must have gone into producing such a marvelous creation more than four and a half millennia past. 

How much work did it REALLY take to build the pyramid?  How many workers were involved?  What was the power of the labor machine that created it?  To put things in perspective, how much would this building cost today?

for W_x is the work involved in transporting the blocks horizontally across the desert, and W_y is the work involved in getting the bricks to their locations in height on the pyramid.

where the force is in opposition to dragging the blocks from their quarry (friction) and the displacement is how far from the pyramid building site the blocks must be dragged (the stones came from various far-away source; the average distance is probably around 500 miles).  The Egyptians had no means of locomotion for these stones except ropes and muscle.  Let us say that the force of friction was approximately equal to the normal force, due to the incredibly high friction generated by rocks on sand without lubrication.  Set F equal to the force of gravity and solve for W_x.

W_x = (2.6 million tonnes)*(9.81 m/s²)*(500 miles)

W_x = 2.1E+16 J

The pyramid has an angle with the normal provided by:

Take the average height of a block (we may be getting into rough territory here) to be 2.0m.  By the total height of the pyramid, we make the deduction that the pyramid is 73 layers high, and that for every layer the angle still holds true (that is to say, the slant height of the pyramid is a straight line).  The height and area of each level depends upon which numerical level it is, so our result is going to be a sum of works required for each level; work will be the volume of the layer multiplied by the density of the pyramid times the gravitational acceleration times the height.  To simplify that expression:

where h is the height at point n, d is one side of the base of the level at point n, ρ is the density of the rock, and g is the gravitational acceleration.  When values are given appropriately:

The sum of the two energies yields:

This amount of energy is approximately equal to what is released from a 5 megaton bomb.  If you wanted to fund a labor force of this size, consider that Egyptologists project that 30,000 workers on average were needed for 20 years, provided that they worked 10 hours a day every day.  If you think you can pay average wages of 10 dollars per day without mutiny, then you too can have your own pyramid for a mere 2.2 billion dollars.

Physical analysis of Planet of the Apes

This was written by me as a freshman undergrad at Case Western Reserve University in 2006 for PHYS 123, Physics I Honors. It is reprinted in full, original form, with no attempt made to make the conclusion more realistic. Some of my assumptions were silly and made the problem trivial, but these were never graded for strict accuracy. 

In the context of the book The Planet of the Apes by Pierre Boulle (also a big-budget 1968 movie by Franklin J. Schaffner and starring Charlton Heston, and a more recent film), we see an early popular understanding of time dilation serve as a major plot device.  The relativity of time was used as a convenient method for allowing travel to a distant star system.  It was imagined in 1963, amidst frenzied advancement in astronomy, and so placed its opening timeframe perhaps only a few decades into the future.

Professor Antelle, a genius scientist, has invented a special spacecraft that is able to move at such a high velocity (via unknown propulsion) that time itself is slowed significantly for the pilot.  This obviates the problem of impossibly low interstellar speed, and allows a huge amount of space to be traversed in even “less” time (from the pilot’s point of view) due to time dilation.  Ulysses, the main character, is part of the expedition, along with the professor, and Levain, a physician. 

Time dilation versus velocity

They intend to use this spaceship to travel to the nearest place they believe that extraterrestrial life may exist- a star system whereof the supergiant Betelgeuse is the local sun.  The time it would take their ship to reach there is 350 years, but for the individuals inside, the time will feel like a mere two years.  What velocity does this entail?  Rearranging the above equation for known values:

In order for time dilation to be that potent, one must get very close to the speed of light.  As we can see, earthly technology brings us nowhere close to even this velocity which would only shrink time by a factor of 175.  In order to reach space millions or billions of light-years away, the only possibility is to get even closer to the speed of light.

The practical difficulty is not so much in what a person would do for years on a spaceship (although this is a bit mind-boggling) but the quantity of energy it would take to transport anything at speeds close to that of light.  The kinetic energy of an object traveling at the speed of light is phenomenal.  The craft described in the book is not miniscule, either.  Let us say, for example, that using miraculous miniaturization technologies that the spacecraft can be able to carry its engines, three passengers, and enough supplies for two years forward, two years back- in a mass no greater than that of the Space Shuttle.

This figure is a bit large, to say the least.  If this spaceship spread out its acceleration over the ridiculously long interval of 20 days, then the power required would be:

which is equal to 3.8 billion horsepower.  If it were to accelerate to that speed in the same time that it took for the Shuttle to clear the atmosphere, then over one trillion horsepower would be required. 

            The conclusion I draw from this is that, in order for humans to attain speeds close to that of light, the mass involved must be infinitesimal enough so that the energy can be produced to power it, or else new methods of power (e.g. not derived from chemical or electrical propulsion) must be found. 

            But audiences would never have suspected this in the optimistic year of 1963, and as The Planet of the Apes shows, it was not a picnic when the light-speed travelers arrived at their destination.  Society involved the subjugation of humans by their primate overlords.  When our hero Ulysse finally fled, and returned to Earth, 700 years had passed and the same fate of human enslavement had befallen his planet.

            The moral of the story, if one can be said to exist, was stated aptly by a student in Physics 123 on the day of the relativity lecture: “Stay the hell away from the speed of light.”

The sacrifice of the HMS Thunder Child

This was written by me as a freshman undergrad at Case Western Reserve University in 2006 for PHYS 123, Physics I Honors. It is reprinted in full, original form, with no attempt made to make the conclusion more realistic. Some of my assumptions were silly and made the problem trivial, but these were never graded for strict accuracy.

“About a couple of miles out lay an ironclad, very low in the water, almost, to my brother's perception, like a water-logged ship. […]It was the torpedo ram, Thunder Child, steaming headlong, coming to the rescue of the threatened shipping."

~H.G. Wells, The War of the Worlds

The prelude:

The year is approximately 1900.  The Dreadnought is not yet conceived, and in the late 1890s, ironclad rammers still represent the pinnacle of naval technology.  Fighting desperately for survival against the Martian war machines, the Royal Navy selects the finest ramming ship they had in their arsenal, and the one with the greatest nimbleness and speed. 

She was the Thunder Child, blessed of agility and formidable guns and armor, yet of size small enough to make a tactical naval battle with the Martians on its own terms.  Indeed, her skirmish would be the single bare victory had by the humans of the Victorian era Earth that attempted to fight for their lives against extraterrestrial invaders.  In the Thunder Child they found a symbol- she was built with amazing care and represented the pinnacle of the technology of the world in 1900.

 

Now was the time.  There would be no other.  Thousands of refugees were fleeing after London fell, and the entire British merchant marine could be destroyed by the horrendous Martian war machines.  Three of these devices were dispatched to the seas around England to intercept any and all human vessels, killing them with black smoke.  The lives of thousands were at stake.

The engagement:

Thunder Child was at full steam when she sighted the Martian war machines.  Not used to water, the Martians were not quite sure what to make of the ramming warship.  They had seen no mechanical device at the humans’ disposal that was as large as a warship.  They made the assumption that the device was organic, and deployed the sinister black smoke against it.  Thunder Child’s crew retreated into the ship and they did not inhale any of the poison.  The smoke clouds gave cover to the Thunder Child, and she steamed on a direct collision course with the first war machine, at full 20 knots:

The Martians finally wised up and attempted to strike it with their Heat Ray.  One hit was successful, and the Thunder Child was extremely damaged; still she steamed on.  The pointed bow, with an edge merely an inch thick and twelve feet high, struck hard and pierced the extraterrestrial metal.  The impact was devastating and Thunder Child cleaved the war machine in half very jarringly, losing half of its momentum within a second. (Assumptions made regarding the dimensions and characteristics of the ramming action are all guesses by me.)

(As we find, the armor of the Martian war machines had a tensile strength of greater than 280MPa- superior to modern rolled homogenous steel.  Steel of this quality was nonexistent in 1900, and may have seemed alien.)

The Thunder Child tried then to open up with her six inch guns, but the range was too short for them to be effective.  Instead she, with foundering keel but usable rudder and engines, accelerates to full speed again, to attack the second war machine.  Persistent and desperate salvos destroy the Thunder Child before she can ram the second ship, but the ships boiler and ammunition explode into a massive hailstorm of steel that crushes the second war machine with thousands of tons of molten iron and wounds the third.

The outcome:

A marginal victory for humanity… the destruction of two Martian war machines.  This raid saved the lives of thousands, but there was to be no respite in the struggle to survive against the extraterrestrial invaders.

Structural failure of a CD

This was written by me as a freshman undergrad at Case Western Reserve University in 2006 for PHYS 123, Physics I Honors. It is reprinted in full, original form, with no attempt made to make the conclusion more realistic. Some of my assumptions were silly and made the problem trivial, but these were never graded for strict accuracy.

Professor Starkman once asked us to use rotational mechanics to find out the properties of a CD spinning at 7200 RPM; this gave us some appreciation of the stress on a CD as it is spun. The Mythbusters once tackled the objective of trying to cause structural failure to a CD by creating unusually high rotational speeds to the CD to investigate the myth that a standard CD drive can under certain circumstances spin fast enough to cause a CD to break apart and turn into a lethal disc of shrapnel.

Let’s mesh these worlds, and see what it would physically take to destroy a standard CD.  The figure we were given in our physics class was 7200 RPM.  In certain disc drives, the regular speed may be more. 

Physical Information of CD-

Thickness (X) = 1.20 mm

Material = 100% Polycarbonate (tensile strength, σ_t, of polycarbonate is about 75 MPa)

Radius (R) = 12.0 cm

Density (d) = 1.20 g/cm^3

Let us make the assumption that since the hole is filled in, we have a complete volume of disc.  Plugging that in to our density:

m/V = d

m =dV

m = dπR²X

Mass (m) = 0.0650 kg

We have a radius and a thickness, which corresponds to a cross-sectional area of a CD on one side.  Recall that it only requires a break at one of these cross-sectional areas to fail.  This material is very brittle; do not expect much strain as a result of stress.  It ought to shatter.  This should simplify things.

I plan to evaluate the centripetal force caused by the spinning of the disc as a function of ω.  This disc must respond to a centripetal force with a normal force.  This normal force is dictated by its structural integrity.  Given that we have a specific area of interest, this force may be divided by area, leaving us with units of N/m^2… the same units as Pa, which is proportional to our tensile strength.  The units of tensile strength and pressure are identical.  Evaluate for the maximum possible ω which will cause a force that exceeds our tensile strength.

σ_t = F/A           (Force required to break divided by area equals tensile strength)

A = XR            (Cross-sectional area is equal to radius times height)

XRσ_t = F         (The force that is required)

F = mv^2 / r

v = Rω

F = mRω²

XRσ_t = mRω²

(XRσ_t / mR) ^½ = ω

(Xσ_t / m) ^½ = ω

This is the maximum possible angular velocity that we can achieve.  After plugging in the appropriate values:

((0.0012m)*(75E+6Pa)/(0.065kg))^ ^½ = 1177 rad/s

We now have a figure in radians per second, but disc drives are never advertised in such figures.  What does this translate to in terms of revolutions per minute, the preferred angular velocity measurement of the West?

1177 rad/s*(radian / second)*(1 revolution / 2π radian)*(60 second / 1 minute) =

11200 RPM

Our ceiling figure for angular speed of a CD is 11200! Um, wasn't it way faster on Mythbusters? =/

In all probability, as we estimate for error in this problem, our estimate is extremely liberal with its notion of structural failure.  In actuality, the polycarbonate material may be higher or lower than the one we listed; but every CD has a bottom and top layer which would likely enhance structural integrity.  Additionally, we did not account for the removed section of the disc (the hole in the center, into which an electric motor pushes a rotor that spins the disc.  

My point in this experiment is to reflect on the magnitude of stress on the CD in your disc drive as it whizzes around at 120 to 170 revolutions per second.  A modern engine will be on the redline when a CD drive is operating properly.

The GAU-8 Avenger Autocannon

This was written by me as a freshman undergrad at Case Western Reserve University in 2006 for PHYS 123, Physics I Honors. It is reprinted in full, original form, with no attempt made to make the conclusion more realistic. Some of my assumptions were silly and made the problem trivial, but these were never graded for strict accuracy.

Those with even a cursory knowledge of national air forces will know that there are several roles to fill- air superiority fighter, heavy bomber, stealth bomber, and close-in ground support.  The A-10, ugly and ungainly as it is, is a magnificent piece of work that fills the role of the last category.  With a low flying speed, “titanium bathtub” armor, immense endurance, and a phenomenally powerful main gun, the A-10 is a wonder of military technology and of physics.

Notice that autocannon in the nose?  This is the GAU-8 Avenger, a scaled-up version of the more familiar 7.62mm Minigun and 20mm Vulcan cannon.  It fires depleted uranium shells of 30 mm diameter at a rate of 4200 rounds per minute.  Here our cannon is shown to scale.

One of the most persistent claims by enthusiastic armchair strategists is that the Avenger is so powerful that its recoil is at least as powerful as the engines of the plane- thus, it is capable of slowing down and even stopping our A-10 in the air.  Is this true?  Let’s consider momentum and force.

(momentum) = (force)*(time)                            p = Ft

What is the momentum of our stream of fire?  Since we have a rate of fire and a mass, we can use dimensional analysis to determine momentum, and hence force.

momentum = (mass of shell)*(velocity)*(rate of fire)*(time)                   p = mrt

(mass of shell)*(rate of fire)*(velocity)*(time)/(time) = force                  (mrt)/t = F

(mass of shell)*(rate of fire)*(velocity) = force                                       F = mrv

Think of r as a frequency instead of rate of fire.  We are firing 4200rpm, which equates to 70 rounds per second or 70Hz.  Doing some additional research, we find that the mass of a round is 0.425kg and our muzzle velocity (highest velocity ever attained by the bullet) is 1036m/s.  In comparison, the makers of the A-10 claim that their two engines will produce 80kN of thrust.  Let’s see if we really have the power to stop a plane.

F = mrv

80.kN < (0.425kg)*(70.Hz)*(1036m/s)

80.kN </ 31kN

We can produce “only” thirty-one thousand newtons of force by our Avenger cannon.  This means that the plane will experience a possible deceleration if it is only using partial power, but this can be overcome by employing a steady, large amount of thrust, which is in practice what the pilots tend to do as they are well aware (and possibly fearful) of the myth of planes stopping in midair.

But the GAU-8 Avenger is still incredibly powerful.  Let’s say we mounted all 281kg of the gun on the Terminator’s back, gave him 1000 rounds of ammo and convinced him that he could fire the weapon from a standing position.  His hydraulic joints lock up (he will not drop the weapon and there will be no force lost to excess motion) and he prepares to fire at the full 4200rpm, perhaps too confident of his abilities after playing around with a 7.62mm Minigun in T2.  The Minigun is well beyond the range of a single soldier to carry and operate, but he manages it without difficulty.  However, this weapon can be mounted on a helicopter door, while the Avenger, as we have seen, generates nearly as much recoil as one the jet engines on the A-10.

The Terminator weighs 200.kg.  Of course, he manages to shoulder the behemothic weapon with a bit of effort, and starts firing at T-1000 who has found himself a nifty T-72 tank.  Arnold digs his heels into the muddy ground and achieves a coefficient of friction of 1.0.  How fast will the Terminator accelerate backwards, or can he actually stand in place?

frictional force = (coefficient of friction)(mass)(gravity)               Ff = μmg

Remember to add up all the weight that is now being held on the Terminator’s hyper-alloy legs.

Ff = (1.0)*(281kg + 425kg + 200.kg)*(9.81m/s^2)

Ff = 8.9kN

The friction force is massive, but it is less than a third of the GAU-8 recoil.  There is a net force and Arnold starts to slide immediately.  The net force, accounting for friction, on the Terminator and his gun is now 22kN.

F = ma

22kN = (281kg + 425kg + 200.kg)a

a = 2.4 m/s^2

Buh-bye Arnie. You'll be all over the place. Good luck aiming the thing.  Was the T-1000 terminated?

At 500m, the Avenger can pierce about 70mm of modern composite armor.  The armor of a T-72 is thicker than 70mm- but its armor is old-fashioned steel, meaning that it would take approximately twice as much armor to achieve the same strength.  Narrowing the range to near point-blank, the effect would be catastrophic to this pensionable commie tank.

At negligible range, the 30mm rounds would turn the tank into a hailstorm of steel confetti, and the pyrophoric depleted uranium rounds, upon piercing the frontal armor, would also set fire to the shrapnel within the tank.  They would have so much kinetic energy left that some of the rounds would indeed pierce the rear armor of the tank as well.