Phys Educ 24 119891 Prlnted In the UK

Physics and the Dambusters

Arthur Stinner

The story of an engineer seeing an idea come to GuyGibson and Michael Redgrave as Dr Barnes fruition against great odds. High-grade scien- Wallis) and on the popular book of the same title tific research is combined with high adventure. written by PaulBrickhill, first published in 1951. This provides a contextthat generates challeng- The headings were chosen from the dialogue in the ing problems for physics students. movie.

The storyof the Dambusters has captured theimagi- ’What can I do to shorten the war?’ nation of every generation since the Second World War.We remember the picture of theLancaster hadbeen working atVickers- bombers flying low above the water at night to drop Armstrong as anaircraft structural engineer since barrel-shapedbombs, spinning and then bouncing beforethe First World War. In the 1920s he on the surface, and hurtling inexorably toward their designed the , the most successful British air- target. The successful attack on the great dams of ship, and in the 1930s he invented the geodetic form Germany dominated the news everywhere on that of aircraft construction, which resulted in the build- day in May 1943. ing of the Wellington bomber. At the outbreak of Behindthe extraordinary human effort of co- the war he decided to investigate the construction operationand heroism, however, there is another and deployment of super- that could destroy story. This is the story of an applied scientist seeing large and significant targets such as the great dams a plausible idea coming to fruition against seemingly of the Ruhr in Germany, with minimal damage to insurmountableodds. It admirably illustrates the people. The destructiveness of such bombs, he rea- relationship between theoretical science (in this case soned, would be due not only to their size but also physics) and technology and contains many of the to their proximity to the dam. When they exploded ingredients of high-grade scientific research. It also close to a dam they should produce very powerful shows that scientific research cannot be captured by shock waves thattravelled through the concrete. aclearly specifiable ‘scientific method’,as science Wallis came to this conclusion when remembering texts claim it can. Rather, it must be understood as something he read about the shattering of concrete part of a social and political contextand seen as piles driven into the bed of the Thames during the governed by human imagination, perseverance and construction of Waterloo Bridge. In an engineering plain good luck. journal of 1935 he found the article that described Thestory of theresearch that preceded the the formation of shock waves in concrete. It seemed famousraid is briefly reconstructedhere and an that the sudden blow produced by thelarge ham- attempt is made to actively involve the scientifically mers sent shock waves down the piles. and part of inclined reader. In fact, this is an example of a large the energywas then reflected upon encountering the contextproblem that the author is usingto teach ground. another solid medium. The reflected shock physics. This type of investigation is especially suit- wave,travelling at an estimated 5000 metresper able for GCSE and A-level students. second, always reached the top before the hammer All information used is based on the well known couldstrike again. Since theshock waves passed 1954 movie The Darnbusters (with Richard Todd as into an .empty’ medium the effect was one of ten- sion onthe concrete. Concrete, however, is very resistantto compression, but not to tension. The piles therefore shattered. Wallis extrapolated from Arthur Stinner is a postdoctoral Fellow of the Social these known facts to a exploding under water Sciences and Humanities Research Council at the neara large dam. He thought that a large bomb Ontario Institute for Studies in Education, University dropped from a great height could destroy the dam of Toronto. His background is in physics and education and he has taught in Canada, Jamaica and as long as it dropped to a depth of about 15 m close Zambia. to the dam. Shockwaves would do the rest.

0031-9120:891050260+08 $02.50 0 1989 IOP Publtshing Ltd 'Only a question of working it out paper' were partof the structure of the dam. Wallis Such bombs, he calculated, would need a mass of at exploded a few ounces ( - 100 g) of gelignite under least 10 tonnes. Furthermore, they would have to be the water 1.2 m from the wall to give the effect of a carried by a 50 tonne bomber at 140 m S-' (320 mph) 10 tonne bomb exploding about 60 m away from the and at a height of 14 km! Wallis knew that both the real dam. Theeffect was less than spectacular, so he bomb and the aircraft could be built with the techno- moved the explosive to 0.9 m and then 0.6 m. Still logy of 1940, so he wrote up aproposal for the there was no damage to thewall. Then he placed the committee appointed to evaluate his idea. This pro- gelignite a distance of 0.3 m from the wall (equiva- posal would havebeen filed underwhat the Air lent to placing a 10 tonne bomb 15 m from the Force called 'revolutionary, complicated and crack- Mohne), but even thenthere were only minor pot theory' if it had not been for Wallis's reputation cracks. He tried larger charges,and finally about and oneman, Arthur Tedder, then an air vice- 150g of the explosive 0.3 m away breached the marshal, who believed in the idea. model dam. He calculated that in the real case a Wallis wasnow given the resources to build a charge of 13 tonnes at a distance of 15 m from the model dam to test out his ideas. He built a scale dam would breach it. A simple calculation showed model of one-fiftieth the size of theMohne, the that it would require 18 tonnes of steel casing, for a largest of theGerman dams, with tiny cubes of total of 31 tonnes! It was clear that the next meeting concrete, scale models of huge masonry blocks that with the committee would be the last one.

136 milllon tonnes

Figure 1. Mohne Dam. Problem I After working out many of his ideas on paper Wallis experimented on a model of the Mohne Dam. The model was scaled down to 150. 1. Find the dimensions of the model dam. 2. If the model had been built faithfully in all directions, how much water should it have been able to hold? 3. Compare the pressure at 15 m for the Mohne dam with the corresponding pressure for the model. 4. About 90 g of geligniteexplosive under water and 1.2 m away from the model was sufficient to break the model. Based on that information alone Wallis claimed that about 10 tonnes of the same explosive under water, and at a distance of about 60 m, would breach the Mohne. Show that this is a good estimate. 5. Wallis then calculated that the steel casing for the sphere that could safely accommodate 10 tonnes of explosive should have a mass of 18 tonnes. The density of steel (iron) is about 7900 kg m-3and the density of the explosive is about 800 kg m-3. Now: (a) determine the mean radius of the sphere; (bl determine the approximate thickness of the shell; (c) considering the dimensions and carrying capacity of a Lancaster bomber (see figure 2). comment on the possibility of accommodating such a bomb.

Figure 2. Initial design of the bomb.

261 Figure 3. Nelson’s skipping cannon balls.

’You can’t hurry this kind of research‘ a bomb on water. The simple apparatus consistedof The next time Wallis met his committee (now more a rubber catapult, placed a metre or so away from a appropriatelycalled ‘The Air Attack on Dams tub full of water, just a few centimetres above the Committee’, focusing the aimof the enterprise more water level. He stretched a string (attached to two definitely) he wanted to present them with a solution sticks) about a metre on the other sideof the tub so to the problem of placinga 5 tonne bomb under that the string was also just above the level of the water close to the dam. The inspiration that led to a water (figure 4). solution seems to have come from two sources: his Wallis borrowed a marble from his daughter and experience as a father skipping stones with his chil- shot it from the catapult at the water. His children drenon water and remembering that Admiral assisted him in making measurements and generally Nelsonoccasionally had cannon balls ricocheton amusedthemselves withthis strangegame. He water for greater accuracy when engaged in a naval decided that, indeed, it was possible to control one battl! (figure 3). He now envisaged bombs skipping skip, but could one control many skips? on water over nets, slithering to stop as they In order to answer that question, clearly a larger reached the wall of the dam. The bombs then would tank was needed with more facilities, including an sink to a preset depth determined by a hydrostatic underwatercamera. Before ‘The Air Attack on trigger that would set off a fuse. DamsCommittee’ granted him permissionto use When pressed for the detailsof his ‘theory’, how- the huge ship-testing tank at Teddington, however, ever, Wallis wasreluctant to disclose how he he had to convince the members that small charges thought the bombs should be released. One member of geligniteplaced under water against the wall of the committee spoke up for him, saving him the would breach it. A new model damwas built for him embarrassment of having to explain his not yet well and he convinced the members of the feasibility of formedidea: ‘If Ihad a partly formed theory I his plan when only a hundred grams or so of the certainly should not want to talk aboutit until it was explosive so placedbreached the wall timeafter clearly established in my own mind.’ time. With this new knowledge he was able to cut In his garden in Dorset, Wallis promptly built a down the required theoretical massof the bomb to a makeshift apparatus to test outhis ideas of skipping manageable 4.3 tonnes consisting of 2.7 tonnes of

Figure 4. Wallis‘s home-made experiment.

W \ -/r I

262 Problem II 1. When Wallis discovered that he needed only 4. Nowfind theimpact force for a drop from about 2.7 tonnes of explosives to do the job, the 18 m. Assume that in this case the bomb sinks to a prospects of building a spherical bomb thatcould depth of3/8 of the diameter. How do the forces be accommodated brightened. Calculate the aver- compare? Comment. age radius of the bomb. 2. The force of impact (average force) of such a , bombon water after falling from a heightof 75 m is n enormous. 1 (a) First, find the speed of impact, assuming that n air resistance is negligible. (b)Then calculate the average force of impact of the sphere if it sinks to a depth of 3/4 of its diameter. "4 ;?/UL 3. What is the buoyant force on the sphere at the momentof maximumimmersion? Figure 5. Spherical bombs skipping over water.

Problem 111 1. All of us have tried skipping stones on water. What is surprising is that the physics of a skipping stone is not well understood. In the August 1968 issue ofthe Scientific American, for example, a Figure 6. A skipping stone. large section of the regular feature 'The Amateur Scientist' was devoted to convincing the reader of just that. It showed, among other unexpected find- ings, thatthe stone does not bounce along the Ih -r-"."r water in a series of successively shorter leaps until dl 4 d3 dL it finally stops, as one would expect. Rather, the stone may skip a short distance first and then skip a Figure 7. Idealised trajectory of a tennis ball. much longer distance. Wallis discovered that even with a perfectly spherical object like a golfball you do not get this expected trajectory of successively diminishing bounces because of the spin imparted to the ball on contact with the water. Look up the article in theScientific American and study it-you may be the first one tofind asolution to this 4. According to thissimple model of a ball puzzling problem. If you do,please contact the projected horizontally over a level surface the hori- author of this article! zontal component of the velocity stays constant. 2. Using a tennis ball we can investigate the The bouncing ball's velocity must, of course, also trajectory of a ball bouncing at a high horizontal diminish. Now account for a progressive reduction speed. Ideally one should have access to a 'gun' of velocity as well. Consider the above problem if like those tennis coaches use. Failing that, rig up a the reduction of the horizontal velocity is to 80% simple slingshot (or catapult) of the typeWallis put after each contact point. together. You could, of course, simply throw the In reality, however, one should also tryto ball horizontally. Use a large area (like the floor of a account for the spin of the ball, a difficult problem school gymnasium) and observe the motion of the to solve. Jearl Walker (19851, for example, claims tennisball. With the aid of several friendsyou that a ball projected without spin will bounce as could mark the places where theball bounces. shown in figure 8. Test this claim. How would you Note, for example, what kind of spin is imparted to give this kind of trajectory a mathematical descrip- the ball on contact. tion? Now consider the following problems. 3. A tennis ball is projected horizontally from a catapult at a height of 1.00 m. The first contact is made at a distance of 5.25 m. Assume that the ball bounces back to 50% of its previous height upon each impact. (a) Predict the bouncing distances dl, d2,d3 , , . . (b)What should be the speed of the ball so that it would land at a spot 20 m away? (c) Write down an equation that relates total distance d, numberof bounces n and theinitial speed vi, for this ideal situation. Figure 8. Trajectory of a tennis ball?

263 RDX explosive and a case 1.6 tonnes. Permission The theory was tested with model barrels using was nowgranted useto the large tank at various weightlsize ratios to make sure that the real Teddington. bomb could be carried under a Lancaster. By trial The experiments at Teddington were begun and anderror he found the speed necessary for each continued for five months, much to the displeasure model to traverse the required distance. The spinof of the committee. The objective was to control a the barrel was also found by trial and error and was skipping missile, to find out how speed, height of shown to be between 450 and 500 revolutions per launching, number of bounces and the desired dis- minute backwards. Would a full-size bomb behave tancewere related. Wallis first thoughtthat the as the theory and the scaled-down experiments pre- missile should be spherical so that identical contact- dicted? Wallis wasconvinced it wouldwork. area hits would be guaranteed. But it soon became However, only after weeks of anxious waiting and obvious that the bounces were too often unpredic- arguing with variousmembers of theMinistry of table, even though he used larger spheres, first golf Suppliesdid hereceive the services of one balls and then 1 kg spheres of steel. It took some Wellington bomber. time but he found the answer: the first bounce gave Thenew bomb was tested as soon as the the sphere a top-spin and that is why the golf balls tended to dig themselves into the water. Would flat discs be preferable? The answer was a clear no, as anyone who has carefully observed the skipping of stones on water or sand would agree. Wallis now rigged up a cradle for the spring-loaded catapult that would give the missiles a pronounced back-spin. The back-spin not only allowed him to control the dis- tance but provided him with a serendipitous effect: R= radlus theresidual back-spin toward the end of the run L =Length "moss made the sphere go under the water at the far end. This was exactly what he wanted! At this pointin the Figure 9. Cylindricalbombs. movie we hear him say to one committee member: 'There is a fine dividing line between inspiration and Problem IV obsession . . . it is sometimesvery hard to know A spinningtop, a bicycle wheel and a spinning which side you are on'. And alittle later he says: cylindrical barrel are very stable. Wallis designed the barrel so that it could withstand a spin of up to 'One goes on and on . . . as if against a brick wall, 500 revolutions per minute. The high rotation was then suddenly a light flashes.' used to fulfil two requirements: to give the bomb stabilitythroughout the trajectory, and to make sure that a little spin was left over when it hit the 'Just when you think you have solved a dam. The lengthof the cylindrical bomb was about problem . . .' 2.1 m, the mass of the RDX explosive 2.7 tonnes and the mass of the case 1.6 tonnes. The size of the sphere required to carry the RDX 1. Find the approximate diameter of the cylindri- explosive, however, turned out to be unmanageably cal bomb. large.The steel case had to be strong enough to 2. You probably know what happens when, sit- withstandthe impact with water at well over ting on a rotating piano stool, you attempt to twist a spinning bicycle wheel. 90ms" (200mph). It was evident that even if the (a) From this knowledge extrapolateto the stabi- monstrous spherical bomb could be attached to the lity (the bomb's tendency to maintain its direction) Lancaster, the aircraft could not be moved without of a rapidly spinning cylindrical bomb. the danger of risking collapse before take-off! What (b)Speculate onhow carrying spinninga he needed was a different shape that would be large cylinder of this size must have affected the stability of the aircraft in flight. enough and make contact with water on identical (c) How, do you think, did the releasing of the surface areas. A large cylinder would do it, provid- spinning bomb affect the stability of the flight? ing that it did not tilt during the trajectory. How do 3. The centripetal forces are very large at these you prevent a cylinder from tilting? By making it high rotational speeds and they must be knownto ensure the safety of the structure of the casing. spin, of course! He knew, for example, thatit took a Calculate the maximum centripetal force on the rim considerable force to tilt a spinning bicycle wheel. of the spinning cylinder. Express this value in terms Thesame gyroscopic principles must apply to a of g, the acceleration due to gravity. spinning cylinder. The answer was now clear: shape 4. Theenergy required tomake the large a missile like a portly barrel withsufficient back-spin cylinder rotate at 480 revolutions per minute isvery large. Calculate this energy. Consider the mass of to keep it gyroscopically ona level axis all the way to the cylinder to be evenly distributed. the end of the trajectory.

264 Wellingtonwas converted. The impact, however, another one under the belly of theaircraft, both damaged the bomb and the trajectory went array. pointingdown, adjusted so thatthe lights would The casing was strengthened, but from a height of meet in a figure 8 at the preset height (figure 10). 45 mthe bombs still shattered.Finally, in despe- The solution to the second problem was equally ration he asked Wing Commander , the simple. It seems that a carpenter put it together in leader of Squadron 617 (later called the 'Dambus- minutes out of bits of spare wood (figure 11). The ters') to tryflying at an altitudeof 18 m. He was sure base was a triangle made of plywood with a small thatat that height the bomb would not shatter. opening to look through and two nails hammered Afterweeks of practising flying atthat altitude partially in the other corners. The idea was to look Gibson was satisfied that it was safe (at least during through the small opening and when the two towers theday) and they made a trial drop. The bomb on the dams were in line with the nails the bomber performed perfectly. Wallis was happy. would press the release button. Assuming that both thespeed and the height were correct the bomb Plywood bombsights and steel spotlights should follow the expected trajectory. There were still twosmall problems to besolved Wallis's last task was to work out the final speed, before an actual attack could be made. One had to altitudeanddropping-range. He decidedon do with keeping the aircraft at the correct height to 100 m S" (230 mph), 18 m and 550 m, respectively. within a metre. The other problem was connected Practice runs were made by Gibson and his crew on with the need to drop the bomb at a certain distance nearby lakes, during the day and at night, using the in front of thedam. Gibson suggested that they plywood bombsights and the steel spotlights. Their dangle a wire with a weight onit from the aircraftso accuracy improved until they were able to drop the that it would touch the water at the correct height. bombsconsistently within metres of thetarget. They tried it, but at the required speed the line was Squadron 617 was ready for the raid on the damsof trailing behind almost horizontally! the Ruhr. On 17 May 1943 the mission was carried The solution came a little later, an idea attributed out successfully, inflicting a significant blow to the to the 'back-room' boys. It was a simple idea but German war machine. very effective: put a spotlight under the nose and The successful raid on the dams of Germany is

Length321 m

Wlngspan I30 m Weght (empty)- 22 tonnes

Figure 10. Lancaster bomber.

Tower- 1

"SI Figure 11. Plywood bombsights. l Toer 2 I

265 Figure 12. Idealised bomb trajectory.

Problem V Just before the raid Wallis gave the following crucial data to Squadron 617: horizontal velocity 100 m S-’; height, 18 m; dropping-range, 550 m. He had faith in the reliability of the hand-made bombsight to establish the dropping-range of 550 m and in the spotlights to guide the pilot at the height of 18 m. The author has made some calculations on the assumption that about eight bounces were required to hitthe dam. This assumption was based on the requirement that the speed of the bouncingbomb must be reduced to about 5 m S”. (At higher speeds we would risk the bomb’s skipping over the dam.) By developing an equation on this assumption, it was found that if the height diminishes to about 90% of the previous height at each impact, and if the horizontal velocity decreases to about 70% of its previous value at each impact, then the bomb makes contact just before hitting the dam with a velocity of about 5 m S”. Remember, however, that our problem is still ’idealised’. We did not account for the effect of the spin on the cylindrical bombs. 1. Calculate the angle required for the bombsight to establish the range of 550 m. 2. Now decide at what angle of separation the spotlights must be placed if they are to come together at an altitude of 18 m. Consider them to be separated by 6 m. 3. Use all this information given and calculate, as for the case of thetennis ball, the last contact point and the speed of the bomb just before it hits the wall and submerges to the preset depth. 4. Now show that the equation that describes the above situation is as follows R- 2(2hv01g)”*(~+ a”’b + ab2+ a3”b3+ a2b4+ aSi2b5+a3b6+. . . a7“b7+a4b8+. . .) where a is the successive diminishing factor for height, O

Problem VI The Mijhne Dam had a capacity of 136 million tonnes of water. Assume that the dam was full before impact and that the surroundingarea wasflooded to an average height of about half a metre. Approximately what area did the flood cover?

one of the most celebrated military adventures of equal enthusiasm the story as an imaginative scien- the Second World War.The idea and its imple- tific adventure. mentation that led to the destruction of the dams There is a sad postlude to this story. When Wallis came from the imaginationand the determined heard of those who died as a result of this raid he effort of a gentle engineeringgenius. In the words of was supposed to have said, with tears in his eyes: one of thecommittee members: ‘He conceived ‘Oh, if I’donly known, I’d never have started this!’ somethingabsolutely unheard-of and carried it Let us conclude with the following compilation throughwith flying colours.’ The story of the of pithy statements made in the movie The Dambusters has been celebratedas a military adven- Dambusters. Eachof these relates to scientific ture for a long time. It is time we celebrated with research. Their thoughtfuldiscussion in a classroom,

266 for example after viewing the movie, can yield information and much clarification about the nature ANSWERS TO PROBLEMS and the social impact of scientific research in Problem I general. 1. 80 cm, 16 cm, 68 cm. 1. ‘It is only aquestion of working it out on 2. about 1000 tonnes. paper. ’ 3.11 50. 2. ‘You can’t hurry this kind of research.’ 4. 1:50, (5013X 90 g 3 10 tonnes. 5. (a) 1.45 m; (b)9 cm; 3. ‘There is a very thin dividing line between (c)bomb is too large and heavy. inspiration and obsession.’ 4. He thought constantly about it. Problem II 5. That is exactly how a full-sized bomb would 1. 93 cm. 2. (a) 38 m S”; (b)6.5~ lo6 N. behave. 3. 35 900 N. 6. ‘The idea came from Nelson.’ 4. about the same as 2(b) above. 7. ‘Everything depends on secrecy.’ 8. It was absurd to think how simple it was. Problem 111 9. ‘Looks clever on paper but can you make it 3. (a)5.25 m, 7.43 m, 5.25 m. (b)10.5 m work?’ S-’ (C) 10. ‘Did you invent this thing in your own head?’ 11. ‘One goes on and on . . . as if against a brick wall, then suddenly a light flashes.’ d=5.25+2 ~,~(\ll+2x5 F 2‘x5 12. ‘Do you mean that a 5 tonne bomb can bounce along like a ping-pong ball?’ 13. ‘If I’d only known, I’d never had started this!’

until equality is reached. 4. r d=5.25+2 vin 0 8 -+0.8‘ 7 (’ 42:5 42 X5

+ . . . 0.8” g) until equality is reached.

Problem IV 1. 1.4m. 2. (a) very stable; Acknowledgments (b) verystable, as long as straightflight is maintained; I wish to thank my wife Ann forpreparing the (c) less stable. sketches, my colleague JohnGooderham for his 3. -200 g. helpful suggestions, and Scott Crawford, a student 4. 3x106J. at Leaside High School, for assisting mewith the Problem V solutions to the problems. 1.19”. 2. 19”. 3. - 5 m S-’, shortly after the eighth bounce. References Problem VI Brickhill 1983 The Darn Busters (London: Pan Books) P 272 km2. Walker J 1985 Roundabout, The Physics of Rotarion in the Eueryday World (San Francisco: Freeman) pp 8-10

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