Bouncing steel balls on water - University of Cambridgehemh1/dambusters/calcs/ricochet...Bouncing...

Bouncing steel balls on water

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2007 Phys Educ 42 466

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SPEC I A L F E A T U RE W A T E R

wwwioporgjournalsphysed

Bouncing steel balls on waterMichael de Podesta

National Physical Laboratory Hampton Road Teddington TW11 0LW UK

E-mail michaeldepodestanplcouk

AbstractThe lsquoskimmingrsquo of stones on water is a subject of perennial fascination forchildren and adults alike In this article I describe the construction of anapparatus for safely and reproducibly demonstrating a similar phenomenonthe bouncing of 25 mm diameter steel balls from a water surface Thelsquobouncingrsquo is technically known as a ricochet and the article recounts the useof the effect in the Second World War in the lsquoDam Bustingrsquo air raids of 1943A number of calculations are made which allow an estimate of the projectilespeed and energy and the use of simple experimental techniques forvalidating these estimates is suggested The role of spin is discussed and aliterature review collates the available empirical and theoretical resultsconcerning the incident angle and speed for successful ricochet of spinningand non-spinning projectiles

IntroductionIn June 2004 I spent several weeks preparing for ademonstration in which I bounced 25 mm diametersteel ball bearings at a shallow angle across thesurface of a 20 m pond Over the course of theirflight the balls bounced three or four times beforestriking an end wall at considerable speed Thedemonstration was part of a local commemorationof the 60th anniversary of lsquoD-Dayrsquo which markedthe beginning of the end of the Second WorldWar in Europe The public who watched thisactivity in the High Street of Teddington seemedto enjoy it greatly and my purpose in describingthis activity here is threefold Firstly duringthe construction I learned a great deal and Ifeel that many teachers or students could beinspired by carrying out similar investigationsSecondly the phenomenon in which a dense objectcan bounce off a less dense fluid is one whichoften fascinates children lsquoskimmingrsquo stones and isrelevant to other physics such as how a meteoritecan lsquobouncersquo off the Earthrsquos atmosphere Andfinally readers above a certain age from the United

Kingdom may recognize this as a lsquobouncingbombrsquo apparatus The story of the lsquobouncingbombrsquo is in itself a tremendous story and it isworth pausing for just a moment to tell it

Background

One of the most famous exploits of the SecondWorld War in Britain at least was an air raidwhich employed a lsquobouncing bombrsquo to attackseveral dams Nowadays guidance systems permitmissiles to land within a metre or so of theirintended target In 1943 this was not possibleBombs were dropped from aircraft at high altitudeand only 50 of the bombs fell within 5 miles oftheir target If they fell within 500 m of their targetthey would be considered accurate To destroya dam a large bomb had to explode underwaterwithin a few centimetres of the wall of the damand there was no conventional way of achievingthis with a heavily guarded dam Inspired by theexperience of lsquoskimming stonesrsquo the aeronauticalengineer Barnes Wallis conceived of a 4 tonnebomb which would bounce along the surface of

466 P H Y S I C S E D U C A T I O N 42 (5) 0031-912007050466+12$3000 copy 2007 IOP Publishing Ltd


the lake behind the dam hit the wall of the damsink against the wall of the dam and explode at apredetermined depth

The exploit was carried out on the night of16 May 1943 and two dams were destroyed Theingenuity of the weapon and the bravery of the aircrew inspired a film The Dam Busters which keptimages of wartime heroism alive for a generationof post-war children of whom I am one

In order to construct such a device a teamlead by Wallis carried out extensive studies atthe ship-testing tanks at the National PhysicalLaboratory (NPL) where I now work Althoughthe lsquoship tanksrsquo no longer exist the collectivememory of the research lives on at the laboratoryand it was this which inspired me to see if I couldrecreate some of his earlier experiments Factorsthat Wallis investigated would have included theeffect of launch speed launch height launch angleand spin The studies would have shown how theoptimal values for these parameters varied with thespeed size and density of the projectile allowingthe experimental results with models to be scaledup to the real thing

The apparatusEven though the phenomenon of lsquoskimmingstonesrsquo is well known it still seems surprisingthat dense objects can bounce off water Thefundamental explanation of this phenomenon isdiscussed in the section lsquoThe literaturersquo but at itssimplest objects will bounce off a water surface ifthey are a simple shape which will not lsquotumblersquo

after the first bounce and are travelling quicklynearly parallel to the surface

I began my research by firing a marble froma catapult out over the River Thames and theseexperiments indicated that it would not be hard toget spheres to bounce I considered several optionsfor reproducibly firing a 25 mm diameter spherebut in the end settled on an lsquoair-poweredrsquo gun

The lsquogunrsquo

The gun (figures 2 and 3) was constructedfrom standard plumbing fixtures connected to apiece of brass pipe with an internal diameterslightly larger than 25 mm The gun hadseveral important features designed to ensuresafety during public demonstration A pressure-limiting valve prevented the pressure rising morethan 3 bar above atmospheric pressure and thevalve could be key-locked in the open positionto prevent pressurization by unauthorized personsIn use the barrel pointed downwards and so thesteel ball was held in place during pressurizationby a magnet taped to the barrel The system waspressurized using a standard car-tyre foot pumpconnected to the system through a modified end-cap

The lsquoship tankrsquo

I constructed a preliminary lsquoship tankrsquo in my backgarden using a combination of light (62 kg) andheavy (132 kg) building blocks each 045 m times01 m times 021 m with polythene laid over them

September 2007 P H Y S I C S E D U C A T I O N 467

M de Podesta

This confirmed the feasibility of the project butonly permitted a single bounce so for the publicdemonstration the tank was lengthened to 20 mto permit multiple bounces to be more easilyobserved (figures 4 and 5)

Public demonstration and safetyGetting the balls to bounce reproducibly

It was possible to make the balls bounce over awide range of speed and launch angles Howeverfor a public demonstration I settled on a single setof standard firing conditions (figure 6)

Firing pressures were typically in the rangefrom 10 to 15 bar above atmospheric pressurecorresponding to a maximum projectile energy inthe range 60ndash70 J and maximum projectile speedsin the range 40ndash50 m sminus1 I conducted extensivetests outside this range to ensure that there wasno possibility of the ball bouncing outside of the

Figure 3 Photograph of the gun in place at the end ofthe lsquoship tankrsquo The air storage tubes (22 mm outsidediameter copper tube) are each around 11 m long andthe 25 mm bore barrel is approximately 16 m long

enclosure Lower pressures consistently failedto produce any bounces with the ball sinkingdirectly underwater Pressures greater than 15 bartypically produced bounces but their increasedprojectile speed caused them to land rather closeto the back wall The pressure was increased to30 bar and the safety release valve was observedto operate and at this pressure the balls still landedsafely in the pond I created waves on the surfacein an effort to induce unusual bounce dynamicsbut waves always seemed always to cause a ballto sink immediately at the first bounce In pre-demonstration tests 60 shots were fired over a 1 hperiod three shots went straight underwater 12shots bounced once and then dived underwater 45shots displayed two or more bounces and hit theend wall with a resounding sound zero shots leftthe pool

Public demonstration

While building the apparatus in my gardenprovided a diversion for my children I wascomforted by the fact that no safety officer would

468 P H Y S I C S E D U C A T I O N September 2007


Figure 4 The arrangement of bricks for the public demonstration pond

allow such an apparatus to be demonstrated inpublic I was wrong The NPL safety officerthought this would make a fantastic demonstrationand so I found myself faced with the task ofworking with the safety officer to make sure thatthe demonstration could be carried out safelyThis was no minor consideration the kineticenergy of the balls is roughly equivalent toa golf ball being hit from a tee which is aserious hazard The safety case then restedon separating balls and people ie ensuring theballs could not leave the pond and that peoplecould not get into the pond I ensured this bya combination of specified operating conditionssafety barriers trained operators and a keyinterlock that prevented the gun being fired byanyone else

In this demonstration I had no choice in theposition of the ship tank and the public had accessto the entire length of the ship tank I placedbarriers along the side of the pond to keep thepublic a distance of 075 m from the edge ofthe pond (If I were to repeat this I would

definitely make sure that the gun was arranged topoint directly away from any audience members)Additionally the final 1 m of the pool was coveredwith weighted polycarbonate sheet to capture anystray bounces off the end wall of the pond

Happily the public demonstration passedwithout any incidents the trained operators alwaysfired the gun with the same operating parameterseach shot was announced with a warning klaxonthe gun was locked in an un-pressurizable statefor the short times it was left unattended andduring the day over 200 balls were fired down thepond and almost all behaved exactly as expectedExcept for two

At the end of the day a ball fired in thestandard conditions bounced in an extraordinarywaymdashbackwardsmdashand left the side of the poolAlthough the ball was travelling slowly when itleft the pool and no one was hurt I was stunnedthat after our extensive testing any such event waspossible We tried again and observed the samestrange behaviour and at that point we stopped thedemonstration


M de Podesta

The rogue ball

At the time I was perplexed by the behaviour ofthe rogue ball but the explanation of its behaviouroccurred to me a few days later it was causedby spin In normal operation I did not expect theball to spin significantly about any axis and so Icould not see how anything had changed duringthe day to make the 200th ball spin when the 199thhad not However as the demonstration day woreon I became rather casual about the reloading ofthe gun The procedure was to fire ten balls thencollect them from the pool dry them and fire themagain Late in the day I skipped the drying stepand placed the balls back in the gun while theywere still wet After a while this must have createda pool of water in the lower part of the tube whichwould have caused extra drag on the lower part ofthe ball and could have initiated a significant topspin

When I began the experiments I had thoughtthat creating a reproducibly spinning ball would beone of the main challenges and so it is somewhatironic that the introduction of spin should havecaused a halt to the demonstration The role ofspin is discussed in the section lsquoSpinrsquo

Calculations and measurements

There are many simple calculations that canbe made about this experiment that can allowstudents and teachers to appreciate more fullysome of the simple physics underlying gasexpansions and particle kinematics

Calculation 1 stored energy

The energy to actuate the device is the excesspressure of gas stored in the 22 mm diametercopper pipe The pipe has wall thickness 1 mmand so the pipe has an internal radius of 10 mmThe storage pipe consists of 2 times 11 m lengths and1 times 035 m length The volume is therefore

volume = length times πr 2

= (2 times 11 + 1 times 035) times π0012

= 80 times 10minus4 m3 (1)

ie just under 1 litre At a typical gauge pressureof 12 bar (12 times 105 Pa) the stored energy(pressuretimesvolume) is 12times105times80times10minus4 m3 =96 J The volume of the barrel is similar to the

volume of the storage tubes

volume of barrel = length times πr 2

= (165) times π001252


If the ball completely seals the barrel thenon firing the gas will have expanded by a factorof approximately two The expansion will beadiabatic ie so quick that the gas will cool asit expands The cooling reduces the pressureof gas below what it would otherwise be and alsquosimple calculationrsquo (below) indicates that for thisgun only approximately 60 of the stored energycould conceivably be utilized in the expansionie at 12 bar the maximum utilizable energy is96 J times 06 = 58 J

Calculation 2 available energy

During an adiabatic expansion the quantity PVγ isa constant K In this expression P is the absolutepressure (ie one atmosphere more than the gaugepressure) V is the volume of the gas and γ is theratio of the principal heat capacities of a gas Thework done on the gas during an expansion from V1

to V2 is given by

work = minusint V2

V1

P dV

= minusint V2

V1

K

V γdV (3)

Removing K from the integral and integrating wefind

work = minusKint V2

V1

V minusγ dV

= minus K

minusγ + 1

[V minusγ +1

]V2

V1 (4)

If we rewrite V2 as a factor f of V1 we find

work = minus K

minusγ + 1

[V minusγ +1] f V1

V1

= minus K V minusγ +11

minusγ + 1

[f minusγ +1

] f

1 (5)

If we remember the value of K then this equationsimplifies

work = minus[P1V γ

1

]V minusγ +1

1

minusγ + 1

[f minusγ +1

] f

1

= minus [P1V1]

minusγ + 1

[f minusγ +1 minus 1minusγ +1]

= minus[stored energy

]minusγ + 1

[f minusγ +1 minus 1minusγ +1

] (6)



Using γ = 14 (appropriate for air) and anexpansion factor f = 2 we find

work = minus[stored energy

]minus04

[2minus04 minus 1

]


]minus04

[2minus04 minus 1

]

= minus060 times [stored energy

] (7)

The minus sign indicates that in fact during theexpansion the gas does work on the environmentmost of which is transferred to the ball Howeveronly approximately 60 of the stored energycould conceivably be accessed in this expansion

Calculation 3 projectile speed

If all the accessible energy of the gas is convertedto kinetic energy of the ball bearing (mass m =0065 kg) the velocity leaving the barrel will be

v =radic

2E

m (8)

At 12 bar (accessible energy = 58 J) themaximum speed possible is 42 m sminus1 (100 mph)In practice we would expect that not all the gascompression energy would be converted to kineticenergy of the ball bearing There is friction inthe barrel and some of the air leaks around theside of the ball Also the rapidity with whichthe valve is opened makes a significant differenceto the speed of the ball Additionally projectilestravelling at such speeds suffer rapid decelerationfrom air resistance

Experiments can determine the exit velocity(technically known as the muzzle velocity)fairly accurately The technique described below(figure 7) requires no sophisticated apparatusbut nowadays a frame-by-frame analysis of adigital video will provide a superior and moreaccessible analysis that could be used by studentsand teachers

First make sure that the barrel is horizontaland then measure the distance d to the firstbounce Although the ball moves quickly theposition of the first bounce can be seen fairlyclearly for a few seconds after the ball has beenfired as the epicentre of a set of ripples

Assuming the ball falls under gravity andstarts with zero initial vertical component of

Figure 5 The ship tank as finally constructed Noticethe polycarbonate enclosure at the far end the klaxonand foot pump and the poppies in remembrance ofthose who died in the original action

velocity the ball hits the water after a time t =radic2hg and the downward speed at impact is

vy = gt = gradic

2hg

= radic2hg (9)

In a time t the ball travels distance d with uniformhorizontal velocity v So

vx = d

t= dradic

2hg

=radic

d2g

2h(10)

and from measurements of d and h it is possibleto determine the exit speed of the ball Fromthese two calculations we can also find the angleof impact as tanminus1[vyvx ]

angle = tanminus1[vyvx

]

= tanminus1

[radic2hg

radicd2g

2h

]

= tanminus1

[2h

d

]

asymp 2h

dif h d (11)


M de Podesta

Table 1 (a) Theoretical and experimental values of some key parameters compare modestly well with each other(b) Typical values for the parameters in the bouncing balls demonstration compared with those used for bouncingbombs

(a)

Theory ExperimentalPressure(bar gauge)

Storedenergy (J)

Availableenergy (J) KE (J) Speed (m sminus1) KE (J) Speed (m sminus1)

12 96 58 58 42 14 (24) 21 (50)15 120 72 72 47 27 (38) 29 (60)30 240 144 144 66 No data No data

(b)

Dam busters Demonstration Factor

Projectile Back-spinning cylinder Non-spinning sphere 63 000127 m diameter times 152 m length 0025 m diameter (in mass)mass 4100 kg mass 0065 kg

Launch height 60 feet 3 in 244(183 m) (0075 m)

Launch speed 250 mph 65 mph 4(111 m sminus1) (29 m sminus1)

Distance to first bounce 214 m 35 m 61

Angle of incidence 10 24 4

Experiments indicate that at 12 bar the muzzlevelocity is approximately 21 m sminus1 (54 mph)which corresponds to an energy of 14 J orapproximately 38 of available energy

If the barrel is pointed down this formula willoverestimate the time to the first bounce and un-derestimate the speed of the ball Also the viscousresistance of the air will cause a rapid decline inspeed of the ball after it has left the muzzle of thegun so we can consider this experimental estimateof the muzzle velocity as a definite lower bound forthe muzzle velocity of the ball

Table 1 shows (a) how the theoretical andexperimental values of some key parameterscompare with each other and (b) typical values forthe demonstration bouncing balls and the actualbouncing bombs

SpinOne of the most persistent questions asked bymembers of the public during the demonstrationconcerns the role of spin Indeed before carryingout any experiments I had been concerned abouthow I would impart a reproducible spin to theballs The implicit assumption that I had madewas that spinning the projectile was somehowassociated with the bouncing phenomenon This is

a natural enough assumption it is essential to spina stone when skimming it and Wallis arrangedfor the bouncing bombs to be spun about theircylindrical axis at 500 rpm However experimentsquickly showed that spin was simply notnecessary After puzzling and calculating aboutthis for several days understanding eventuallydawned My understanding was later confirmedby a literature review described in the section lsquoTheliteraturersquo

The stones used for lsquoskimmingrsquo are ideallyflat-bottomed stones in the shape of extremelyoblate ellipsoids These are spun about their shortaxis and bounce off their large flat surface In thiscase the role of spin is to stabilize the angle of theimpact of the stone from one bounce to the nextIf the stone is not spun then after the first bounceit is likely to tumble strike the water at a randomangle and sink at its next bounce In this sense thespinning amounts to gyroscopic stabilization of theattitude of the stone (figure 8)

In the bouncing bombs situation the spinningof the cylinder performed two functions one todo with the bouncing and the other to do with thebombs As with skimming stones the spinning ofthe bombs caused them to gyroscopically maintaintheir attitude relative to the water from one bounceto the next After the final construction of full-



scale bombs the key step in Wallisrsquos achievementof making a 4200 kg cylinder bounce was not thespin but his ability to persuade pilots to fly at250 miles per hour at a height of just 183 m abovethe water (rather than 80 m in previous trials)This was done in order to stop the bomb casingdisintegrating when it hit the water and it alsoreduced the angle of incidence on the water toaround 10 But although the spin was not thekey parameter in causing the bounces it did haveanother effect When the forward motion of thebombs was stopped (by for example a dam wall)the spinning caused the bomb to roll down the wallsurface (figure 1) increasing the effectiveness ofexplosive which was triggered at a predetermineddepth

For the steel balls in this demonstration nomatter whether the balls spin or not the spheresstill hit the water with the same attitude because

the spheres are homogeneous and lacking in anyasymmetry Thus even if the balls tumble or beginto spin after the first collision they still hit thewater in the same way So for a perfect sphereno gyroscopic stabilization of attitude is required

The final question concerning spin is why (ifspin is not important) the flight of the balls wasso affected when I did induce spin on them HereI have to confess that I have over-simplified thediscussion above It is well known that in manyball sports spin plays a minor but significantrole in ball dynamics and the dynamics of thebouncing balls is no different And as in sporton occasion spin can play an extremely importantrole

The literatureThis experiment was embarked upon lightlywas performed quickly and it was not until


M de Podesta

writing this paper that I consulted the scientificliterature In additional to the semi-technical [1]and educational publications [2] I quickly learnedfrom the sparse and sophisticated literature on thistopic

Ricochet

Johnson and Reid [3] point out that the correctterm for this phenomenon is not bouncing butricochet This describes the case in which thewater surface is lsquoploughedrsquo by the sphere for ashort distance which transiently leaves a lsquofurrowrsquobehind it and piles water up in front of it (figure 9)

Johnson and Reid also point out that thephenomenon of ricochet has been well known fora very long time They include a table from NavalGunnery [4] published in 1855 which describesextensive tests by HMS Excellent including onein which a hollow cannonball (20 cm in diameter

and weighing 25 kg) ricocheted 32 times over arange of 25 km The Oxford English Dictionarycontains references to the word ricochet in thiscontext dating back to 1769 and thus the practicecertainly predates Admiral Nelson to whom itsinvention has been ascribed

Historically the empirical relationship of thelargest angle (the critical angle) θC at which aricochet can occur has been known to be given by

θC asymp 18radic

σ(12)

where σ is the ratio of the density of the materialof the ball to the density of the liquid in this casewater For solid steel balls σ takes the valueof roughly 8 yielding a critical angle of roughlyθC asymp 64 For the bouncing bombs (table 1)σ took the value 217 yielding a critical angleof θC asymp 122 just larger than the 10 angle ofincidence



Theory

From a theoretical point of view the ricochetproblem consists of trying to estimate the forcesexerted on the sphere as it strikes the watersurface and then lsquoploughs its furrowrsquo throughthe water (figure 9) Summing all the forces iscomplex and even then estimating the pressureforces and contact area between the ball and thewater depends on several assumptions Despitethis early works by Birkhoff [5] modifiedby Johnson and Reid [3] and Hutchings [6]were all able to yield an expression similarto equation (12) More recently Miloh andShukron [7] have developed an approach whichallows the calculation of previously unpredictedquantities of interest the variation of the criticalangle with projectile velocity the minimumvelocity required to ricochet and the dependenceof the critical angle on spin

One feature which all the theories share is thatrather than calculate directly in terms of the spherespeed v its radius a and the acceleration dueto gravity g they calculate the problem in termsof a dimensionless parameter called the Froudenumber F given by

F asymp vradicga

(13)

Use of the Froude number allows results obtainedwith spheres of one diameter to be simply scaledto spheres with different diameters if the sphere

size is doubled its speed must be increased by afactor

radic2 in order to maintain equivalent ricochet

characteristicsMiloh and Shukron do not produce a specific

formula for the dependence of the critical angleθC on speed but they do show that equation (12)is essentially the limiting case for a large Froudenumber (ie high speed) Miloh and Shukroncalculate that the Froude number must fall belowabout 50 before θC is significantly reduced For125 mm radius spheres travelling at 30 m sminus1the Froude number is around 85 so equation (12)should apply to these bouncing steel balls

Miloh and Shukron calculate that the mini-mum velocity for a ricochet of any kind at essen-tially glancing incidence is given by

vMIN asymp radicga

[20

π

radicσ + 25

](14)

which for these bouncing steel balls evaluates toa plausible 7 m sminus1 Finally Miloh and Shukrondevelop an expression for the dependence of θC onspin albeit for a cylinder rather than a sphere andfind

θC asymp 187radic

σ

radic1 + 16aω

v(15)

where the rotation rate ω (measured in rad sminus1) isdefined as positive for a sphere with back spinEquation (15) is clearly similar to the empiricallyderived equation (12) but it includes an extra


M de Podesta

factor which reduces to unity in the limit ofextremely fast projectiles but which can becomelarge at lower speeds and high spin rates

For the bouncing bombs which were back-spun at a rate of 500 rpm (asymp52 rad sminus1) the criticalangle would have been enhanced from roughly122 to approximately 15 Given that the actualimpact angle was around 10 this would haveincreased the chances of a successful first bouncebut it seems the bombs would have bounced (atleast once) whether they had been spun or not

My final discovery in the literature wasthe most personally astounding Schlien [8]reports that when extreme spin (gt20 000 rad sminus1)was accidentally introduced in their ricochetexperiments they observed that the critical anglefor a ricochet could exceed 45 and in one case(recorded on video) the sphere left the surfacetravelling backwards ie back towards the gunThis was exactly the kind of anomalous behaviourthat I had observed during my demonstrationFrom a safety point of view this confirms that(as far as public demonstrations are concerned)uncontrolled spinning of the projectiles is to beavoided

But how can steel bounce off water

So given that spin does not somehow lsquomagicallyrsquocause bouncing we now need to understand thebasic phenomenon we observed the bouncingof steel balls off water The physics of thisphenomenon is known as fluid dynamics and isas complicated as physics can get However one

can gain some understanding by considering themicroscopic arrangement of water molecules inthe liquid state

In a liquid the molecules are arranged in amanner similar to that in a solid Indeed ifone could take a snapshot of a liquid structure(figure 10) one would be unable to distinguishit from a disordered solid Each moleculein the liquid is trapped by its neighbours andvibrates within its available space typically every01 ps (10minus13 s) However after typically1000 oscillations the molecules surrounding aparticular molecule (let us call it lsquoStephaniersquo)rearrange themselves and allow Stephanie to hopto a new space Thus naturally the liquid canchange its structure Stephanie (who is typical ofthe molecules in the liquid) can move roughly onemolecular spacing (100 pm) in a time of 1000 times01 ps ie 100 ps In other words molecules ina liquid can naturally rearrange their positions ifthey can move at a speed of slower than 100 pmper 100 ps ie 1 m sminus1 This fact is very familiarto us If we try walking through a swimming pool(ie rearranging the positions of water moleculesaround our legs) it is easy if we walk slowly butrunning is almost impossible

Similarly when steel balls (or stones orbombs) hit the water at a speed much faster than(say) 10 m sminus1 then the liquid is unable to easilyflow out of the way and it resists the passage of thesphere strongly giving rise to a transient reactionforce which has an upward component and a lsquodragrsquocomponent If this upward force is sufficient to



lift the sphere then it can ricochet If the ball ismoving too slowly horizontally then the reactionforce is never sufficiently large and the spheresinks

Acknowledgments

A great many people helped with the originalpublic demonstration and I should particularlylike to thank Stephanie Bell Maxwell Christianand Charles de Podesta Jacquie Elkin JohnMakepeace Janine Avison Stephan SchottlGavin Sutton Angela Dawson David Griffithsand Fulwell Fire Brigade Additionally I shouldlike to thank NPL and Fiona Auty for the resourcesand flexibility which made the event possibleFinally I am grateful to Fiona Auty and StephanieBell for carefully reading the manuscript

Received 19 June 2007doi1010880031-9120425003

References[1] Johnson W 1998 The ricochet of spinning and

non-spinning spherical projectiles mainly fromwater Part II an outline of theory and warlikeapplications Int J Impact Eng 21 25ndash34

[2] Stinner A 1989 Physics and the bouncing bombPhys Educ 21 260ndash7

[3] Johnson W and Reid S R 1976 Ricochet of spheresoff water J Mech Eng Sci 17 71ndash81

[4] Douglas H 1855 Naval Gunnery 4th edn[5] Birkhoff G Birkhoff G D Bleick W E

Handler E H Murnaghan F D and Smith T L1944 Ricochet off water AMP Memo 424 M

[6] Hutchings I M 1976 The ricochet of spheres andcylinders from the surface of water Int J MechSci 18 243ndash7

[7] Miloh T and Shukron Y 1976 Ricochet off water ofspherical projectiles J Ship Res 35 91ndash100

[8] Shlien D J 1994 Unexpected ricochet of spheres offwater Exp Fluids 17 267ndash72

Michael de Podesta gained a PhD inlow-temperature condensed matterphysics from Sussex University and wasa lecturer at Birkbeck College andUniversity College until 2000 Since thenhe has worked on a variety ofthermometry projects at the NationalPhysical Laboratory Currently he isworking to redetermine the Boltzmannconstant He is an NPL ScienceAmbassador and runs the popularlsquoProtons for Breakfastrsquo introductorycourse for people interested in scienceand its impact on society


Acknowledgments

References

SPEC I A L F E A T U RE W A T E R

wwwioporgjournalsphysed

Bouncing steel balls on waterMichael de Podesta

National Physical Laboratory Hampton Road Teddington TW11 0LW UK

E-mail michaeldepodestanplcouk

AbstractThe lsquoskimmingrsquo of stones on water is a subject of perennial fascination forchildren and adults alike In this article I describe the construction of anapparatus for safely and reproducibly demonstrating a similar phenomenonthe bouncing of 25 mm diameter steel balls from a water surface Thelsquobouncingrsquo is technically known as a ricochet and the article recounts the useof the effect in the Second World War in the lsquoDam Bustingrsquo air raids of 1943A number of calculations are made which allow an estimate of the projectilespeed and energy and the use of simple experimental techniques forvalidating these estimates is suggested The role of spin is discussed and aliterature review collates the available empirical and theoretical resultsconcerning the incident angle and speed for successful ricochet of spinningand non-spinning projectiles

IntroductionIn June 2004 I spent several weeks preparing for ademonstration in which I bounced 25 mm diametersteel ball bearings at a shallow angle across thesurface of a 20 m pond Over the course of theirflight the balls bounced three or four times beforestriking an end wall at considerable speed Thedemonstration was part of a local commemorationof the 60th anniversary of lsquoD-Dayrsquo which markedthe beginning of the end of the Second WorldWar in Europe The public who watched thisactivity in the High Street of Teddington seemedto enjoy it greatly and my purpose in describingthis activity here is threefold Firstly duringthe construction I learned a great deal and Ifeel that many teachers or students could beinspired by carrying out similar investigationsSecondly the phenomenon in which a dense objectcan bounce off a less dense fluid is one whichoften fascinates children lsquoskimmingrsquo stones and isrelevant to other physics such as how a meteoritecan lsquobouncersquo off the Earthrsquos atmosphere Andfinally readers above a certain age from the United

Kingdom may recognize this as a lsquobouncingbombrsquo apparatus The story of the lsquobouncingbombrsquo is in itself a tremendous story and it isworth pausing for just a moment to tell it

Background

One of the most famous exploits of the SecondWorld War in Britain at least was an air raidwhich employed a lsquobouncing bombrsquo to attackseveral dams Nowadays guidance systems permitmissiles to land within a metre or so of theirintended target In 1943 this was not possibleBombs were dropped from aircraft at high altitudeand only 50 of the bombs fell within 5 miles oftheir target If they fell within 500 m of their targetthey would be considered accurate To destroya dam a large bomb had to explode underwaterwithin a few centimetres of the wall of the damand there was no conventional way of achievingthis with a heavily guarded dam Inspired by theexperience of lsquoskimming stonesrsquo the aeronauticalengineer Barnes Wallis conceived of a 4 tonnebomb which would bounce along the surface of

466 P H Y S I C S E D U C A T I O N 42 (5) 0031-912007050466+12$3000 copy 2007 IOP Publishing Ltd








The lsquogunrsquo





M de Podesta


















M de Podesta

The rogue ball













= (165) times π001252





to V2 is given by

work = minusint V2

V1

P dV

= minusint V2

V1

K

V γdV (3)


work = minusKint V2

V1

V minusγ dV

= minus K

minusγ + 1

[V minusγ +1

]V2

V1 (4)


work = minus K

minusγ + 1

[V minusγ +1] f V1

V1


minusγ + 1

[f minusγ +1

] f

1 (5)


work = minus[P1V γ

1

]V minusγ +1

1

minusγ + 1

[f minusγ +1

] f

1

= minus [P1V1]

minusγ + 1



]minusγ + 1


] (6)





]minus04

[2minus04 minus 1

]


]minus04

[2minus04 minus 1

]


] (7)




v =radic

2E

m (8)







vy = gt = gradic

2hg

= radic2hg (9)


vx = d

t= dradic

2hg

=radic

d2g

2h(10)



]

= tanminus1

[radic2hg

radicd2g

2h

]

= tanminus1

[2h

d

]

asymp 2h

dif h d (11)


M de Podesta


(a)


Storedenergy (J)


12 96 58 58 42 14 (24) 21 (50)15 120 72 72 47 27 (38) 29 (60)30 240 144 144 66 No data No data

(b)






















M de Podesta


Ricochet





θC asymp 18radic

σ(12)




Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References








The lsquogunrsquo





M de Podesta


















M de Podesta

The rogue ball













= (165) times π001252





to V2 is given by

work = minusint V2

V1

P dV

= minusint V2

V1

K

V γdV (3)


work = minusKint V2

V1

V minusγ dV

= minus K

minusγ + 1

[V minusγ +1

]V2

V1 (4)


work = minus K

minusγ + 1

[V minusγ +1] f V1

V1


minusγ + 1

[f minusγ +1

] f

1 (5)


work = minus[P1V γ

1

]V minusγ +1

1

minusγ + 1

[f minusγ +1

] f

1

= minus [P1V1]

minusγ + 1



]minusγ + 1


] (6)





]minus04

[2minus04 minus 1

]


]minus04

[2minus04 minus 1

]


] (7)




v =radic

2E

m (8)







vy = gt = gradic

2hg

= radic2hg (9)


vx = d

t= dradic

2hg

=radic

d2g

2h(10)



]

= tanminus1

[radic2hg

radicd2g

2h

]

= tanminus1

[2h

d

]

asymp 2h

dif h d (11)


M de Podesta


(a)


Storedenergy (J)


12 96 58 58 42 14 (24) 21 (50)15 120 72 72 47 27 (38) 29 (60)30 240 144 144 66 No data No data

(b)






















M de Podesta


Ricochet





θC asymp 18radic

σ(12)




Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References

M de Podesta


















M de Podesta

The rogue ball













= (165) times π001252





to V2 is given by

work = minusint V2

V1

P dV

= minusint V2

V1

K

V γdV (3)


work = minusKint V2

V1

V minusγ dV

= minus K

minusγ + 1

[V minusγ +1

]V2

V1 (4)


work = minus K

minusγ + 1

[V minusγ +1] f V1

V1


minusγ + 1

[f minusγ +1

] f

1 (5)


work = minus[P1V γ

1

]V minusγ +1

1

minusγ + 1

[f minusγ +1

] f

1

= minus [P1V1]

minusγ + 1



]minusγ + 1


] (6)





]minus04

[2minus04 minus 1

]


]minus04

[2minus04 minus 1

]


] (7)




v =radic

2E

m (8)







vy = gt = gradic

2hg

= radic2hg (9)


vx = d

t= dradic

2hg

=radic

d2g

2h(10)



]

= tanminus1

[radic2hg

radicd2g

2h

]

= tanminus1

[2h

d

]

asymp 2h

dif h d (11)


M de Podesta


(a)


Storedenergy (J)


12 96 58 58 42 14 (24) 21 (50)15 120 72 72 47 27 (38) 29 (60)30 240 144 144 66 No data No data

(b)






















M de Podesta


Ricochet





θC asymp 18radic

σ(12)




Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References









M de Podesta

The rogue ball













= (165) times π001252





to V2 is given by

work = minusint V2

V1

P dV

= minusint V2

V1

K

V γdV (3)


work = minusKint V2

V1

V minusγ dV

= minus K

minusγ + 1

[V minusγ +1

]V2

V1 (4)


work = minus K

minusγ + 1

[V minusγ +1] f V1

V1


minusγ + 1

[f minusγ +1

] f

1 (5)


work = minus[P1V γ

1

]V minusγ +1

1

minusγ + 1

[f minusγ +1

] f

1

= minus [P1V1]

minusγ + 1



]minusγ + 1


] (6)





]minus04

[2minus04 minus 1

]


]minus04

[2minus04 minus 1

]


] (7)




v =radic

2E

m (8)







vy = gt = gradic

2hg

= radic2hg (9)


vx = d

t= dradic

2hg

=radic

d2g

2h(10)



]

= tanminus1

[radic2hg

radicd2g

2h

]

= tanminus1

[2h

d

]

asymp 2h

dif h d (11)


M de Podesta


(a)


Storedenergy (J)


12 96 58 58 42 14 (24) 21 (50)15 120 72 72 47 27 (38) 29 (60)30 240 144 144 66 No data No data

(b)






















M de Podesta


Ricochet





θC asymp 18radic

σ(12)




Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References




]minus04

[2minus04 minus 1

]


]minus04

[2minus04 minus 1

]


] (7)




v =radic

2E

m (8)







vy = gt = gradic

2hg

= radic2hg (9)


vx = d

t= dradic

2hg

=radic

d2g

2h(10)



]

= tanminus1

[radic2hg

radicd2g

2h

]

= tanminus1

[2h

d

]

asymp 2h

dif h d (11)


M de Podesta


(a)


Storedenergy (J)


12 96 58 58 42 14 (24) 21 (50)15 120 72 72 47 27 (38) 29 (60)30 240 144 144 66 No data No data

(b)






















M de Podesta


Ricochet





θC asymp 18radic

σ(12)




Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References

M de Podesta


(a)


Storedenergy (J)


12 96 58 58 42 14 (24) 21 (50)15 120 72 72 47 27 (38) 29 (60)30 240 144 144 66 No data No data

(b)






















M de Podesta


Ricochet





θC asymp 18radic

σ(12)




Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References








M de Podesta


Ricochet





θC asymp 18radic

σ(12)




Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References


Theory



F asymp vradicga

(13)







vMIN asymp radicga

[20

π

radicσ + 25

](14)


θC asymp 187radic

σ

radic1 + 16aω

v(15)



M de Podesta












Acknowledgments














Acknowledgments

References

M de Podesta












Acknowledgments














Acknowledgments

References



Acknowledgments














Acknowledgments

References

Bouncing steel balls on water - University of Cambridgehemh1/dambusters/calcs/ricochet...Bouncing...

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