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IOPPUBLISHING EUROPEANJOURNALOFPHYSICS Eur.J.Phys.30(2009)453–457 doi:10.1088/0143-0807/30/3/003 Internal ballistics of a pneumatic potato cannon Carl E Mungan PhysicsDepartment,USNavalAcademy,Annapolis,MD21402-5002,USA E-mail:[email protected] Received29December2008,infinalform29January2009 Published9March2009 Onlineatstacks.iop.org/EJP/30/453 Abstract Basiclawsofthermodynamicsandmechanicsareusedtoanalyseanairgun. Suchdevicesareoftenemployedinoutdoorphysicsdemonstrationstolaunch potatoesusingcompressedgasthatishereassumedtoexpandreversiblyand adiabatically. Reasonableagreementisfoundwithreportedmuzzlespeedsfor suchhomebuiltcannons. Thetreatmentisaccessibletoundergraduatestudents whohavetakencalculus-basedintroductoryphysics. Potatocannonsareapopularconstructionprojectforphysicsdemonstrationsandsciencefairs [1]. They can be powered in three different ways: by a compressed gas (which is the type analysedinthepresentpaper),byanexplosivepropellant[2]orbyasuddenvacuumbreaking [3, 4]. In the first, pneumatic case, the projectile can be modelled as a piston accelerating downtheboreofahorizontalcylinderundertheactionofpressurizedgas, takentobeideal forsimplicity. Threeequationsareusedfortheanalysis: Newton’ssecondlaw,theidealgas lawandthefirstlawofthermodynamics. Additionalsimplifyingapproximationsarethatthe piston slides frictionlessly and no gas leaks around its edges (as should be appropriate for a thick, wet potato slice that is forcibly fit to the bore); atmospheric pressure is negligible compared to the gas pressure (while the piston is in the bore); the gas expansion occurs quasistatically(whichwillbevalidprovidedthepiston’sspeedissmallcomparedtothespeed ofsound)andadiabatically(i.e.,withoutheatleakagetothesurroundings)[5]. Someofthese assumptionscanbeliftedbyperforminganumericalratherthanananalyticanalysis[6],and such extensions could become attractive classroom projects with different possible levels of sophistication. Theidealgaslawfornmolesofgasatpressurep,volumeVandtemperatureTis pV =nRT (1) whereR = 8.314 Jmol−1 K−1 istheuniversalgasconstant. Theforceonapistonofcross- sectionalareaAisF =pA,assketchedinfigure1. Solvingforpandsubstitutingitintothe left-handsideofequation(1)alongwiththevolumeofgas,V = Ax wherexisthedistance fromthesealedendofthecylindertothepiston,leadsto max =nRT, (2) 0143-0807/09/030453+05$30.00 (cid:2)c 2009IOPPublishingLtd PrintedintheUK 453 Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED JAN 2009 2. REPORT TYPE 00-00-2009 to 00-00-2009 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Internal ballistics of a pneumatic potato cannon 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION US Naval Academy,Physics Department,Annapolis,MD,21402-5002 REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT Basic laws of thermodynamics and mechanics are used to analyse an air gun. Such devices are often employed in outdoor physics demonstrations to launch potatoes using compressed gas that is here assumed to expand reversibly and adiabatically. Reasonable agreement is found with reported muzzle speeds for such homebuilt cannons. The treatment is accessible to undergraduate students who have taken calculus-based introductory physics. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE Same as 5 unclassified unclassified unclassified Report (SAR) Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 454 CEMungan p A m F x Figure1.Sketchoftheprojectilemovingdowntheboreofthecannon. afterreplacingFbymawheremisthemassandaistheaccelerationofthepiston. Theinitial conditions are that the piston is held fixed in place (say by a pin) at position x (so that its 0 initial speed is υ = 0) with the gas at temperature T . Therefore the initial acceleration of 0 0 thepistonafterthepinispulledoutis nRT a = 0 (3) 0 mx 0 fromequation(2). Nowaccordingtothefirstlawofthermodynamics,intheabsenceofheatexchangewiththe surroundings,theinternalenergyUofthegasdecreasesasitdoesworkonthepiston. Noting thatU = ncT foranidealgas, wherecisthemolarspecificheatatconstantvolume(equal forexampleto3R/2foramonatomicandto5R/2foradiatomicgasnearroomtemperature), andthattheinfinitesimalworkdoneonthepistonisF dx =pdV =(nRT/x)dx,thebalance betweentheratesofenergylostandworkdoneimplies dT nRT nc =− υ. (4) dt x ReplacingnRT bymax fromequation(2)onbothsidesofequation(4)gives c d dx (ax)=−a . (5) Rdt dt Thisequationcanbeseparatedandintegratedtoget a =kx−(R/c+1). (6) Fittheconstantofintegrationktoequation(3),sothat nRT k = 0xR/c. (7) m 0 Next,writinga =υdυ/dx,onecanagainseparateandintegrateequation(6)tofind (cid:2) (cid:3) (cid:4) (cid:5) 2ncT x R/c υ2 = 0 1− 0 (8) m x using the initial condition υ = 0. Take the positive square root because the piston always 0 movesawayfromthesealedendofthecylindershowninfigure1andseparateyetagainto obtaintheintegralresult, (cid:6) (cid:8) x dx 2ncT (cid:7) = 0 t. (9) x0 1−(x0/x)R/c m Internalballisticsofapneumaticpotatocannon 455 Makethechangeofvariablex =x csc2c/Rθ,wherecscθ ≡1/sinθ isthecosecantfunction, 0 toobtain (cid:8) (cid:6) 2ncT 2cx π/2 0 t = 0 csc2c/R+1θdθ where sinφ ≡(x /x)R/2c. (10) 0 m R φ Thisresultcanberewrittenintermsofdimensionlesspositionandtimevariables, (cid:9) x 2ncT x˜ ≡ and t˜≡ 0 t, (11) x mx2 0 0 and in terms of the integer N ≡ 2c/R (equal to the number of degrees of freedom per gas molecule)as (cid:6) π/2 t˜=N cscN+1θdθ where φ ≡csc−1x˜1/N. (12) φ A recursive analytic solution for the indefinite integral exists. But rather than treating the general situation, consider two cases of practical interest: a monatomic and a diatomic gas. Forthemonatomiccase,therequiredintegralis (cid:6) −cotθ csc4θdθ = (2+csc2θ). (13) 3 Substitutingthelimitsofintegration,equation(12)remarkablysimplifiesto (cid:7) t˜= x˜2+3x˜4/3−4, (14) which is a solution for the time it takes the projectile to move any given distance down the bore. Recognizing the argument of the square root as a cubic polynomial in x˜2/3, equation(14)canbeinvertedtoobtain (cid:12) (cid:12) (cid:10)(cid:11) (cid:13) (cid:11) (cid:13) (cid:14) x˜ = 1+ 1t˜2+ t˜2+ 1t˜4 1/3+ 1+ 1t˜2− t˜2+ 1t˜4 1/3−1 3/2. (15) 2 4 2 4 Inthediatomiccase,theindefiniteintegralinequation(12)is (cid:6) −cotθ csc6θdθ = (8+4csc2θ +3csc4θ) (16) 15 sothat (cid:7) t˜= 1 9x˜2+15x˜8/5+40x˜6/5−64 (17) 3 andtheradicandisaquinticpolynomialinx˜2/5. Equation(17)cannotbeanalyticallyinverted tofindthepositionasafunctionoftime. However,timeincreasesmonotonicallywithdistance, as expected intuitively, and hence one can always find the inverse graphically (cf figure 2). Byseparatingandintegratingequation(4),thetemperatureofthepropellantgascanbe writtenindimensionlessformas T T˜ ≡ =x˜−R/c (18) T 0 andthereforesincepx ∝T fromequation(1),thedimensionlesspressureis p p˜ ≡ =x˜−(R/c+1), (19) p 0 which can alternatively be written in the familiar adiabatic form pVγ = constant, where γ = (c+R)/c istheratioofthespecificheatsatconstantpressureandatconstantvolume. Note that p˜ = a/a since ma = pA and hence the pressure is directly proportional to the 0 accelerationoftheprojectile. Equation(19)isausefulconversionformulabetweenpressure 456 CEMungan 0.8 speed 0.6 0.4 time/20 0.2 acceleration 0 1 3 5 7 9 dimensionless distance Figure2. Dimensionlessspeed,accelerationandtimeasafunctionofdistanceforaprojectile propelledalongacylinderbyadiatomicgas.Thetimehasbeendividedby20tokeepitonscale. (Thisfigureisincolouronlyintheelectronicversion) and distance (and thus speed according to the next equation) since in practice pneumatic cannons are pressurized to some desired (and hopefully safe) initial value p . For example, 0 supposetheinitialpressureisp =10atm(i.e.,1MPa). Thenequation(19)impliesthatthe 0 lowestusefulpressureofp = 1 atmisattainedwhenx˜ = 5.2foradiatomicgassuchasdry air. Butequation(8)canberewrittenas (cid:7) υ =υ∞ 1−x˜−R/c (20) wherethelimitingmuzzlespeedis (cid:8) 2ncT υ∞ = 0. (21) m Substituting say n = 0.1 mol, c = 5R/2, T = 300 K, m = 100 g and x˜ = 5.2, these two 0 equations imply υ = 78 ms−1, comparable to reported values for homemade air cannons [7]. Ignoringairdrag,thatmuzzlespeedgivesanoptimalrangeofυ2/gwhichisalittleover 600m. Definingυ˜ ≡dx˜/dt˜=υ/υ∞anda˜ =d2x˜/dt˜2 =a/Na0,theprecedingresultscorrectly imply at t˜ = 0 that x˜ = 1, υ˜ = 0 and a˜ = R/2c. On the other hand, as t˜ → ∞ 0 0 0 one obtains x˜∞ → t˜, υ˜∞ = 1 and a˜∞ = 0. These initial and final values of the position, speedandaccelerationgivephysicalmeaningtothescalingconstantsinequation(11). The dimensionless time, speed and acceleration are plotted versus distance along the bore in figure2forthediatomiccase. Thekeyresultisthatthespeedoftheprojectilelevelsoffafterthevolumeofthegashas expandedtoafewtimesitsinitial, compressedvalueandsothereisnopointinmakingthe borelongerthanthat. (Infact,alongerborewouldbedisadvantageousbecauseoftheeffects of friction and atmospheric pressure.) To enhance the muzzle speed, one should maximize theproductoftheinitialpressureandvolumeofthepropellantgasanddecreasetheprojectile mass,accordingtoequation(21). Tocomparethetheoreticalresultsobtainedinthispapertoexperiment,onewouldneedto makemeasurementswhiletheprojectileisinternallytravellingdowntheboreofthecannon. Onepossibilityistomeasurethetemperatureorpressureofthepropellantgasforcomparisonto equations(18)and(19). Thatwouldrequireafastsensorbecausethepotatoleavesthebarrelin Internalballisticsofapneumaticpotatocannon 457 lessthan50msafterfiring[2]. Amoredirecttechniqueconsistsinmeasuringthespeedofthe projectileatseveralpointsalongtheboreforcomparisontoequation(20). Thesimplestway todothatwouldbetousephotogatesmountedalongthelengthofatransparentcannon,such asonemadeoutofanacrylictube[4]. Additionalusefulexperimentalmeasurementsinclude estimating the frictional loss between a potato and the bore, and the leakage of compressed gasaroundthepotato. Onthetheoreticalside,onecouldaccountfortheaccelerationofthe aircolumn behind theprojectile[3]or determine byhow much thegas pressure(andhence pistonspeed)increasesiftheexpansionoccursnotadiabaticallybutisothermally[8](asmight beappropriateifthecannonismadeoutofmetalwithalargethermalconductivity). ©USGovernment References [1] GurstelleW2001BackyardBallistics:BuildPotatoCannons,PaperMatchRockets,CincinnatiFireKites,Tennis BallMortars,andMoreDynamiteDevices(Chicago:ChicagoReviewPress) [2] CourtneyMandCourtneyA2007AcousticmeasurementofpotatocannonvelocityPhys.Teach.45496–7 [3] AyarsEandBuchholtzL2004AnalysisofthevacuumcannonAm.J.Phys.72961–3 [4] PetersonRW,PulfordBNandSteinKR2005Theping-pongcannon:acloserlookPhys.Teach.4322–5 [5] MunganCE2003IrreversibleadiabaticcompressionofanidealgasPhys.Teach.41450–3 [6] Severn J 1999 Use of spreadsheets for demonstrating the solutions of simple differential equations Phys. 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