{"text_format":"1","workbook":"xlsx","n_sheets":7,"sheets":["ReadMe","Air Resistance Model","Stationary Turret No Air Model","Moving Turret No Air Model","Raw Data","Parameters","4th Order Polynomial Solver"],"not_rendered":{"charts":6,"images":31,"macros":1,"external_links":0,"defined_names":5,"comments":0,"pivot_tables":0},"unopened":[],"bounds":{"document_bytes":16777216,"sheet_cells":100000,"rows_first":2000,"rows_last":500,"rows_between":2000,"halvings":0,"cell_chars":16384,"cell_head_chars":4096},"cells_cut":0,"cells_raw":0,"read":"whole","date_system":"1900"}

{"sheet":"ReadMe","index":0,"n_rows":26,"n_cols":3,"rows_kept":26,"cut":false,"hidden":false,"merged":[[12,0,12,1],[0,0,0,2],[17,0,17,1],[22,0,22,1]]}
["Revision HIstory"]
["Date","Who","Comments"]
["2026-03-22","Dean Banerjee","Initial Worksheet"]
["2026-04-15","Dean Banerjee","Updated Calculations for Gamma"]
[]
[]
[]
[]
[]
[]
[]
[]
["Stationary No Air Model"]
["Sheet","Stationary No Air Model"]
["Supporting Sheets","None"]
["Comments","This is the simplest case with no air resistance and simple closed-form solutions.  Calculations are easily visible.  It has the option to choose to solve for the angle or velocity.  Also it includes manual mode that simply plots the trajectory and does the calculations."]
[]
["Moving No Air Model"]
["Sheet","Stationary No Air Model"]
["Supporting Sheets","4th Order Polynomial Solver"]
["Comments","This accounts for the case of a moving vehicle.  The math is more involved for this as it requires the 4th Order Polynomial Solver sheet to calculate."]
[]
["Air Resistance Model"]
["Sheets","Air Resistance Model"]
["Supporting Sheets","Parameters, Raw Data"]
["Comments","This covers all 4 cases, but requires numerical methods and also excel macros to be enabled.  The approach is that it allows the user to calculate the trajectories and also tune coeficients to hit the target or match measured data.  The Raw Data sheet is for the case of matching or comparing to raw data.  The parameters has the information on the constants and the target.  Although the air constants are also on the main sheet, the parameters tab shows some more intermediate steps on how to calculate them.  This sheet covers all 4 cases of with/without air resistance and with/without motion."]

{"sheet":"Air Resistance Model","index":1,"n_rows":670,"n_cols":23,"rows_kept":670,"cut":false,"hidden":false,"merged":[[16,0,16,3],[26,0,26,3],[9,4,14,4],[6,4,8,4]]}
["Parameter","Symbol","Value","Unit","Options",null,null,"x","With Air Resistance","No Air Resistance","Measurement","Target",null,"x","With Air Resistance","No Air Resistance","Target","Target",null,null,null,null,"Drop Menu Options (Do not Delete!)"]
["Initial Velocity","v0",6.4,"m/s",1,null,null,0,0,0,null,null,null,0,null,null,null,null,null,null,null,null,"Simulate"]
["Launch Angle","q",72.400000000000006,"deg",true,null,null,6.1306538807920146E-3,1.0964655017099791E-2,1.0958339610665254E-2,null,null,null,6.1306538807920146E-3,null,null,null,null,null,null,null,null,"Calculate Velocity"]
["Acceleration Due to Gravity","g",9.8000000000000007,"m/s2",false,null,null,1.2253994406586985E-2,2.1897116744536614E-2,2.1871949494636021E-2,null,null,null,1.2253994406586985E-2,null,null,null,null,null,null,null,null,"Calculate Theta"]
["Target Distance","d",3,"m",false,null,null,1.8370045527466662E-2,3.2797399877621002E-2,3.2740985513877684E-2,null,null,null,1.8370045527466662E-2,null,null,null,null,null,null,null,null,"Calculate Phi"]
["Target Height","h",1.2,"m",true,null,null,2.4478831156482805E-2,4.3665519060428856E-2,4.356560283563525E-2,null,null,null,2.4478831156482805E-2,null,null,null,null,null,null,null,null,"Calculate W0"]
["Azimuth Angle","f",-30,"deg",true,null,null,3.0580375169770344E-2,5.4501488885897988E-2,5.4345955935056908E-2,null,null,null,3.0580375169770344E-2,null,null,null,null,null,null,null,null,"Calculate Tau"]
["Vehicle Heading","fm",30,"deg",null,null,null,3.6674701406660029E-2,6.5305323895924347E-2,6.5082198597804994E-2,null,null,null,3.6674701406660029E-2,null,null,null,null,null,null,null,null,"Calculate Alpha"]
["Vehicle Speed","vm",2,"m/s",null,null,null,4.2085840159248149E-2,7.4881767133618049E-2,7.4588618207735935E-2,null,null,null,4.2085840159248149E-2,null,null,null,null,null,null,null,null,"Calculate Beta"]
["Mass of Ball","m",0.23,"kg",true,null,null,4.8166597731925248E-2,8.5624942543733845E-2,8.5241958317453306E-2,null,null,null,4.8166597731925248E-2,null,null,null,null,null,null,null,null,"Calculate Gamma"]
["Rotational velocity","w / (2p)",15,"rev/s",null,null,null,5.4240206191668153E-2,9.6336024862487546E-2,9.5851628410884779E-2,null,null,null,5.4240206191668153E-2,null,null,null,null,null,null,null,null,"Calculate Theta and Phi"]
["Spin Decay Coefficient","t",10,"s",null,null,null,6.0306689235742957E-2,0.1070150284341866,0.10641777961694807,null,null,null,6.0306689235742957E-2]
["Quadratic Drag Term","a",{"$f":"Parameters!C18","v":5.775386124634966E-3},"N\u009e(s/m)2",null,null,null,6.6366070525155918E-2,0.11766196755256167,0.11694056238766184,null,null,null,6.6366070525155918E-2]
["Linear Drag Term","b",{"$f":"Parameters!C19","v":2.5446900494077325E-5},"N\u009e(s/m)",null,null,null,7.2418373684763365E-2,0.12827685646086059,0.12742012650068399,null,null,null,7.2418373684763365E-2]
["Magnus Force Drag Term","g",{"$f":"Parameters!C20","v":6.6639070668864987E-4},"N\u009e(s/m)",null,null,null,7.8463622303381372E-2,0.1388597093519422,0.13785662106183816,null,null,null,7.8463622303381372E-2]
[null,null,null,null,null,null,null,8.4501839933894707E-2,0.14941054036836954,0.14825019450762786,null,null,null,8.4501839933894707E-2]
["Parameters with No Air Resistance",null,null,null,null,null,null,9.0533050093365527E-2,0.1599293636025029,0.15860099460773877,null,null,null,9.0533050093365527E-2]
["Parameter","Formula","Value","Unit",null,null,null,9.655727626314152E-2,0.17041619309659251,0.16890916846752896,null,null,null,9.655727626314152E-2]
["Peak Time",null,0.62249186836724968,"s",null,null,null,0.10257454188896366,0.18087104284287073,0.17917486253050743,null,null,null,0.10257454188896366]
["Peak Height",null,1.8987310182984121,"m",null,null,null,0.1085848703810734,0.19129392678364407,0.18939822258080033,null,null,null,0.1085848703810734]
["Time to Target",null,0.88029338171899763,"s",null,null,null,0.11458828511431957,0.20168485881138479,0.19957939374560604,null,null,null,0.11458828511431957]
["Y Target Error",null,0.3730690788947994,"m",null,null,null,0.12058480942826467,0.21204385276882204,0.20971852049763801,null,null,null,0.12058480942826467]
["Z Target Error",null,2.8535948516393028E-2,"m",null,null,null,0.12657446662729085,0.22237092244903287,0.21981574665755596,null,null,null,0.12657446662729085]
["Time to Hit Ground",null,1.2449837367344994,"s",null,null,null,0.13255727998070521,0.23266608159553287,0.22987121539638558,null,null,null,0.13255727998070521]
["X Range",null,4.2428482228391315,"m",null,null,null,0.13853327272284499,0.2429293439023662,0.23988506923792685,null,null,null,0.13853327272284499]
[null,null,null,null,null,null,null,0.14450246805318204,0.25316072301419562,0.24985745006115034,null,null,null,0.14450246805318204]
["Parameters with Air Resistance",null,null,null,null,null,null,0.15046488913642689,0.26336023252639218,0.25978849910258245,null,null,null,0.15046488913642689]
["Parameter","Formula",null,"Unit",null,null,null,0.15642055910263247,0.2735278859851244,0.26967835695867909,null,null,null,0.15642055910263247]
["Peak Time",null,0.64819999999996125,"s",null,null,null,0.16236950104729722,0.28366369688744703,0.27952716358818763,null,null,null,0.16236950104729722]
["Peak Height",null,1.9491088588347567,"m",null,null,null,0.16831173803146796,0.29376767868138975,0.28933505831449857,null,null,null,0.16831173803146796]
["Time to Target",null,1.0569999999999162,"s",null,null,null,0.17424729308184206,0.30383984476604564,0.2991021798279847,null,null,null,0.17424729308184206]
["Y Target Error",null,-4.7998075766879644E-3,"m",null,null,null,0.18017618919086947,0.3138802084916586,0.30882866618832983,null,null,null,0.18017618919086947]
["Z Target Error",null,3.2137286917771263E-2,"m",null,null,null,0.18609844931685382,0.32388878315971159,0.31851465482684593,null,null,null,0.18609844931685382]
["Time to Hit Ground",null,1.3091999999998885,"s",null,null,null,0.1920140963840537,0.3338655820230137,0.32816028254877977,null,null,null,0.1920140963840537]
["Range",null,3.720406693660955,"m",null,null,null,0.19857930935307352,0.34491366223256303,0.33883047542031258,null,null,null,0.19857930935307352]
[null,null,null,null,null,null,null,0.20448107064789428,0.35482342214441792,0.34839134322021559,null,null,null,0.20448107064789428]
["Model Fitting Constant","Calculated from Simulator","n/a","n/a",null,null,null,0.21037628998721739,0.36470144717215719,0.3579122717410757,null,null,null,0.21037628998721739]
[null,null,null,null,null,null,null,0.21626499015638881,0.37454775036939697,0.36739339523830905,null,null,null,0.21626499015638881]
["Analog Derivations",null,null,null,null,null,null,0.22214719390721827,0.38436234474152109,0.37683484735288203,null,null,null,0.22214719390721827]
[null,null,null,"Vo",{"$f":"C2","v":6.4},null,null,0.22802292395807749,0.39414524324576639,0.38623676111355082,null,null,null,0.22802292395807749]
[null,null,null,"Theta",{"$f":"PI()/180*C3","v":1.2636183784438948},null,null,0.23454394788258154,0.40497797200619895,0.39663712085276748,null,null,null,0.23454394788258154]
[null,null,null,"Vox",{"$f":"E40*COS(E41)","v":1.9351673008028438},null,null,0.24040608568959718,0.41469399934633161,0.40595599887277722,null,null,null,0.24040608568959718]
[null,null,null,"Voy",{"$f":"E40*SIN(E41)","v":6.1004202738748612},null,null,0.24626182026324536,0.42437837082301211,0.41523574851522382,null,null,null,0.24626182026324536]
[null,null,null,null,null,null,null,0.25211117418545254,0.43403109919643118,0.42447650031529827,null,null,null,0.25211117418545254]
[null,null,null,"tp",{"$f":"E43/9.8","v":0.62249186468110829},null,null,0.25860300026587496,0.44471925612336682,0.43469842212113169,null,null,null,0.25860300026587496]
[null,null,null,"h",{"$f":"E45*E43-4.9*E45^2","v":1.8987309958113996},null,null,0.26443895793372729,0.454305224079114,0.44385727111237705,null,null,null,0.26443895793372729]
[null,null,null,"tt",{"$f":"3/E42","v":1.5502535614132114},null,null,0.27026860499477601,0.46385958832520807,0.45297752453751494,null,null,null,0.27026860499477601]
["Discrete Derivations with Air Resistance",null,null,"d",{"$f":"E47*E42","v":3},null,null,0.27609196389612306,0.47338236142621248,0.46205931047510168,null,null,null,0.27609196389612306]
[null,null,null,"yt",{"$f":"E43*E47-4.9*E47^2","v":-2.3189036572124326},null,null,0.28255501493450302,0.48392618430710804,0.47210522258906307,null,null,null,0.28255501493450302]
[null,null,null,null,null,null,null,0.2883651724008276,0.49338230603902272,0.48110621732818426,null,null,null,0.2883651724008276]
[null,null,null,null,null,null,null,0.2941691113351167,0.50280687541302105,0.4900691392697053,null,null,null,0.2941691113351167]
[null,null,null,null,null,null,null,0.30061066617195215,0.51324162835420872,0.49998344096130848,null,null,null,0.30061066617195215]
[null,null,null,null,null,null,null,0.30640155033267674,0.52259962776119484,0.5088663989424097,null,null,null,0.30640155033267674]
[null,null,null,null,null,null,null,0.31218628527674636,0.53192611314139127,0.5177116731491217,null,null,null,0.31218628527674636]
[null,null,null,null,null,null,null,0.31860658352112631,0.54225192864585658,0.52749570919830568,null,null,null,0.31860658352112631]
[null,null,null,null,null,null,null,0.32437840969674425,0.55151192450094633,0.53626183478615741,null,null,null,0.32437840969674425]
[null,null,null,null,null,null,null,0.33014415566156363,0.56074044420677782,0.54499066001900798,null,null,null,0.33014415566156363]
[null,null,null,null,null,null,null,0.33654343585261137,0.57095745320951885,0.55464575583434661,null,null,null,0.33654343585261137]
[null,null,null,null,null,null,null,0.34229641840353225,0.58011956287712552,0.56329623602524137,null,null,null,0.34229641840353225]
[null,null,null,null,null,null,null,0.34804338944270596,0.58925023382183661,0.57190979373867812,null,null,null,0.34804338944270596]
[null,null,null,null,null,null,null,0.35442188905897876,0.59935856569825252,0.58143725557162784,null,null,null,0.35442188905897876]
[null,null,null,null,null,null,null,0.36015624139394292,0.60842290514169595,0.58997326018522145,null,null,null,0.36015624139394292]
[null,null,null,null,null,null,null,0.36652077461861499,0.61845756471735647,0.59941484753208019,null,null,null,0.36652077461861499]
[null,null,null,null,null,null,null,0.37224260608082377,0.62745562541403954,0.60787382987405614,null,null,null,0.37224260608082377]
[null,null,null,null,null,null,null,0.37859328145270804,0.63741667106401168,0.61723012961697377,null,null,null,0.37859328145270804]
[null,null,null,null,null,null,null,0.38430268958099212,0.64634850533493138,0.62561261525662049,null,null,null,0.38430268958099212]
[null,null,null,null,null,null,null,0.39000624693238944,0.65524898733089521,0.63395904755218047,null,null,null,0.39000624693238944]
[null,null,null,null,null,null,null,0.39633669735699095,0.66510165468766713,0.64319071774297609,null,null,null,0.39633669735699095]
[null,null,null,null,null,null,null,0.40202797458613004,0.6739359865756086,0.65146141835617632,null,null,null,0.40202797458613004]
[null,null,null,null,null,null,null,0.40834483425983914,0.68371518234604323,0.66060922778366027,null,null,null,0.40834483425983914]
[null,null,null,null,null,null,null,0.4140239281889917,0.69248341538125335,0.66880470944184334,null,null,null,0.4140239281889917]
[null,null,null,null,null,null,null,0.42032730453256661,0.70218919627760601,0.6778692249507049,null,null,null,0.42032730453256661]
[null,null,null,null,null,null,null,0.42662362742589949,0.71185636038501832,0.68689012731438126,null,null,null,0.42662362742589949]
[null,null,null,null,null,null,null,0.43228431181370802,0.72052380354688172,0.69497177801782783,null,null,null,0.43228431181370802]
[null,null,null,null,null,null,null,0.43856731472295846,0.73011763842605104,0.70391024072284947,null,null,null,0.43856731472295846]
[null,null,null,null,null,null,null,0.44421605914075574,0.73871911031858883,0.71191794503980821,null,null,null,0.44421605914075574]
[null,null,null,null,null,null,null,0.45048584868750147,0.74823967148534454,0.72077452067832637,null,null,null,0.45048584868750147]
[null,null,null,null,null,null,null,0.45612274895357108,0.75677522186083079,0.72870877343105023,null,null,null,0.45612274895357108]
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[null,null,null,null,null,null,null,0.47424826453323149,0.78406642095685464,0.7540397370902655,null,null,null,0.47424826453323149]
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[null,null,null,null,null,null,null,0.48609254889355563,0.8017713442918597,0.7704427257482438,null,null,null,0.48609254889355563]
[null,null,null,null,null,null,null,0.49231659026868368,0.81103395535644462,0.77901487025991389,null,null,null,0.49231659026868368]
[null,null,null,null,null,null,null,0.49853391609954495,0.82025812385095964,0.78754513095635692,null,null,null,0.49853391609954495]
[null,null,null,null,null,null,null,0.50412379161037157,0.82852701926965822,0.79518667510348429,null,null,null,0.50412379161037157]
[null,null,null,null,null,null,null,0.51032843804531414,0.83767818739078637,0.80363775645678848,null,null,null,0.51032843804531414]
[null,null,null,null,null,null,null,0.5165264532082835,0.84679095545663274,0.81204737347311096,null,null,null,0.5165264532082835]
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[null,null,null,null,null,null,null,0.52828451659502429,0.86399947461112847,0.82791192353771437,null,null,null,0.52828451659502429]
[null,null,null,null,null,null,null,0.53446346475395912,0.87300096443818209,0.836202100581384,null,null,null,0.53446346475395912]
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[null,null,null,null,null,null,null,0.54618553152226834,0.88999816981441748,0.85184084430943996,null,null,null,0.54618553152226834]
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[null,null,null,null,null,null,null,0.55849934389385369,0.90774053152668366,0.86814418190358178,null,null,null,0.55849934389385369]
[null,null,null,null,null,null,null,0.56403218532953781,0.91567462106876529,0.87542793712900369,null,null,null,0.56403218532953781]
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[null,null,null,null,null,null,null,0.58243778022125303,0.94189772930995674,0.8994722467716223,null,null,null,0.58243778022125303]
[null,null,null,null,null,null,null,0.58856035044875743,0.95056227846444052,0.90740716101410623,null,null,null,0.58856035044875743]
[null,null,null,null,null,null,null,0.5940653049361202,0.9583276937952141,0.91451463355178475,null,null,null,0.5940653049361202]
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{"sheet":"Stationary Turret No Air Model","index":2,"n_rows":1007,"n_cols":12,"rows_kept":1007,"cut":false,"hidden":false,"merged":[[1,4,3,5],[7,0,7,4],[30,0,30,2],[39,0,39,3],[22,0,22,2],[46,4,46,5],[41,4,41,5],[42,4,42,5],[43,4,43,5],[44,4,44,5],[45,4,45,5]]}
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["Target Height","h",1.2,"m"]
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["Acceleration Due to Gravity","g",9.8000000000000007,"m/s2",null,null,null,null,"x","High Angle","Low Angle","Target"]
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["Calculated Values",null,null,null,null,null,null,null,{"$f":"I7+0.01","v":0.01},2.9676791083897425E-2,8.4876306848100623E-3,{"$f":"IF(ABS($I8-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I8-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I8-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I8-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I8-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Parameter","Symbol","High Value","Low Value","Unit",null,null,null,{"$f":"I8+0.01","v":0.02},5.9181831056865773E-2,1.694524377306621E-2,{"$f":"IF(ABS($I9-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I9-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I9-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I9-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I9-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Velocity to Target","V1,2",{"$f":"IF(F1=2,B26,V)","v":7.5},{"$f":"C10","v":7.5},"m/s",null,null,null,0.03,8.8515119918905039E-2,2.5372839264768448E-2,{"$f":"IF(ABS($I10-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I10-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I10-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I10-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I10-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Angle to Target","q1,2",{"$f":"IF(F1=1,B37,C3)","v":71.428094303997156},{"$f":"IF(F1=1,B38,C3)","v":40.373315182354659},"deg",null,null,null,0.04,0.11767665767001523,3.3770417159916771E-2,{"$f":"IF(ABS($I11-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I11-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I11-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I11-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I11-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Target Strike Angle","f1,2",{"$f":"B44","v":-124.69089493855965},-2.8799120997523797,"deg",null,null,null,0.05,0.14666644431019635,4.2137977458511187E-2,{"$f":"IF(ABS($I12-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I12-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I12-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I12-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I12-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Time to Target","t1,2",1.2559084842365162,0.52504457438103336,"s",null,null,null,6.0000000000000005E-2,0.17548447983944837,5.0475520160551686E-2,{"$f":"IF(ABS($I13-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I13-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I13-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I13-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I13-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Peak Time","tp1,2",0.72545257191597012,0.49573863789024503,"s",null,null,null,7.0000000000000007E-2,0.20413076425777132,5.8783045266038267E-2,{"$f":"IF(ABS($I14-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I14-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I14-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I14-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I14-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Peak Height","h1,2",2.5787790270875304,1.2042083057766499,"m/s",null,null,null,0.08,0.23260529756516518,6.706055277497093E-2,{"$f":"IF(ABS($I15-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I15-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I15-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I15-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I15-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
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["General Derivations",null,null,null,null,null,null,null,9.9999999999999992E-2,0.2890391108471656,8.3525515003174525E-2,{"$f":"IF(ABS($I17-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I17-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I17-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I17-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I17-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
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["Calculate Velocity if angle is known",null,null,null,null,null,null,null,0.16,0.45421852403086926,0.13219997937049138,{"$f":"IF(ABS($I23-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I23-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I23-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I23-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I23-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["Formula","Value","Unit",null,null,null,null,null,0.17,0.48114729733990141,0.14020732851043885,{"$f":"IF(ABS($I24-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I24-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I24-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I24-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I24-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
["q",{"$f":"C3*PI()/180","v":1.0471975511965976},"rad",null,null,null,null,null,0.18000000000000002,0.50790431953800452,0.14818466005383241,{"$f":"IF(ABS($I25-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I25-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I25-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I25-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I25-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
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["Calculate Angle if Velocity is known",null,null,null,null,null,null,null,0.21000000000000005,0.58714487946673932,0.17193654910468956,{"$f":"IF(ABS($I28-$C$5-Parameters!$C$21)>Parameters!$C$22,-100,IF($I28-$C$5-Parameters!$C$21<-Parameters!$C$23,Parameters!$C$24 + Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I28-$C$5-Parameters!$C$21) + Parameters!$C$23),IF($I28-$C$5-Parameters!$C$21<Parameters!$C$23,Parameters!$C$24,Parameters!$C$24 - Parameters!$C$25 / (Parameters!$C$23 - Parameters!$C$22) * (($I28-$C$5-Parameters!$C$21) - Parameters!$C$23))))","v":-100}]
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{"sheet":"Moving Turret No Air Model","index":3,"n_rows":1007,"n_cols":20,"rows_kept":1007,"cut":false,"hidden":false,"merged":[[66,0,66,2],[22,3,22,5],[23,3,23,5],[24,3,24,5],[18,3,18,5],[61,0,61,2],[1,4,6,5],[10,0,12,0],[13,0,13,5],[20,3,20,5],[21,3,21,5],[19,3,19,5],[14,2,16,5]]}
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["Launch Angle","q",70,"deg",null,null,null,null,null,"V0y",null,7.4144332763591061,4.9327615003811225,null,null,"v0y",{"$f":"L3","v":7.4144332763591061},4.9327615003811225]
["Launch Azimuth","f",-50,"deg",null,null,null,null,null,"V0z",null,0,0,null,null,"vox",0,0]
["Vehicle Speed","vm",2,"m/s"]
["Vehicle heading angle","fm",30,"deg",null,null,null,null,"th","tl","x","High Angle","Low Angle","Target",null,"x","High Angle","Low Angle","Target","Target"]
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{"sheet":"Raw Data","index":4,"n_rows":71,"n_cols":19,"rows_kept":71,"cut":false,"hidden":false,"merged":[]}
["X (m)","Y (m)","t (s)",null,null,1,"Put X and Y data in columns A and B",null,null,null,null,null,null,null,null,null,"X (m)","Y (m)","t (s)"]
[0,0.1,0.6,null,null,null,"Ensure that the Y data stays positive",null,null,null,null,null,null,null,null,null,5.5511151231257827E-17,2.2204460492503131E-16,0]
[5.6481900000000002E-2,0.1232332,0.61666670000000001,null,null,2,"Time data in column C is optional",null,null,null,null,null,null,null,null,null,9.3000000000000055E-2,0.2240000000000002,55]
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{"sheet":"Parameters","index":5,"n_rows":25,"n_cols":4,"rows_kept":25,"cut":false,"hidden":false,"merged":[[0,0,0,3],[20,0,24,0]]}
["Parameter Calculator"]
["Parameter","Formula","Value","Unit"]
["Roller Rotation Speed",null,46.331000000000003,"rev/s"]
["Roller radius",null,0.05,"m"]
["Ball Radius","r",7.4999999999999997E-2,"m"]
["Roller to Ball Slip factor","k1",1,"n/a"]
["Ball to wall Slip factor","k2",1,"n/a"]
["Air Density","r",1.2569999999999999,"kg/m^3"]
["Air Viscosity","m",1.8E-5,"N*s/m^2"]
["Ball drag coefficient",null,0.52,"n/a"]
["Surface-Air Coupling","h",0.4,"n/a"]
["Roller Tangental Speed",null,{"$f":"C3*C4*2*PI()","v":14.555312923346873},"m/s"]
["Ball Speed",null,{"$f":"C6/(1+C7)*C12","v":7.2776564616734367},"m/s"]
["Ball Rotation",null,{"$f":"C6*C7/(1+C7)*C12/C5/(2*PI())","v":15.443666666666671},"rev/s"]
["Cross Sectional Area (Ball)",null,{"$f":"PI()*C5^2","v":1.7671458676442587E-2},"m^2"]
["Length (Ball)",null,{"$f":"C5*2","v":0.15},"m"]
["Reynold's Number",null,{"$f":"C8*C13*C16/C9","v":76233.451436029238},"m"]
["Quadratic Drag Coefficient",null,{"$f":"0.5*C8*C10*C15","v":5.775386124634966E-3},"N*(s/m)^2"]
["Linear Drag Coefficient","C_d",{"$f":"6*PI()*C9*C5","v":2.5446900494077325E-5},"N*s/m"]
["Magnus Force Coefficient",null,{"$f":"PI()*C8*C11*C5^3","v":6.6639070668864987E-4},"kg"]
["Target Drawing","Offset",0,"m"]
[null,"Outer",0.5,"m"]
[null,"Inner",0.3,"m"]
[null,"Height",1.2,"m"]
[null,"Rim",0.38784227201975024,"m"]

{"sheet":"4th Order Polynomial Solver","index":6,"n_rows":45,"n_cols":7,"rows_kept":45,"cut":false,"hidden":false,"merged":[]}
["Quartic Equation Solution Method"]
["A",{"$f":"'Moving Turret No Air Model'!B15","v":-1.6863806747188668}]
["B",{"$f":"'Moving Turret No Air Model'!B16","v":-0.43283235507760359}]
["C",{"$f":"'Moving Turret No Air Model'!B17","v":0.43481882548937933}]
[]
["First Solve the Cubic Equation"]
[]
["a2",{"$f":"-B2/2","v":0.84319033735943338}]
["a1",{"$f":"-B4","v":-0.43481882548937933}]
["a0",{"$f":"(4*B2*B4-B3^2)/8","v":-0.39005301310487545}]
[]
["Q",{"$f":"(3*B9-B8^2)/9","v":-0.2239362690538281}]
["R",{"$f":"(9*B9*B8-2*B8^3-27*B10)/54","v":0.11171759423812198}]
["D",{"$f":"B13^2+B12^3","v":1.2509874250601696E-3}]
[]
["D>=0 Case"]
["S",{"$f":"(B13+SQRT(B14))^(1/3)","v":0.52786717900866875}]
["T",{"$f":"(B13-SQRT(B14))^(1/3)","v":0.42422843843858415}]
["u0",{"$f":"B17+B18-B8/3","v":0.67103217166077511}]
[]
["D<0 Case",null,null,null,null,null,"Choose Root that makes 2u-A>0"]
[null,{"$f":"ACOS(B13/SQRT(-B12^3))","v":{"$error":"#NUM!"}}]
[null,"u0","2u-A"]
["Choice1",{"$f":"-B$8/3+2*SQRT(-B$12)*COS(B$22/3)","v":{"$error":"#NUM!"}},{"$f":"2*B24-B2","v":{"$error":"#NUM!"}}]
["Choice2",{"$f":"-B$8/3+2*SQRT(-B$12)*COS(B$22/3+2*PI()/3)","v":{"$error":"#NUM!"}},{"$f":"2*B25-B3","v":{"$error":"#NUM!"}}]
["Choice3",{"$f":"-B$8/3+2*SQRT(-B$12)*COS(B$22/3+4*PI()/3)","v":{"$error":"#NUM!"}},{"$f":"2*B26-B4","v":{"$error":"#NUM!"}}]
["u0",null,{"$f":"IF(C24>0,B24,IF(B25>0,C25,B26))","v":{"$error":"#NUM!"}}]
[]
["Cubic Root Choice"]
["p0",{"$f":"IF(B14<0,C27,B19)","v":0.67103217166077511}]
["p",{"$f":"SQRT(2*B30-B2)","v":1.740242804335193}]
[]
["Quadratic Solution"]
["a",1,1]
["b",{"$f":"B31","v":1.740242804335193},{"$f":"-B35","v":-1.740242804335193}]
["c",{"$f":"B30-0.5*B3/B31","v":0.79539193168947819},{"$f":"B30+0.5*B3/B31","v":0.54667241163207203}]
[]
["Deteminant",{"$f":"B35^2-4*B34*B36","v":-0.153122708717496},{"$f":"C35^2-4*C34*C36","v":0.84175537151212865}]
[]
["t1",{"$f":"IF(B38<0,\"\",(-B35+SQRT(B38))/2)","v":""},{"$f":"IF(C36<0,\"\",(-C35+SQRT(C38))/2)","v":1.3288575386099222}]
["t2",{"$f":"IF(B38<0,\"\",(-B35-SQRT(B38))/2)","v":""},{"$f":"IF(C38<0,\"\",(-C35-SQRT(C38))/2)","v":0.41138526572527068}]
[]
["Final Answer"]
["t1",{"$f":"MAX(B40:C41)","v":1.3288575386099222},null,"Longest Time"]
["t2",{"$f":"MIN(B40:C41)","v":0.41138526572527068},null,"Shortest Time"]