Numerical Integration of Newton s Second Law of Motion


 Esmond French
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1 Nueicl Itegtio of Newto s Secod Lw of Motio Oe of the biggest thigs we ve doe this seeste is develop Newto s Secod Lw of Motio: F. Oe of the eso s this is ipott is opetiol: usig Newto s lws of otio, it becoes possible to deteie the positio of give object t y tie (give it s positio d velocity t soe iitil tie. (A oe tivil eso fo the ipotce is tht Newto s echics deies the existece of fee will but I digess. Soe of the elie successes of Newto s echics wee desciptios of the plety otio tht people hd obseved fo cetuies befoe. Othe oe ude pplictios e otio of pojectiles d othe objects hee o Eth. It is the pupose of this lb to develop the es to lyze the pojectile otio of object subject to dg foce tht vies s the sque of the velocity (s well s the foce of gvity, of couse, d to exploe the behvio of such object. Theoeticl lysis Fist, coside the tsk of fidig positio s fuctio of tie. Newto s Secod tells us tht: dv ( ( t d ( t F, v, t (1 dt dt Tht is, we c fid positio s fuctio of tie if we kow the foces ivolved d c solve diffeetil equtio ( equtio with deivtives ll i it. Notice tht the foce ight deped o positio d tie (d eve velocity, the te of chge of positio. I this cse, the poble gdutes fo beig eely Hd to beig Relly Hd. Becuse i ode to wite dow the foce you eed to kow the positio d velocity but to get the positio d velocity, I eed to wite dow the foce d solve the deivtives! Ctch. Reebe tht the cocept of eegy ws iveted to get oud soe of this. I othe cses, it is possible to septe the vious depedeces (fo exple, if the dg foce depeds oly upo the velocity to the fist powe clled Stokes dg it is possible to septe the vibles d solve the poble. But the ube of pobles you c solve i this wy is ctully petty sll (it ws, to ceti extet, lucky fo Newto tht the ipott pobles wee pobles it is possible to solve. I the vst joity of cses, it is ecessy to solve this poble ueiclly. We discuss wht ueiclly es below, but i shot it ivolves doig LOTS d LOTS of siple clcultios. It is possible to do this by hd, but it oly bece coo with the dvet of coputes (d pticully coputes tht c do lots of clcultios elly fst. So how does tht help? Igie tht I kow the positio (t d velocity v(t of object t oe pticul istt of tie, t. (Neve id how we ll get to tht i oet. Newto s Secod tells e tht: dv v( t dt v( t F(, v, t ( dt dt So we see how to fid the velocity t soe istt lte:
2 F ( ( (, v, t v t dt v t dt (3 Ad the positio c be foud siply fo: v ( ( ( ( t dt v( t t dt t v AVGdt t dt (4 1 F ( ( (, v, t ( t v t dt dt Hopefully, these equtios will look fili: they e the equtios we developed fo the specil cse of costt cceletio. Ad tht is effectively wht we e doig: ssuig tht the cceletio does ot chge. This is exctly tue fo ifiitely shot tie peiod, d is APPROIMATEL tue fo petty shot tie peiod. Most of the hd wok i ueicl clcultios of the type we e udetkig is figuig out whe the tie peiods e shot eough to give you ppoxite swe close eough to the ight swe. Note tht ll the equtios give so f e vecto equtios. Tht es tht they e elly thee equtios i oe (oe fo ech diesio x, y, d z. Fo the pobles we will coside, the pobles e ctully two diesiol (the pojectile objects will sty i the ple cotiig thei iitil velocity we will ot pove tht hee, but you SHOULD. So ou pobles will elly be diesiol, which kes life esie. The two equtios of otio c be developed i the followig wy. The cceletio is obtied fo the foce i ech diectio: F ( (, v, t t (5 F ( (, v, t t The velocity d positio s idicted bove: v ( t dt v ( t ( t dt (6 v t dt v t t ( ( ( dt 1 x( t dt x( t v ( t dt ( dt (7 1 y( t dt y( t v ( t dt ( dt The lst step is to elize tht if we hve the iitil vlues (positio d velocity t t0 we c the step though tie d geete the etie tjectoy of the pticle. Exple: Siple pojectile otio A siple cse is tht of pojectile subject oly to gvity (eig x 0, y 9.8 /s. Note tht this is poble we do t eed to clculte ueiclly. We discuss it hee i ode to clify the ethod d to llow us to ssess its ccucy. We wt to clculte the tjectoy of pticle fo give iitil coditios. We stt the pojectile t the oigi (x0, y0, with iitil speed of 10 /s t iitil gle of 30 degees (eig v x ( /s, v y (0 5 /s. We choose tie step of 0.1 secod. The t the ext tie step (t0.1 s, v x ( /s d v y (0.1 5/s9.8 /s (0.1s
3 4.0 /s. The positio is x( /s (0.1 s d y(0.1 0 (5 /s(0.1s ½ (9.8 /s (0.1 s Kowig the vlues fo t0.1 s llows us to clculte the positio t t0. sec: v x ( /s d v y ( /s9.8 /s (0.1s 3.04 /s. The positio is x( /s (0.1 s 1.73 d y( (4.0 /s(0.1s ½ (9.8 /s (0.1 s This is ledy tedious d thee e still bout 10 oe itetios to go tht s why people do this with coputes the th by hd (with clculto. But if we cy this out, we get tble of vlues tht looks like: tie(sec x positio y positio vx vy x y I stopped hee becuse the ypositio is becoig egtive (eig tht the object s ledy hit the floo. Ad we c look t the esults d deteie tht the ge is betwee 7.8 d 8.6 (pobbly close to 7.8, the xiu height is bout 1.03, the tie to the xiu height is betwee 0.4 sec d 0.5 sec (pobbly bout 0.45 sec, d the tie bck to the goud is betwee 0.9 sec d 1.0 sec (pobbly close to 0.9 sec. Ad we c check these esults becuse we kow how to clculte ll these qutities fo siple pojectile otio: R (v 0 /g si Θ 8.837, y MA (v 0 si Θ /g 1.76, tie fo y MA is t(v 0 si Θ/g sec d tie fo R is t(v 0 si Θ/g 1.00 sec. Notice tht ll the esults fo the ueicl clcultio e close (but o cig. Ad they e ll little too sll. This is becuse ou ppoxitio (tht cceletio is costt fils, t lest little. Fo exple, we clculte the positio t the ed of 0.5 secod s x 4.33 d y But exct clcultio gives esult of x 4.33 d y It s coplicted to expli, but thee e two esos ou ueicl esults e diffeet fo lyticl esults: ou tie steps e too big d ou ethod of clcultig the positio is too sipleided. Thee is etie bch of th d physics devoted just to the study of oe itelliget wys to solve diffeetil equtios. But fotutely fo us, we do t hve to wok ste, we c ke the copute wok hde. If we do the se clcultio s descibed bove, but usig steps of 0.01 sec, we get: R 8.747, y MA 1.505, tie fo y MA is t 0.51 sec d tie fo R is t 1.01 sec. Note tht the esults e uch close this go oud (d we c get close still by shoteig the step size oe. ou c fid the spedsheet (i Excel fot tht I used i these clcultios d scew oud with it soe. Fo exple, you could clculte R while vyig the gle, plot R vesus Θ d coclude tht the xiu ge is fo Θ 45º. (With soe effot you ight eve be ble to figue out
4 tht R~si( Θ, but it s hde th with the lyticl techique. But the REAL poit of this lb is to do soethig you CAN T do lyticlly: pojectile udegoig qudtic dg. Pojectile with Qudtic Dg A el object flyig though the i feels dg foce tht depeds o the speed of the object. Fo low speeds, tht foce is lie i the speed (i.e., F~v, while fo high speeds, the foce is qudtic (i.e., F~v. Fo bsebll, fo exple, low speeds e speeds uch, uch less th bout 1 /sec (tht s illietes. High speeds e uch, uch gete tht 1 /sec. So fstbll is DEFINITEL i the qudtic egie. (Aside: This depeds oly o the SIZE of the object, so co bll is the se s bsebll fo these puposes ybody who wts to kow how this is clculted should sig out. Ufotutely fo us, it is possible to solve the poble of lie dg lyticlly (d lso qudtic dg i oe diesio we did this i clss, but NOT qudtic dg i two diesios. So we ll do it ueiclly. ou tsk is to ipleet the solutio i whteve clcultig pog you wt (icludig Fot, if tht s you bg. I ll sketch out the solutio hee. Fist, the foces e: 1 F (, v, t g( ˆj ( vˆ (7 Note tht I ve defied dow s egtive (ight? d the dg foce poits i diectio opposite the velocity (ight?. Thee s o explicit tie depedece (good! d the oly spce depedece is tivil. The cceletio i ech diectio is the: ( t cosθ (8 ( t g siθ But ote tht we ust be VER CAREFUL tht the diectios e coect we hve witte the dg foce log y s egtive, but this is oly tue if the object is goig up. O the wy dow, the dg foce is ctully UP (gist the velocity. If we wite the tig fuctios s cosθ v /v d siθ v /v, the the poble will tke ce of itself (you should pove this: v v ( t v (9 v v ( t g g v The velocity d positio e the clculted i pecisely the se wy s we did fo costt cceletio. But ote tht ow we do t hve the beefit of kowig the swe to cope to ou clcultio. So how do we kow ou swe is ight? We do t. All we c do is wok hd d ty to be ceful d hope d if possible, cope ou swe to soe expeiet. (Tht s why theoeticl physicists e lwys despete fo expeietl dt to cope to thei clcultios if ot, they e elly theticis. Hee s gph of the esults I hd fo the followig coditios:
5 gvity plus qudtic i dg ss v0 45 dius 0.03 thet0 75 i desity 1. k (the k thig is just the costt pts of the dg foce (A/ to ke the clcultio slightly esie to code The gph looks like: Tjectoy positio [] positio [] The blue lie is the tjectoy fo the object with i dg d the pik lie (excuse e get is without i dg. Coclusio The tough pt is gettig the clcultios to wok. Oce it does, if gets kid of fu (if you id woks tht wy becuse you c the stt plyig fo exple, ty vyig the dius without vyig the ss (you ll see why you c thow bsebll uch fthe th bech bll, eve though they weigh bout the se. Also, you c vy the gle without chgig ythig else you ll see tht with i dg, the xiu ge does NOT occu t luch gle of 45º. (Fo the coditios I show you bove, the xiu ge occus fo luch gle of 38º. ou could eve costuct plot of the ge s fuctio of luch gle (tht would be good ide fo epot. ou should discuss i epot (if you do oe why defiig the si d cosie the wy we did is coect d gives the ight diectio fo the cceletio. Filly, suppose soe itwit i New Oles decides it would be get ide to fie off his pistol i the i o New es Eve. How high would it go? How fst would it be goig whe it lded? How f would it go to the side (if he shoots i the Migy, will he kill soebody i the Qute? How bout Uptow?? Filly (elly this tie, if you dop pey of the Epie Stte Buildig, would it be goig fst eough to kill soebody if it hit the? How bout if you spit?
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