MA56 utorial Solutions Qustion a Intgrating fator is ln p p in gnral, multipl b p So b ln p p sin his kin is all a Brnoulli quation -- st
Sin w fin Y, Y Y, Y Y p Qustion Dfin v / hn our quation is v μ v Noti that v Now b th funamntal thorm of alulus μ v v his is a sparabl iffrntial quation with initial onition v Sparating th variabls an intgrating [rmmbr that osh sinh ] w fin that v sinhμ/ Intgrating, w gt osh μ C, μ whr C must b /μ sin an osh hus w hav μ osh h shap of th graph of this funtion is in -shap μ μ [Draw it!] Qustion h onstant C has units of /tim Solving th quation [ithr as linar first orr or as a sparabl quation] w gt M M Ct Clarl will approah M mor rapil if C is larg; that is, C masurs how rapil th stunt is abl to larn hus th quation in prsss th ia that th stunt s
prforman improvs mor slowl as sh approahs hr maimum possibl prforman As th ars go b an th stunt boms mor familiar with th mthos of mastring mathmatis, hr rat of larning nw things might b pt to improv; but surl thr is an uppr boun to how muh sh an improv h tanh funtion is a simpl wa of rprsnting this sin it alwas inrass but is boun abov [Rmin th stunts of th shap of tanh if nssar] hn K rprsnts hr maimum possibl sp of larning [sin tanh is asmptoti to ], an masurs th amount of tim rquir for hr to ralis hr maimum potntial [Not that K has units of /tim, whil of ours has units of tim, so K is imnsionlss] h quation an now b writtn as K tanh t / M It s onvnint now to fin Q M, so th quation is Q K tanh t / Q his is a first-orr quation with intgrating fator osh K t /, so w hav Q osh K t / A, whr A is a onstant Sin w ar assuming that, w hav Q - M, so w hav A - M, thus finall K [ sh t / ] M Stunts ar nourag to graph ampls of suh qustions using [for ampl] th softwar availabl fr at http://wwwgraphmatiaom/ Qustion 4 h onstant K masurs th rapiit with whih th rumour will spra It pns on how intrsting th rumour is, how muh th stunts lik to gossip, how gullibl th ar, t h right han si of th quation is sign to b small both nar R an nar R, whn in th rumour an b pt to spra slowl ithr baus not nough or too man popl hav har it W hav R KR KR
his is a Brnoulli quation as isuss in th nots W solv it b fining Z /R, whih transforms th quation into a linar on: Z KZ K, with solution R C p Kt h problm sas that this highl intrsting rumour was start b on stunt, so R hus C 99/ Hn R 99 p Kt Of ours as t tns to infinit, R tns to Qustion 5 Suppos w writ th quation govrning th ranium as k whr rprsnts th numbr of ranium atoms, t, as usual h half-lif of ranium an b us to omput th a rat onstant, as in th ltur nots, an similarl for horium From th ltur nots w hav k k k [ p k k t] an so if / is w know vrthing in this quation pt t Solving for t, ou shoul fin that th answr is approimatl 95 thousan ars In th son part of th problm, th a rat onstant is assum to b a givn funtion of tim a Now th quation for boms t / b h units of a ar larl /ars an thos of b must b ars [rsumabl b is vr larg; othrwis w woul hav noti this fft hat is, w gt bak th usual quations for raioativit in th limit as b tns to infinit] Sparating th variabls an oing th intgration w gt [using an a bit of simpl algbra]
p[ at ] t / b aking th limit as t tns to infinit, w fin that tns to p[-ab]; that is, ranium woul nvr ompltl isappar if this thor wr right [Sin b is supposl vr larg, howvr, th surviving amount is vr small] OIONAL NOE ON HE EQAION IN QESION : Consir a small lmnt of th abl, with th lft n loat at a point,, an th right at, Lt tan θ /, an lt not th tnsion in th abl h horiontal fors ating ar just os θ an os θ os θ at th two ns of th small lmnt, so if ths fors ar to balan w must hav os θ onstant h vrtial fors ar sin θ an sin θ sin θ an also μ s, whr s is th lngth of th lmnt For ths to balan w must hav sin θ μ s, or an θ μ s Intgrating this quation from [whr / ] w obtain th stat quation