Physics 302 Exam Find the curve that passes through endpoints (0,0) and (1,1) and minimizes 1

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1 Physis Exam 6. Fid th urv that passs through dpoits (, ad (, ad miimizs J [ y' y ]dx Solutio: Si th itgrad f dos ot dpd upo th variabl of itgratio x, w will us th sod form of Eulr s quatio: f f y' y' y si x ( y' y y' ( y' y + y' Not that w hos th si x solutio rathr tha th os x solutio, ad th valu, i ordr to satisfy th d poit oditios. J si x si [ os x si x] dx os xdx. 455 mi For a straight li, yx ad [ ] x J straight x dx x

2 7. A bad of mass m is ostraid to mov without fritio o a irular wir hoop of radius R, whih itslf is rotatig about a vrtial diamtr with ostat agular vloity ω. Fid th quilibrium positio of th bad. Now suppos that it is displad a small dista from its quilibrium positio. St up Lagrag s quatios for its motio o th hoop, ad alulat th agular frquy of small-agl osillatios. Fid ad itrprt physially a ritial agular vloity ω that divids th bad s motio ito two distit typs. Costrut phas diagrams for th two ass ω>ω, ω<ω. Solutio: Dfi th loatio of th bad by its agl θ with rspt to th bottom of th hoop. Equilibrium orrspods to th oditio that th for i th dirtio i whih th wir is fr to mov is just what is dd to produ th irular motio of th bad: g siθ ω Rsiθ ( g Rω Now displa th bad from quilibrium by a agl dθ: T mω ( Rsiθ + mr & θ U mgr( L d L mω R siθ mgrsiθ mr && θ θ dt & θ && g θ ω θ R iωt g θ θ Ω ω R Th motio hags haratr wh th radiad hags sig; th xpot i th θ motio hags from imagiary to ral, ad th motio hags from harmoi osillatio about stabl quilibrium to xpotial divrg from a ustabl quilibrium. Th ritial valu of ω is ω g R. Th phas diagrams ar: / θ & ω δ Ω θ & ω + δ Ω θ θ

3 8. Som folks hav proposd disposig of ular wast by ithr arryig it out of th solar systm or by rashig it ito th su. Assum that o slig-shot sarios ar prmittd ad that thrusts our oly i th orbital pla. Calulat th miimum amout of impuls Δv that is rquird i ah as. G 6.67 x - m /s kg, M su x kg, R arth-su.5x m, R su 7 x 8 m, M arth 6 x 4 kg. Solutio: Esap th solar systm: Th miimum sap vloity Δv to lav th solar systm from Earth orbit is obtaid by rgy osrvatio: GM s vo R R GM R v s + s s + ( v + Δv ( v + Δv s Δv ( + v Δv To rash ito th su, w simply rquir to stop it from orbitig th su; i.. w giv it a boost Δv s -v So it taks lss boost vloity to jt from solar orbit tha to rash ito th su.

4 9. a A tis ball of mass m sits atop a basktball of mass m >> m. Th balls ar rlasd from rst wh th bottom of th basktball is a hight h abov a horizotal surfa. To what hight dos th tis ball bou? Igor wid vloity, assum that th ball diamtrs ar gligibl, assum that th balls bou lastially. For simpliity, assum that th balls ar sparatd by a vry small dista, so that th rlvat bous happ a short tim apart. This assumptio is t ssary, but it maks for a slightly lar solutio. Just bfor th basktball hits th groud, both balls ar movig dowward with spd v gh. Wh th basktball bous off th groud, it movs upward with spd v, whil th tis ball still movs dowward with spd v. Th rlativ spd is thrfor v. Th tr-of-mass fram is approximatly that of th basktball (m >>m so aftr th basktball bous th tr-of-mass fram is movig upwards with vloity Vv. I that fram th bak-to-bak sattrig simply rvrss th sigs of th two momtum vtors. Th tis ball thus has a upwards lab-fram vloity of v +Vv. y osrvatio of rgy, it will thrfor ris to a hight of ( v h f h + h g b Now osidr balls,,,, havig masss m >>m >> >>m, sittig i a vrtial stak. Th stak is rlasd from rst wh th bottom of ball is h abov th groud. I trms of, to what hight dos th top ball bou? If h m, what is th miimum umbr of balls rquird for th top o to rah sap vloity from Earth?

5 Solutio: Just bfor ball th groud, all of th balls ar movig dowward with spd v gh. W will idutivly dtrmi th spd of ah ball aftr it bous off th o blow it. If ball i ahivs a spd of v i aftr bouig off ball i-, th what is th spd of ball i+ aftr it bous off ball i? Th rlativ spd of balls i+ ad i (right bfor thy bou is v + v i. This is also th rlativ spd aftr thy bou. Si ball i is still movig upwards at sstially spd v i, th fial upward spd of ball i+ is thrfor (v + v i + v i. Thus, i+ v 7v i, + v 4 5v., t. From osrvatio of rgy, ball will bou to a hight of ( v [ + ( ] h h + h g If h is mtr, ad w wat this hight to qual mtrs, th (assumig th rst >. Six balls ar suffiit. of th balls ar ot vry larg w d ( Esap vloity from th arth is gh > gr ( wh > log ( + R / ~ h v gr km s. That spd is rahd / So a stak of balls should suffi to sd th top o to th moo!

1985 AP Calculus BC: Section I

1985 AP Calculus BC: Section I 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b