Homework 3. problems: 4.5, 4.31, 4.49, 4.67

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Rolf White
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1 Hoewok 3 poble: 4.5, 4.3, 4.49, 4.67

2 Poble 4.5 The veco poiion of a paicle vaie in ie accoding o he expeion ( 3. î 6. ˆj ). (a) Find expeion fo he velociy and acceleaion a funcion of ie. (b) Deeine he paicle poiion and velociy a.. Noe. The poiion funcion i no defined unabiguouly. I i no clea wha he ie uni ae. By defaul, one hould aue ha econd bu ineed wihou he uni ino he funcion. I would be cleae, if he funcion wee peened in he following way () ( ) ˆ 3. i 6. ˆj a) By definiion, he velociy funcion i equal o he deivaive of he poiion funcion. Auing a fixed efeence fae (wih ie independen uni veco) d d v() ˆ i ˆ j î ˆj ˆj d d By definiion, he acceleaion funcion i equal o he deivaive of he velociy funcion. a () dv d d d ˆ j ˆ j b) The above funcion allow one o deeine pecific value a an abiay inan. Hence a, he paicle i a ( ) ( ) ˆ 3. i 6. ( ) ˆj ( 3. î 6. ˆj ) y x The velociy of he paicle a hi inan i v( ) ˆj ˆj v

3 Poble 4.3 A ain low down a i ound a hap hoizonal cuve, lowing fo 9. k/h o 5 k/h in he 5. ha i ake o ound he cuve. The adiu of he cuve i 5. Copue he acceleaion a he oen he ain peed eache 5 k/h. Aue he ain low down a a conan ae duing he 5- ineval fo 9 k/h o 5 k/h in 5. 5 In a cicula oion Caeian cala coponen v of acceleaion ae igonoeic funcion of ie. Fo he infoaion given, i i oe convenien o deeine he angenial (along he diecion of he ajecoy) and cenipeal (anvee o he ajecoy) coponen of he acceleaion. In any ype of oion, he angenial coponen of acceleaion i equal o he ae a he peed of he paicle change. Since in hi poble i i aued ha he peed change a a conan ae, angenial acceleaion a an inan unde conideaion i equal o i aveage value (deivaive of he funcion i equal o he diffeence quoien). Hence a () dv d Δv Δ k h 5 ( 5 9) ( 5 9) 3 k h 36 5 / k / h.74 The anvee coponen of acceleaion affec only he diecion of velociy. In a cicula oion, he anvee coponen i alway dieced owad he cene of he cicula pah of he paicle and i efeed a he cenipeal acceleaion. In a cicula oion, he cenipeal acceleaion depend only on he peed of he objec and he adiu of pah cuvaue. 3 k / k k 5 5 v h h 36 / h a c() The wod acceleaion afen efe o he agniude of he acceleaion veco, heefoe one ay include hi eaning alo ino he anwe a f a f

4 Poble 4.39 Heahe Lia in he Laboghini acceleae a ae of ( ˆi ˆj ) Jagua acceleae a ( ˆi 3ˆj ) 3 /, while Jill in he + /. They boh a fo he e a he oigin on a xy coodinae ye. Afe 5, (a) wha i Lia peed wih epec o Jill, (b) how fa apa ae hey, and (c) wha i Lia acceleaion elaive o Jill? a) The iniial poiion and iniial velociy of boh ca ae given in he efeence fae of he oigin. Addiionally, he acceleaion of each ca i given in hi efeence fae. Uing invee elaion he velociy and he poiion funcion can be found fo boh ca. In he efeence fae of he oigin, Jill velociy a ie i u () u + a d' [,3] J while Heahe velociy in hi efeence fae i () v + a d' [ 3, ] v H In he efeence fae of he oigin Jill poiion a ie i () R + u() d' [,3] R while Heahe poiion in hi efeence fae i () + v( ' ) d' [ 3, ] Looking a he figue, Heahe poiion wih epec o Jill i elaed o he poiion of boh gil wih epec o he oigin. Wih he appopiae choice of Jill coodinae ye ' () () R() [ 3, ] [,3] [, 5] O y y J R H x x

5 a) Diecly fo he definiion of velociy, Heahe velociy wih epec o Jill i decibed by he following funcion v ' d ' d () [, 5] A 5, he velociy i heefoe v ' ( 5) [, 5] 5 [, 5] and fo he definiion of peed i value i v' + ( 5) 6.9 b) A he conideed inan, Heahe poiion wih epec o Jill i ' ( 5) [, 5] ( 5) [ 5, 6.5] Magniude of he poiion veco epeen he diance of he objec fo he oigin of he efeence fae. Theefoe he diance fo Jill o Heae i ' 5 + ( 6.5) 67.3 c) Diecly fo he definiion, Heahe acceleaion in Jill efeence fae i a ' ' dv d () [, 5]

6 Poble 4.59 A kie leave he ap of a ki jup wih a velociy of /, 5 above he hoizonal, a in figue 4.8. The lope i inclined a 5, and ai eiance i negligible. Find (a) he diance fo he ap ha o whee he jupe land and (b) he velociy coponen ju befoe landing. (How do you hink he eul igh be affeced if ai eiance wee included? Noe ha jupe lean fowad in he hape of an aifoil wih hei hand a hei ide o inceae hei diance. Why doe i wok?) y 5 / 5 x - iniial poiion - locaion of he landing v - iniial peed α - angle beween iniial velociy wih he hoizonal β - angle beween he diecion of he lope wih he hoizonal In ode o olve he poble I will fi decibe he oion of he kie. Thi funcion will allow e o deeine he inan when he kie jup and land, and deeine he iniial and final poiion, fo which he agniude of he diplaceen can be found. Finally I can find he velociy a he inan of landing. Le' pecify he efeence ie and he efeence fae in which I will decibe he oion. I decided o chooe he inan when he kie, eaed a a paicle, leave he ap a he efeence inan and he end of he ap a he oigin of a Caeian ye. The axe ae in he hoizonal and veical diecion. The z-axi i pependicula o he oion, heefoe we know ha he coponen of all veco elaed o he oion ae zeo in he z-diecion. We can heefoe decibe he oion by wo dienional veco R. Wih he above aupion he iniial poiion and iniial velociy ae:,, v v [ coα,in α] 9.7,.6. Wih good appoxiaion we can aue ha he kie ove wih conan (fee fall) acceleaion

7 a () g, 9.8 Thi infoaion allow u o pedic he kie' velociy and poiion a any inan. () v () v + a( ' ) d' v + g 9.7, () () + v( ' ) d' + v + g [,] + [ 9.7,.6] + [, 4.9] 9.7, When he kie land, hi o he poiion i oewhee along he lope of he ounain. Theefoe, in hi coodinae ye he coponen of hi o he landing poiion u aify y (3) anβ x Ue equaion () o find ou if hi i poible. Thee u be uch an inan ha vy + g anβ vx Thi equaion ha he oluion o v ( ) 9.7 an5.5 x anβ + v β + y.87 g 9.8 a) A hi inan he poiion of he kie (he landing locaion) i (.87) , (.87) [ 7.9, 3.9] The diplaceen i defined in uch a way ha i agniude i equal o he diance Δ beween he iniial and he final poiion. Theefoe Δ x, y, x, y [ ] [ ] ( ) ( )

8 b) Equaion () allow one o find he velociy of he kie a any inan ha he/he i in he ai. Ju befoe he landing, hi o he velociy i heefoe v () 9.7, , 8. The peence of ineacion wih he ai ake he copeiion oe exciing. The kie can influence he lengh of he jup. I would be upiing bu heoeically jup can be longe in he peence of ineacion wih he ai! The idea i o educe he y-coponen of he acceleaion by uing ai a a ouce of an addiional foce (lif), while ainaining alo conan hoizonal coponen of he velociy. The pinciple i applicable in hand gliding (when hee i no ai convecion). Soe anial alo leaned how o inceae he ange of hei jup by aking advanage of ineacion wih he ai.

Worked Examples. v max =?

Worked Examples. v max =? Exaple iction + Unifo Cicula Motion Cicula Hill A ca i diing oe a ei-cicula hill of adiu. What i the fatet the ca can die oe the top of the hill without it tie lifting off of the gound? ax? (1) Copehend