Paricle Moion Soluions We have inenionally included more maerial han can be covered in mos Suden Sudy Sessions o accoun for groups ha are able o answer he quesions a a faser rae. Use your own judgmen, based on he group of sudens, o deermine he order and selecion of quesions o work in he session. Be sure o include a variey of ypes of quesions (muliple choice, free response, calculaor, and non-calculaor) in he ime alloed. Muliple Choice Soluions. E ( AB5) () 7 x v () x() 6 47 6 7 ( 4),4. A (8 AB/BC) V is increasing when v() a() which occurs when x() is concave up, so.. B (8 AB7) Using Fundamenal Theorem of Calculus: x() x() ( 6 ) d () ( ) x x() (4 ) 6 Alernaively: v () 6 x () c x() 6( ) c c x () x() 6 4. D (985 AB4) v () for all herefore, 4 4 x() v() d 5 d 4 5 6 64 8 meers Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a www.nms.org
5. C (985 AB8) Average velociy of he paricle is s 5() 5() 5. Paricle Moion 6. B (988 BC appropriae for AB) v () dcand v() () C 4 C Disance raveled from v() 4 and v() x() (4) d 4 68 4 meers 7. C (8 AB86) v() x(), so x() has a horizonal angen a ; herefore, he only possible graphs are C and E. From he able, v() x(), so x() is increasing a, so he answer is C. 8. C ( AB76) Using he derivaive funcion on he calculaor: v() a() a(4).6 9. E ( AB9/BC9) Using he Fundamenal Theorem of Calculus and he inegral funcion on he calculaor: v() v() ln d v() ln d.46. A ( AB8) Average velociy of a funcion on [, ]: fee ( e e ) d.86 second Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a www.nms.org
Paricle Moion Free Response. ( AB/BC) (a) Runner A: velociy m 6.666 or 6.667 sec : velociy for Runner A : velociy for Runner B Runner B : 48 m v() 6.857 7 sec (b) Runner A: acceleraion. meers / sec 7 Runner B: a() v() ( ) 7.469 meers / sec 49 (c) Runner A: disance ()() 7() 85 meers Runner B: disance 4 d 8.6 meers 4 : acceleraion for Runner A : acceleraion for Runner B : disance for Runner A : mehod : answer : disance for Runner B : inegral : answer Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a www.nms.org
. (999 AB) (a) v(.5).5sin(.5 ).67 Up, because v(.5) (b) a () v() sin cos a(.5) v(.5).48 or.49 No, v is decreasing a.5 because v(.5) (c) y() v() d cos sin d C 7 y() C C 7 y () cos 7 y() cos 4.86 or.87 (d) disance = v ( ) d.7 or v () sin or.77 y () ; y 4; y ().86 or.87 y y() y y().7 or.74 : answer and reason : a (.5) : conclusion and reason : y() v() d : y () cos C : y () Paricle Moion : limis of and on an inegral of v () or v ( ) or uses y () and y () o compue disance : handles change of direcion a suden s urning poin : answer / if incorrec urning poin Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a www.nms.org
. (5 Form B AB) Paricle Moion (a) 5 a(4) v(4) : answer 7 (b) v ( ) ( )( ), v () for v () for v () for 5 : ses v ( ) : direcion change a, : inerval wih reason (c) () () ln( ) s s u u du () 8 ln( ) s u u du 8.68 or 8.69 : ln( u u ) du : handles iniial condiion : answer (d) v ( ) d.7 or.7 : inegral : answer Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a www.nms.org
4. (8 AB4/BC4) (a) Since v () for and 5 6, and v () for 5, we consider and 6. x() v( ) d 8 6 x(6) v( ) d 8 9 Therefore, he paricle is farhes lef a ime when is posiion is x() (b) The paricle moves coninuously and monoonically from x() o x(). Similarly, he paricle moves coninuously and monoonically from x() o x(5) 7and also from x(5) 7 o x(6) 9. By he Inermediae Value Theorem, here are hree values of for which he paricle is a x ( ) 8. (c) The speed is decreasing on he inerval since on his inerval v and v is increasing. (d) The acceleraion is negaive on he inervals x and 4 x 6 since velociy is decreasing on hese inervals. Paricle Moion : idenifies as a candidae : considers 6 v () d : conclusion : posiion a, 5, and 6 : descripion of moion : conclusion : answer wih reason : answer : jusificaion Copyrigh Naional Mah + Science Iniiaive, Dallas, TX. All righs reserved. Visi us online a www.nms.org