2017 AP CALCULUS AB FREE-RESPONSE QUESTIONS

Size: px
Start display at page:

Download "2017 AP CALCULUS AB FREE-RESPONSE QUESTIONS"

Transcription

1 17 FREE-RESPONSE QUESTIONS 5. Two paricles move along he x-axis. For 8, he posiion of paricle P a ime is given by xp () ln ( 1 ), while he velociy of paricle Q a ime is given by vq () Paricle Q is a posiion x 5 a ime. (a) For 8, when is paricle P moving o he lef? (b) For 8, find all imes during which he wo paricles ravel in he same direcion. (c) Find he acceleraion of paricle Q a ime. Is he speed of paricle Q increasing, decreasing, or neiher a ime? Explain your reasoning. (d) Find he posiion of paricle Q he firs ime i changes direcion. 17 The College Board. Visi he College Board on he Web: GO ON TO THE NEXT PAGE.

2 17 SCORING GUIDELINES Quesion 5 ( 1) (a) x P ( ) = = > for all. 1 : x P( ) : 1 : inerval x P ( ) = = 1 xp( ) for 1. Therefore, he paricle is moving o he lef for 1. (b) v ( ) = ( 5)( ) Q vq( ) = =, = 5 : 1 : inervals 1 : analysis using vp( ) and vq( ) Noe: 1 if only one inerval wih analysis Noe: if no analysis Boh paricles move in he same direcion for 1 and 5 8 v x v have he same sign since ( ) = ( ) and ( ) on hese inervals. P P Q (c) a ( ) = v ( ) = 8 Q Q a ( ) = 8 = Q : 1 : a Q ( ) 1 : speed decreasing wih reason a ( ) and v ( ) = > Q Q A ime =, he speed of he paricle is decreasing because velociy and acceleraion have opposie signs. (d) Paricle Q firs changes direcion a ime =. Q ( ) = Q( ) + Q( ) = 5 + ( ) x x v d d = 1 = = 5 + ( ) = = 1 : aniderivaive : 1 : uses iniial condiion 1 : answer 17 The College Board. Visi he College Board on he Web:

3 16 SCORING GUIDELINES Quesion For, a paricle moves along he x-axis. The velociy of he paricle a ime is given by v ( ) = 1+ sin. The paricle is a posiion x = a ime. = (a) A ime =, is he paricle speeding up or slowing down? (b) Find all imes in he inerval < < when he paricle changes direcion. Jusify your answer. (c) Find he posiion of he paricle a ime =. (d) Find he oal disance he paricle ravels from ime = o ime =. (a) v( ) = > v ( ) = 1.16 < : conclusion wih reason The paricle is slowing down since he velociy and acceleraion have differen signs. (b) v ( ) = =.7768 v ( ) changes from posiive o negaive a =.77. Therefore, he paricle changes direcion a his ime. (c) x( ) x( ) v( ) d = + = + ( ) =.815 { 1 : =.77 : 1 : jusificaion 1 : inegral : 1 : uses iniial condiion 1 : answer = : { 1 : inegral (d) Disance v( ) d = : answer 16 The College Board. Visi he College Board on he Web:

4

5

6

7

8 7 SCORING GUIDELINES Quesion A paricle moves along he x-axis wih posiion a ime given by x() = e sin for π. (a) Find he ime a which he paricle is farhes o he lef. Jusify your answer. (b) Find he value of he consan A for which x() saisfies he equaion Ax () + x () + x() = for < < π. (a) () sin x = e + e cos = e ( cos sin ) x () = when cos = sin. Therefore, x () = on π 5 π π for = and =. The candidaes for he absolue minimum are a π 5π,,, and π. 5 : : x () 1 : ses x () = 1 : answer 1: jusificaion x() e sin ( ) = π 5π e e π π ( ) > π ( ) sin 5π 5 sin < π e π sin ( π ) = The paricle is farhes o he lef when = 5 π. (b) x () = e ( cos sin ) + e ( sin cos ) = e cos Ax () + x () + x() ( ) ( ) = A e cos + e cos sin + e sin = ( A + 1) e cos = : : x () 1 : subsiues x (), x (), and x() ino Ax () + x () + x() 1 : answer Therefore, A = 1. 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens). 1

9 7 SCORING GUIDELINES (Form B) Quesion A paricle moves along he x-axis so ha is velociy v a ime is given by v sin. The graph of v is shown above for 5. The posiion of he paricle a ime is x and is posiion a ime is x 5. (a) Find he acceleraion of he paricle a ime. (b) Find he oal disance raveled by he paricle from ime o. (c) Find he posiion of he paricle a ime. (d) For 5, find he ime a which he paricle is farhes o he righ. Explain your answer. (a) a v 6cos or 5.67 (b) Disance v d 1.7 OR For, v when and.566 x 5 x x 5 v d v d 5.1 x 5 v d x x x x x x : a : 1 : seup 1 : answer 1 : inegrand (c) x 5 v d 5.77 or 5.77 : 1 : uses x 5 1 : answer (d) The paricle s righmos posiion occurs a ime The paricle changes from moving righ o moving lef a hose imes for which v wih v changing from posiive o negaive, namely a,, ,.7,.96. T Using xt 5 v d, he paricle s posiions a he imes i changes from righward o lefward movemen are: T: 5 xt : The paricle is farhes o he righ when T. : 1 : ses v 1 : answer 1 : reason 7 The College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for sudens and parens). 1

10 5 SCORING GUIDELINES (Form B) Quesion A paricle moves along he x-axis so ha is velociy v a ime, for 5, is given by () ( ) v = ln +. The paricle is a posiion x = 8 a ime =. (a) Find he acceleraion of he paricle a ime =. (b) Find all imes in he open inerval < < 5 a which he paricle changes direcion. During which ime inervals, for 5, does he paricle ravel o he lef? (c) Find he posiion of he paricle a ime =. (d) Find he average speed of he paricle over he inerval. 5 (a) a( ) = v ( ) = 1 : answer 7 (b) v () = + = 1 + = ( ) ( 1) = = 1, 1 : ses v () = : 1 : direcion change a = 1, 1 : inerval wih reason v () > for < < 1 v () < for 1 < < v () > for < < 5 The paricle changes direcion when = 1 and =. The paricle ravels o he lef when 1 < <. (c) () = ( ) + ln( + ) ( ) = 8 + ln( + ) s s u u du s u u du = 8.68 or 8.69 : ( ) 1 : ln u u + du 1 : handles iniial condiion 1 : answer 1 (d) () v d=.7 or.71 1 : inegral : 1 : answer Copyrigh 5 by College Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens). 1

11 SCORING GUIDELINES Quesion A paricle moves along he y-axis so ha is velociy v a ime A ime =, he paricle is a y = 1. (Noe: an (a) Find he acceleraion of he paricle a ime =. 1 x = arcan x ) 1 is given by v () = ( e) 1 an. (b) Is he speed of he paricle increasing or decreasing a ime =? Give a reason for your answer. (c) Find he ime a which he paricle reaches is highes poin. Jusify your answer. (d) Find he posiion of he paricle a ime =. Is he paricle moving oward he origin or away from he origin a ime =? Jusify your answer. (a) a( ) = v ( ) =.1 or.1 1 : answer (b) v ( ) =.6 Speed is increasing since a ( ) < and v ( ) <. 1 : answer wih reason 1 v = when ( e ) (c) () an = 1 = ln ( an () 1 ) =. is he only criical value for y. v () > for < < ln ( an () 1 ) v () < for > ln ( an () 1 ) 1 : ses v () = : 1 : idenifies =. as a candidae 1 : jusifies absolue maximum y () has an absolue maximum a =.. (d) y( ) = 1+ v( ) d = 1.6 or 1.61 The paricle is moving away from he origin since v ( ) < and y ( ) <. : 1 : v () d 1 : handles iniial condiion 1 : value of y( ) 1 : answer wih reason Copyrigh by College Enrance Examinaion Board. All righs reserved. Visi apcenral.collegeboard.com (for AP professionals) and (for AP sudens and parens). 11

12 SCORING GUIDELINES (Form B) Quesion A paricle moves along he x-axis wih velociy a ime given by v( ) = 1 + e1. (a) Find he acceleraion of he paricle a ime =. (b) Is he speed of he paricle increasing a ime =? Give a reason for your answer. (c) Find all values of a which he paricle changes direcion. Jusify your answer. (d) Find he oal disance raveled by he paricle over he ime inerval. 1 (a) a () = v() = e a() = e : 1 : v( ) 1 : a() (b) a () < e v() = 1 + < Speed is increasing since v () < and a () <. 1 : answer wih reason 1 (c) v () = when 1 = e, so = 1. v () > for < 1 and v () < for > 1. Therefore, he paricle changes direcion a = 1. : 1 : solves v ( ) = o ge = 1 1 : jusifies change in direcion a = 1 (d) Disance = v () d = ( ) + ( + ) = ( 1+ ) + ( 1 ) e d e d e e 1 = ( 1 1+ e) + ( + e 1 1) : 1 : limis 1 : inegrand 1 : anidiffereniaion 1 : evaluaion = e + 1 e OR OR 1 () = x e x() = e x (1) = e x() = Disance = ( x(1) x() ) + ( x(1) x() ) = ( + e) + ( 1+ e ) = e + 1 e : 1 : any aniderivaive 1 : evaluaes x ( ) when =, 1, 1 : evaluaes disance beween poins 1 : evaluaes oal disance Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com. 99 5

13 SCORING GUIDELINES Quesion A paricle moves along he x-axis so ha is velociy a ime is given by A ime =, he paricle is a posiion x = 1. v( ) = ( + 1) sin. (a) Find he acceleraion of he paricle a ime =. Is he speed of he paricle increasing a =? Why or why no? (b) Find all imes in he open inerval < < when he paricle changes direcion. Jusify your answer. (c) Find he oal disance raveled by he paricle from ime = unil ime =. (d) During he ime inerval, wha is he greaes disance beween he paricle and he origin? Show he work ha leads o your answer. (a) a() = v() = or v () = sin() < Speed is decreasing since a () > and v () <. : 1: a() 1: speed decreasing wih reason (b) v () = when = = or.56 or.57 Since v () < for < < and v () > for < <, he paricle changes direcions a =. : 1: = only 1: jusificaion (c) Disance = v () d =. or. : 1: limis 1: inegrand 1: answer (d) v () d=.65 x( ) = x() + v( ) d =.65 Since he oal disance from = o = is : 1: ± (disance paricle ravels while velociy is negaive) 1 : answer., he paricle is sill o he lef of he origin a =. Hence he greaes disance from he origin is.65. Copyrigh by College Enrance Examinaion Board. All righs reserved. Available a apcenral.collegeboard.com. 1

14 SCORING GUIDELINES (Form B) Quesion )F=HJE?ALAI=CJDAN=NEIIJD=JEJILA?EJOL=J=OJEAJBH > J > $ EICELA>O IEJ LJ A Г)JJEAJJDAF=HJE?AEI=JJDAHECE = JDA=NAIFHLE@A@IAJ?DJDACH=FDBLJ BH > J > $ >,KHECMD=JEJAHL=IBJEAEIJDAF=HJE?ALECJJDAABJ/ELA=HA=IBHOKH 1IJDAHA=OJEAJ J > $ =JMDE?DJDAF=HJE?AHAJKHIJJDAHECEKIJEBOOKH = LJ =IMAH > =HJE?AEILECJJDAABJMDA LJ EA A IE?! " =@# $ H " LJ IEJ LJ Г JHJ N $#$ N" " J N ГN N" ГN 6DAHAEIIK?DJEA>A?=KIA 6 BH=6 J CH=FD Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered rademarks of he College Enrance Examinaion Board. JDHAA]DKFI^ FAHE@E?>AD=LEH IJ=HJI=JHECE HA=I=>AHA=JELA=N=@EL=KAI EJAHL=I! Г A=?DEIIECHE?HHA?JEJAHL= HA=I EEJIB=@"=EJACH=B LJ H LJ H KIAIN=@N"J?FKJA@EIJ=?A! D=@AI?D=CAB@EHA?JE=JIJK@AJ\I JKHECFEJ =IMAH JAEBE?HHA?JJKHECFEJ IK?DJEA HA=I 97

15 SCORING GUIDELINES Quesion )>A?JLAI=CJDA N=NEIMEJDEEJE=FIEJE N 6DALA?EJOBJDA>A?J=JJEA J F EICELA >O L J IE J! = 9D=JEIJDA=??AAH=JEBJDA>A?J=JJEA J " > +IE@AHJDABMECJMIJ=JAAJI 5J=JAAJ1.H! J "#JDALA?EJOBJDA>A?JEI@A?HA=IEC 5J=JAAJ11.H! J "#JDAIFAA@BJDA>A?JEIE?HA=IEC )HAAEJDAHH>JDBJDAIAIJ=JAAJI?HHA?J.HA=?DIJ=JAAJFHLE@A=HA=IMDOEJEI?HHA?JHJ?HHA?J? 9D=JEIJDAJJ=@EIJ=?AJH=LAA@>OJDA>A?JLAHJDAJEAEJAHL= > J > 9D=JEIJDAFIEJEBJDA>A?J=JJEA J " " = = " L= "?I!! Г HГ#! HГ# " $ =IMAH >! J "# =J L= J?I J!! 5J=JAAJ1EI?HHA?JIE?A=J 5J=JAAJ11EI?HHA?JIE?A LJ =@ =J?,EIJ=?A " NJ Г?I J! N N" '!"!!' LJ MDAJ! $ N!!''&$ N! ГN N" ГN! N" N NJ Г?I J! N" '!"! " Copyrigh by College Enrance Examinaion Board. All righs reserved. Advanced Placemen Program and AP are regisered 98 rademarks of he College Enrance Examinaion Board. 1?HHA?JMEJDHA=I! 11?HHA?J HA=IBH11 EEJIB=@"=EJACH= BLJ H LJ H D=@AI?D=CAB@EHA?JE=J IJK@AJ\IJKHECFEJ =IMAH EBE?HHA?JJKHECFEJH JKHECFEJ EJACH= =IMAH! NJ Г?I J +! =IMAH EB?IJ=JBEJACH=JE

16

AP CALCULUS AB 2004 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2004 SCORING GUIDELINES (Form B) 4 SCORING GUIDELINES (Form B) Quesion A es plane flies in a sraigh line wih (min) 5 1 15 5 5 4 posiive velociy v (), in miles per v ()(mpm) 7. 9. 9.5 7. 4.5.4.4 4. 7. minue a ime minues, where v is a differeniable

More information

AP CALCULUS AB 2017 SCORING GUIDELINES

AP CALCULUS AB 2017 SCORING GUIDELINES AP CALCULUS AB 17 SCORING GUIDELINES 16 SCORING GUIDELINES Quesion For, a paricle moves along he x-axis. The velociy of he paricle a ime is given by v ( ) = 1 + sin. The paricle is a posiion x = a ime.

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B) SCING GUIDELINES (Form B) Quesion 4 A paricle moves along he x-axis wih velociy a ime given by v( ) = 1 + e1. (a) Find he acceleraion of he paricle a ime =. (b) Is he speed of he paricle increasing a ime

More information

AP CALCULUS BC 2016 SCORING GUIDELINES

AP CALCULUS BC 2016 SCORING GUIDELINES 6 SCORING GUIDELINES Quesion A ime, he posiion of a paricle moving in he xy-plane is given by he parameric funcions ( x ( ), y ( )), where = + sin ( ). The graph of y, consising of hree line segmens, is

More information

AP Calculus BC - Parametric equations and vectors Chapter 9- AP Exam Problems solutions

AP Calculus BC - Parametric equations and vectors Chapter 9- AP Exam Problems solutions AP Calculus BC - Parameric equaions and vecors Chaper 9- AP Exam Problems soluions. A 5 and 5. B A, 4 + 8. C A, 4 + 4 8 ; he poin a is (,). y + ( x ) x + 4 4. e + e D A, slope.5 6 e e e 5. A d hus d d

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B) SCORING GUIDELINES (Form B) Quesion A blood vessel is 6 millimeers (mm) long Disance wih circular cross secions of varying diameer. x (mm) 6 8 4 6 Diameer The able above gives he measuremens of he B(x)

More information

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr AP CALCULUS AB/CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) 1 3 6 8 134 119 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for 8, where is measured

More information

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B)

AP CALCULUS AB 2003 SCORING GUIDELINES (Form B) SCORING GUIDELINES (Form B) Quesion A ank conains 15 gallons of heaing oil a ime =. During he ime inerval 1 hours, heaing oil is pumped ino he ank a he rae 1 H ( ) = + ( 1 + ln( + 1) ) gallons per hour.

More information

AP CALCULUS AB 2017 SCORING GUIDELINES

AP CALCULUS AB 2017 SCORING GUIDELINES AP CALCULUS AB 07 SCORING GUIDELINES 06 SCORING GUIDELINES Question 6 f ( ) f ( ) g( ) g ( ) 6 8 0 8 7 6 6 5 The functions f and g have continuous second derivatives. The table above gives values of the

More information

AP CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES. Question 1

AP CALCULUS AB/CALCULUS BC 2015 SCORING GUIDELINES. Question 1 AP CALCULUS AB/CALCULUS BC 15 SCORING GUIDELINES Quesion 1 The rae a which rainwaer flows ino a drainpipe is modeled by he funcion R, where R ( ) = sin 5 cubic fee per hour, is measured in hours, and 8.

More information

1998 Calculus AB Scoring Guidelines

1998 Calculus AB Scoring Guidelines AB{ / BC{ 1999. The rae a which waer ows ou of a pipe, in gallons per hour, is given by a diereniable funcion R of ime. The able above shows he rae as measured every hours for a {hour period. (a) Use a

More information

Multiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1

Multiple Choice Solutions 1. E (2003 AB25) () xt t t t 2. A (2008 AB21/BC21) 3. B (2008 AB7) Using Fundamental Theorem of Calculus: 1 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,

More information

AP CALCULUS AB 2017 SCORING GUIDELINES

AP CALCULUS AB 2017 SCORING GUIDELINES AP CALCULUS AB 17 SCORING GUIDELINES /CALCULUS BC 15 SCORING GUIDELINES Quesion (minues) v ( ) (meers per minue) 1 4 4 4 15 Johanna jogs along a sraigh pah. For 4, Johanna s velociy is given by a differeniable

More information

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1

AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1 AP CALCULUS AB/CALCULUS BC 6 SCORING GUIDELINES Quesion (hours) R ( ) (liers / hour) 6 4 9 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for, where is measured in hours. Waer

More information

AP Calculus BC 2004 Free-Response Questions Form B

AP Calculus BC 2004 Free-Response Questions Form B AP Calculus BC 200 Free-Response Quesions Form B The maerials included in hese files are inended for noncommercial use by AP eachers for course and exam preparaion; permission for any oher use mus be sough

More information

AP CALCULUS AB 2017 SCORING GUIDELINES

AP CALCULUS AB 2017 SCORING GUIDELINES AP CALCULUS AB 17 SCORING GUIDELINES /CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) 1 6 8 14 119 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for

More information

!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)

!!#$%&#'()!#&'(*%)+,&',-)./0)1-*23) "#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5

More information

MEI Mechanics 1 General motion. Section 1: Using calculus

MEI Mechanics 1 General motion. Section 1: Using calculus Soluions o Exercise MEI Mechanics General moion Secion : Using calculus. s 4 v a 6 4 4 When =, v 4 a 6 4 6. (i) When = 0, s = -, so he iniial displacemen = - m. s v 4 When = 0, v = so he iniial velociy

More information

Parametrics and Vectors (BC Only)

Parametrics and Vectors (BC Only) Paramerics and Vecors (BC Only) The following relaionships should be learned and memorized. The paricle s posiion vecor is r() x(), y(). The velociy vecor is v(),. The speed is he magniude of he velociy

More information

AP Calculus BC Chapter 10 Part 1 AP Exam Problems

AP Calculus BC Chapter 10 Part 1 AP Exam Problems AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a

More information

Answers to 1 Homework

Answers to 1 Homework Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:

More information

Q2.1 This is the x t graph of the motion of a particle. Of the four points P, Q, R, and S, the velocity v x is greatest (most positive) at

Q2.1 This is the x t graph of the motion of a particle. Of the four points P, Q, R, and S, the velocity v x is greatest (most positive) at Q2.1 This is he x graph of he moion of a paricle. Of he four poins P, Q, R, and S, he velociy is greaes (mos posiive) a A. poin P. B. poin Q. C. poin R. D. poin S. E. no enough informaion in he graph o

More information

Physics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008

Physics 221 Fall 2008 Homework #2 Solutions Ch. 2 Due Tues, Sept 9, 2008 Physics 221 Fall 28 Homework #2 Soluions Ch. 2 Due Tues, Sep 9, 28 2.1 A paricle moving along he x-axis moves direcly from posiion x =. m a ime =. s o posiion x = 1. m by ime = 1. s, and hen moves direcly

More information

SPH3U1 Lesson 03 Kinematics

SPH3U1 Lesson 03 Kinematics SPH3U1 Lesson 03 Kinemaics GRAPHICAL ANALYSIS LEARNING GOALS Sudens will Learn how o read values, find slopes and calculae areas on graphs. Learn wha hese values mean on boh posiion-ime and velociy-ime

More information

Practicing Problem Solving and Graphing

Practicing Problem Solving and Graphing Pracicing Problem Solving and Graphing Tes 1: Jan 30, 7pm, Ming Hsieh G20 The Bes Ways To Pracice for Tes Bes If need more, ry suggesed problems from each new opic: Suden Response Examples A pas opic ha

More information

Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan

Ground Rules. PC1221 Fundamentals of Physics I. Kinematics. Position. Lectures 3 and 4 Motion in One Dimension. A/Prof Tay Seng Chuan Ground Rules PC11 Fundamenals of Physics I Lecures 3 and 4 Moion in One Dimension A/Prof Tay Seng Chuan 1 Swich off your handphone and pager Swich off your lapop compuer and keep i No alking while lecure

More information

Speed and Velocity. Overview. Velocity & Speed. Speed & Velocity. Instantaneous Velocity. Instantaneous and Average

Speed and Velocity. Overview. Velocity & Speed. Speed & Velocity. Instantaneous Velocity. Instantaneous and Average Overview Kinemaics: Descripion of Moion Posiion and displacemen velociy»insananeous acceleraion»insananeous Speed Velociy Speed and Velociy Speed & Velociy Velociy & Speed A physics eacher walks 4 meers

More information

4.5 Constant Acceleration

4.5 Constant Acceleration 4.5 Consan Acceleraion v() v() = v 0 + a a() a a() = a v 0 Area = a (a) (b) Figure 4.8 Consan acceleraion: (a) velociy, (b) acceleraion When he x -componen of he velociy is a linear funcion (Figure 4.8(a)),

More information

2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance

2.1: What is physics? Ch02: Motion along a straight line. 2.2: Motion. 2.3: Position, Displacement, Distance Ch: Moion along a sraigh line Moion Posiion and Displacemen Average Velociy and Average Speed Insananeous Velociy and Speed Acceleraion Consan Acceleraion: A Special Case Anoher Look a Consan Acceleraion

More information

KINEMATICS IN ONE DIMENSION

KINEMATICS IN ONE DIMENSION KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec

More information

IB Physics Kinematics Worksheet

IB Physics Kinematics Worksheet IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?

More information

Welcome Back to Physics 215!

Welcome Back to Physics 215! Welcome Back o Physics 215! (General Physics I) Thurs. Jan 19 h, 2017 Lecure01-2 1 Las ime: Syllabus Unis and dimensional analysis Today: Displacemen, velociy, acceleraion graphs Nex ime: More acceleraion

More information

Kinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8.

Kinematics Vocabulary. Kinematics and One Dimensional Motion. Position. Coordinate System in One Dimension. Kinema means movement 8. Kinemaics Vocabulary Kinemaics and One Dimensional Moion 8.1 WD1 Kinema means movemen Mahemaical descripion of moion Posiion Time Inerval Displacemen Velociy; absolue value: speed Acceleraion Averages

More information

MEI STRUCTURED MATHEMATICS 4758

MEI STRUCTURED MATHEMATICS 4758 OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Cerificae of Educaion Advanced General Cerificae of Educaion MEI STRUCTURED MATHEMATICS 4758 Differenial Equaions Thursday 5 JUNE 006 Afernoon

More information

x(m) t(sec ) Homework #2. Ph 231 Introductory Physics, Sp-03 Page 1 of 4

x(m) t(sec ) Homework #2. Ph 231 Introductory Physics, Sp-03 Page 1 of 4 Homework #2. Ph 231 Inroducory Physics, Sp-03 Page 1 of 4 2-1A. A person walks 2 miles Eas (E) in 40 minues and hen back 1 mile Wes (W) in 20 minues. Wha are her average speed and average velociy (in ha

More information

3.6 Derivatives as Rates of Change

3.6 Derivatives as Rates of Change 3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe

More information

4.6 One Dimensional Kinematics and Integration

4.6 One Dimensional Kinematics and Integration 4.6 One Dimensional Kinemaics and Inegraion When he acceleraion a( of an objec is a non-consan funcion of ime, we would like o deermine he ime dependence of he posiion funcion x( and he x -componen of

More information

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures.

HOMEWORK # 2: MATH 211, SPRING Note: This is the last solution set where I will describe the MATLAB I used to make my pictures. HOMEWORK # 2: MATH 2, SPRING 25 TJ HITCHMAN Noe: This is he las soluion se where I will describe he MATLAB I used o make my picures.. Exercises from he ex.. Chaper 2.. Problem 6. We are o show ha y() =

More information

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version):

CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS. Second Fundamental Theorem of Calculus (Chain Rule Version): CALCULUS EXPLORATION OF THE SECOND FUNDAMENTAL THEOREM OF CALCULUS 6 cos Secon Funamenal Theorem of Calculus: f a 4 a f 6 cos Secon Funamenal Theorem of Calculus (Chain Rule Version): g f a E. Use he Secon

More information

Robotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.

Robotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1. Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L 1 0 1 0 0 L 3 0 0 L 3 3 able 1: able of DH parameers of

More information

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile

More information

UCLA: Math 3B Problem set 3 (solutions) Fall, 2018

UCLA: Math 3B Problem set 3 (solutions) Fall, 2018 UCLA: Mah 3B Problem se 3 (soluions) Fall, 28 This problem se concenraes on pracice wih aniderivaives. You will ge los of pracice finding simple aniderivaives as well as finding aniderivaives graphically

More information

1. VELOCITY AND ACCELERATION

1. VELOCITY AND ACCELERATION 1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under

More information

2. What is the displacement of the bug between t = 0.00 s and t = 20.0 s? A) cm B) 39.9 cm C) cm D) 16.1 cm E) +16.

2. What is the displacement of the bug between t = 0.00 s and t = 20.0 s? A) cm B) 39.9 cm C) cm D) 16.1 cm E) +16. 1. For which one of he following siuaions will he pah lengh equal he magniude of he displacemen? A) A jogger is running around a circular pah. B) A ball is rolling down an inclined plane. C) A rain ravels

More information

Topics covered in tutorial 01: 1. Review of definite integrals 2. Physical Application 3. Area between curves. 1. Review of definite integrals

Topics covered in tutorial 01: 1. Review of definite integrals 2. Physical Application 3. Area between curves. 1. Review of definite integrals MATH4 Calculus II (8 Spring) MATH 4 Tuorial Noes Tuorial Noes (Phyllis LIANG) IA: Phyllis LIANG Email: masliang@us.hk Homepage: hps://masliang.people.us.hk Office: Room 3 (Lif/Lif 3) Phone number: 3587453

More information

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.

1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx. . Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.

More information

ln y t 2 t c where c is an arbitrary real constant

ln y t 2 t c where c is an arbitrary real constant SOLUTION TO THE PROBLEM.A y y subjec o condiion y 0 8 We recognize is as a linear firs order differenial equaion wi consan coefficiens. Firs we sall find e general soluion, and en we sall find one a saisfies

More information

a 10.0 (m/s 2 ) 5.0 Name: Date: 1. The graph below describes the motion of a fly that starts out going right V(m/s)

a 10.0 (m/s 2 ) 5.0 Name: Date: 1. The graph below describes the motion of a fly that starts out going right V(m/s) Name: Dae: Kinemaics Review (Honors. Physics) Complee he following on a separae shee of paper o be urned in on he day of he es. ALL WORK MUST BE SHOWN TO RECEIVE CREDIT. 1. The graph below describes he

More information

MOMENTUM CONSERVATION LAW

MOMENTUM CONSERVATION LAW 1 AAST/AEDT AP PHYSICS B: Impulse and Momenum Le us run an experimen: The ball is moving wih a velociy of V o and a force of F is applied on i for he ime inerval of. As he resul he ball s velociy changes

More information

Variable acceleration, Mixed Exercise 11

Variable acceleration, Mixed Exercise 11 Variable acceleraion, Mixed Exercise 11 1 a v 1 P is a res when v 0. 0 1 b s 0 0 v d (1 ) 1 0 1 0 7. The disance ravelled by P is 7. m. 1 a v 6+ a d v 6 + When, a 6+ 0 The acceleraion of P when is 0 m

More information

15. Vector Valued Functions

15. Vector Valued Functions 1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,

More information

Physics 20 Lesson 5 Graphical Analysis Acceleration

Physics 20 Lesson 5 Graphical Analysis Acceleration Physics 2 Lesson 5 Graphical Analysis Acceleraion I. Insananeous Velociy From our previous work wih consan speed and consan velociy, we know ha he slope of a posiion-ime graph is equal o he velociy of

More information

1. Kinematics I: Position and Velocity

1. Kinematics I: Position and Velocity 1. Kinemaics I: Posiion and Velociy Inroducion The purpose of his eperimen is o undersand and describe moion. We describe he moion of an objec by specifying is posiion, velociy, and acceleraion. In his

More information

Phys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole

Phys 221 Fall Chapter 2. Motion in One Dimension. 2014, 2005 A. Dzyubenko Brooks/Cole Phys 221 Fall 2014 Chaper 2 Moion in One Dimension 2014, 2005 A. Dzyubenko 2004 Brooks/Cole 1 Kinemaics Kinemaics, a par of classical mechanics: Describes moion in erms of space and ime Ignores he agen

More information

PHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections

PHYSICS 220 Lecture 02 Motion, Forces, and Newton s Laws Textbook Sections PHYSICS 220 Lecure 02 Moion, Forces, and Newon s Laws Texbook Secions 2.2-2.4 Lecure 2 Purdue Universiy, Physics 220 1 Overview Las Lecure Unis Scienific Noaion Significan Figures Moion Displacemen: Δx

More information

Kinematics Motion in 1 Dimension and Graphs

Kinematics Motion in 1 Dimension and Graphs Kinemaics Moion in 1 Dimension and Graphs Lana Sheridan De Anza College Sep 27, 2017 Las ime moion in 1-dimension some kinemaic quaniies graphs Overview velociy and speed acceleraion more graphs Kinemaics

More information

PHYS 100: Lecture 2. Motion at Constant Acceleration. Relative Motion: Reference Frames. x x = v t + a t. x = vdt. v = adt. x Tortoise.

PHYS 100: Lecture 2. Motion at Constant Acceleration. Relative Motion: Reference Frames. x x = v t + a t. x = vdt. v = adt. x Tortoise. a PHYS 100: Lecure 2 Moion a Consan Acceleraion a 0 0 Area a 0 a 0 v ad v v0 a0 v 0 x vd 0 A(1/2)( v) Area v 0 v v-v 0 v 0 x x v + a 1 0 0 2 0 2 Relaive Moion: Reference Frames x d Achilles Toroise x Toroise

More information

Chapter 15 Oscillatory Motion I

Chapter 15 Oscillatory Motion I Chaper 15 Oscillaory Moion I Level : AP Physics Insrucor : Kim Inroducion A very special kind of moion occurs when he force acing on a body is proporional o he displacemen of he body from some equilibrium

More information

Suggested Practice Problems (set #2) for the Physics Placement Test

Suggested Practice Problems (set #2) for the Physics Placement Test Deparmen of Physics College of Ars and Sciences American Universiy of Sharjah (AUS) Fall 014 Suggesed Pracice Problems (se #) for he Physics Placemen Tes This documen conains 5 suggesed problems ha are

More information

The Fundamental Theorem of Calculus Solutions

The Fundamental Theorem of Calculus Solutions The Fundamenal Theorem of Calculus 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

More information

72 Calculus and Structures

72 Calculus and Structures 72 Calculus and Srucures CHAPTER 5 DISTANCE AND ACCUMULATED CHANGE Calculus and Srucures 73 Copyrigh Chaper 5 DISTANCE AND ACCUMULATED CHANGE 5. DISTANCE a. Consan velociy Le s ake anoher look a Mary s

More information

Instructor: Barry McQuarrie Page 1 of 5

Instructor: Barry McQuarrie Page 1 of 5 Procedure for Solving radical equaions 1. Algebraically isolae one radical by iself on one side of equal sign. 2. Raise each side of he equaion o an appropriae power o remove he radical. 3. Simplify. 4.

More information

Solution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration

Solution: b All the terms must have the dimension of acceleration. We see that, indeed, each term has the units of acceleration PHYS 54 Tes Pracice Soluions Spring 8 Q: [4] Knowing ha in he ne epression a is acceleraion, v is speed, is posiion and is ime, from a dimensional v poin of view, he equaion a is a) incorrec b) correc

More information

a. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines.

a. Show that these lines intersect by finding the point of intersection. b. Find an equation for the plane containing these lines. Mah A Final Eam Problems for onsideraion. Show all work for credi. Be sure o show wha you know. Given poins A(,,, B(,,, (,, 4 and (,,, find he volume of he parallelepiped wih adjacen edges AB, A, and A.

More information

Math 115 Final Exam December 14, 2017

Math 115 Final Exam December 14, 2017 On my honor, as a suden, I have neiher given nor received unauhorized aid on his academic work. Your Iniials Only: Iniials: Do no wrie in his area Mah 5 Final Exam December, 07 Your U-M ID # (no uniqname):

More information

PHYSICS 149: Lecture 9

PHYSICS 149: Lecture 9 PHYSICS 149: Lecure 9 Chaper 3 3.2 Velociy and Acceleraion 3.3 Newon s Second Law of Moion 3.4 Applying Newon s Second Law 3.5 Relaive Velociy Lecure 9 Purdue Universiy, Physics 149 1 Velociy (m/s) The

More information

Displacement ( x) x x x

Displacement ( x) x x x Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh

More information

Mechanics Acceleration The Kinematics Equations

Mechanics Acceleration The Kinematics Equations Mechanics Acceleraion The Kinemaics Equaions Lana Sheridan De Anza College Sep 27, 2018 Las ime kinemaic quaniies graphs of kinemaic quaniies Overview acceleraion he kinemaics equaions (consan acceleraion)

More information

02. MOTION. Questions and Answers

02. MOTION. Questions and Answers CLASS-09 02. MOTION Quesions and Answers PHYSICAL SCIENCE 1. Se moves a a consan speed in a consan direcion.. Reprase e same senence in fewer words using conceps relaed o moion. Se moves wi uniform velociy.

More information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in

More information

Math 116 Second Midterm March 21, 2016

Math 116 Second Midterm March 21, 2016 Mah 6 Second Miderm March, 06 UMID: EXAM SOLUTIONS Iniials: Insrucor: Secion:. Do no open his exam unil you are old o do so.. Do no wrie your name anywhere on his exam. 3. This exam has pages including

More information

PHYS 100: Lecture 2. Motion at Constant Acceleration. Relative Motion: Reference Frames. x Tortoise. Tortoise. d Achilles. Reference frame = Earth

PHYS 100: Lecture 2. Motion at Constant Acceleration. Relative Motion: Reference Frames. x Tortoise. Tortoise. d Achilles. Reference frame = Earth a PHYS 1: Lecure 2 Moion a Consan Acceleraion a Area = a a v = ad v v = a v x = vd A=(1/2)( v) Area = v v = v-v v x x = v + a 1 2 2 Relaive Moion: Reference Frames x d Achilles Toroise x Toroise Reference

More information

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:

Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities: Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial

More information

CHAPTER 12 DIRECT CURRENT CIRCUITS

CHAPTER 12 DIRECT CURRENT CIRCUITS CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As

More information

2002 November 14 Exam III Physics 191

2002 November 14 Exam III Physics 191 November 4 Exam III Physics 9 Physical onsans: Earh s free-fall acceleraion = g = 9.8 m/s ircle he leer of he single bes answer. quesion is worh poin Each 3. Four differen objecs wih masses: m = kg, m

More information

Week #13 - Integration by Parts & Numerical Integration Section 7.2

Week #13 - Integration by Parts & Numerical Integration Section 7.2 Week #3 - Inegraion by Pars & Numerical Inegraion Secion 7. From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 5 by John Wiley & Sons, Inc. This maerial is used by permission

More information

Module 3: The Damped Oscillator-II Lecture 3: The Damped Oscillator-II

Module 3: The Damped Oscillator-II Lecture 3: The Damped Oscillator-II Module 3: The Damped Oscillaor-II Lecure 3: The Damped Oscillaor-II 3. Over-damped Oscillaions. This refers o he siuaion where β > ω (3.) The wo roos are and α = β + α 2 = β β 2 ω 2 = (3.2) β 2 ω 2 = 2

More information

University Physics with Modern Physics 14th Edition Young TEST BANK

University Physics with Modern Physics 14th Edition Young TEST BANK Universi Phsics wih Modern Phsics 14h Ediion Young SOLUTIONS MANUAL Full clear download (no formaing errors) a: hps://esbankreal.com/download/universi-phsics-modern-phsics- 14h-ediion-oung-soluions-manual-/

More information

SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.

SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15. SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a

More information

k 1 k 2 x (1) x 2 = k 1 x 1 = k 2 k 1 +k 2 x (2) x k series x (3) k 2 x 2 = k 1 k 2 = k 1+k 2 = 1 k k 2 k series

k 1 k 2 x (1) x 2 = k 1 x 1 = k 2 k 1 +k 2 x (2) x k series x (3) k 2 x 2 = k 1 k 2 = k 1+k 2 = 1 k k 2 k series Final Review A Puzzle... Consider wo massless springs wih spring consans k 1 and k and he same equilibrium lengh. 1. If hese springs ac on a mass m in parallel, hey would be equivalen o a single spring

More information

Giambattista, Ch 3 Problems: 9, 15, 21, 27, 35, 37, 42, 43, 47, 55, 63, 76

Giambattista, Ch 3 Problems: 9, 15, 21, 27, 35, 37, 42, 43, 47, 55, 63, 76 Giambaisa, Ch 3 Problems: 9, 15, 21, 27, 35, 37, 42, 43, 47, 55, 63, 76 9. Sraeg Le be direced along he +x-axis and le be 60.0 CCW from Find he magniude of 6.0 B 60.0 4.0 A x 15. (a) Sraeg Since he angle

More information

Of all of the intellectual hurdles which the human mind has confronted and has overcome in the last fifteen hundred years, the one which seems to me

Of all of the intellectual hurdles which the human mind has confronted and has overcome in the last fifteen hundred years, the one which seems to me Of all of he inellecual hurdles which he human mind has confroned and has overcome in he las fifeen hundred years, he one which seems o me o have been he mos amazing in characer and he mos supendous in

More information

Principle of Least Action

Principle of Least Action The Based on par of Chaper 19, Volume II of The Feynman Lecures on Physics Addison-Wesley, 1964: pages 19-1 hru 19-3 & 19-8 hru 19-9. Edwin F. Taylor July. The Acion Sofware The se of exercises on Acion

More information

In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should

In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion

More information

ACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1

ACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1 ACCUMULATION Secion 7.5 Calculus AP/Dual, Revised 2019 vie.dang@humbleisd.ne 12/26/2018 7:27 PM 7.5A: Accumulaion 1 APPLICATION PROBLEMS A. Undersand he quesion. I is ofen no necessary o as much compuaion

More information

WELCOME TO 1103 PERIOD 3. Homework Exercise #2 is due at the beginning of class. Please put it on the stool in the front of the classroom.

WELCOME TO 1103 PERIOD 3. Homework Exercise #2 is due at the beginning of class. Please put it on the stool in the front of the classroom. WELCOME TO 1103 PERIOD 3 Homework Exercise #2 is due a he beginning of class. Please pu i on he sool in he fron of he classroom. Ring of Truh: Change 1) Give examples of some energy ransformaions in he

More information

Kinematics in One Dimension

Kinematics in One Dimension Kinemaics in One Dimension PHY 7 - d-kinemaics - J. Hedberg - 7. Inroducion. Differen Types of Moion We'll look a:. Dimensionaliy in physics 3. One dimensional kinemaics 4. Paricle model. Displacemen Vecor.

More information

PROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES

PROBLEMS FOR MATH 162 If a problem is starred, all subproblems are due. If only subproblems are starred, only those are due. SLOPES OF TANGENT LINES PROBLEMS FOR MATH 6 If a problem is sarred, all subproblems are due. If onl subproblems are sarred, onl hose are due. 00. Shor answer quesions. SLOPES OF TANGENT LINES (a) A ball is hrown ino he air. Is

More information

A. Using Newton s second law in one dimension, F net. , write down the differential equation that governs the motion of the block.

A. Using Newton s second law in one dimension, F net. , write down the differential equation that governs the motion of the block. Simple SIMPLE harmonic HARMONIC moion MOTION I. Differenial equaion of moion A block is conneced o a spring, one end of which is aached o a wall. (Neglec he mass of he spring, and assume he surface is

More information

Physics 101 Fall 2006: Exam #1- PROBLEM #1

Physics 101 Fall 2006: Exam #1- PROBLEM #1 Physics 101 Fall 2006: Exam #1- PROBLEM #1 1. Problem 1. (+20 ps) (a) (+10 ps) i. +5 ps graph for x of he rain vs. ime. The graph needs o be parabolic and concave upward. ii. +3 ps graph for x of he person

More information

Ch.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC

Ch.1. Group Work Units. Continuum Mechanics Course (MMC) - ETSECCPB - UPC Ch.. Group Work Unis Coninuum Mechanics Course (MMC) - ETSECCPB - UPC Uni 2 Jusify wheher he following saemens are rue or false: a) Two sreamlines, corresponding o a same insan of ime, can never inersec

More information

Section 3.8, Mechanical and Electrical Vibrations

Section 3.8, Mechanical and Electrical Vibrations Secion 3.8, Mechanical and Elecrical Vibraions Mechanical Unis in he U.S. Cusomary and Meric Sysems Disance Mass Time Force g (Earh) Uni U.S. Cusomary MKS Sysem CGS Sysem fee f slugs seconds sec pounds

More information

t + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that

t + t sin t t cos t sin t. t cos t sin t dt t 2 = exp 2 log t log(t cos t sin t) = Multiplying by this factor and then integrating, we conclude that ODEs, Homework #4 Soluions. Check ha y ( = is a soluion of he second-order ODE ( cos sin y + y sin y sin = 0 and hen use his fac o find all soluions of he ODE. When y =, we have y = and also y = 0, so

More information

Physics for Scientists and Engineers I

Physics for Scientists and Engineers I Physics for Scieniss and Engineers I PHY 48, Secion 4 Dr. Beariz Roldán Cuenya Universiy of Cenral Florida, Physics Deparmen, Orlando, FL Chaper - Inroducion I. General II. Inernaional Sysem of Unis III.

More information

CHEAPEST PMT ONLINE TEST SERIES AIIMS/NEET TOPPER PREPARE QUESTIONS

CHEAPEST PMT ONLINE TEST SERIES AIIMS/NEET TOPPER PREPARE QUESTIONS CHEAPEST PMT ONLINE TEST SERIES AIIMS/NEET TOPPER PREPARE QUESTIONS For more deails see las page or conac @aimaiims.in Physics Mock Tes Paper AIIMS/NEET 07 Physics 06 Saurday Augus 0 Uni es : Moion in

More information

Math 23 Spring Differential Equations. Final Exam Due Date: Tuesday, June 6, 5pm

Math 23 Spring Differential Equations. Final Exam Due Date: Tuesday, June 6, 5pm Mah Spring 6 Differenial Equaions Final Exam Due Dae: Tuesday, June 6, 5pm Your name (please prin): Insrucions: This is an open book, open noes exam. You are free o use a calculaor or compuer o check your

More information

v x + v 0 x v y + a y + v 0 y + 2a y + v y Today: Projectile motion Soccer problem Firefighter example

v x + v 0 x v y + a y + v 0 y + 2a y + v y Today: Projectile motion Soccer problem Firefighter example Thurs Sep 10 Assign 2 Friday SI Sessions: Moron 227 Mon 8:10-9:10 PM Tues 8:10-9:10 PM Thur 7:05-8:05 PM Read Read Draw/Image lay ou coordinae sysem Wha know? Don' know? Wan o know? Physical Processes?

More information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,

More information

Dynamics. Option topic: Dynamics

Dynamics. Option topic: Dynamics Dynamics 11 syllabusref Opion opic: Dynamics eferenceence In his cha chaper 11A Differeniaion and displacemen, velociy and acceleraion 11B Inerpreing graphs 11C Algebraic links beween displacemen, velociy

More information