Fundamental Units. Then, watch out. Page 1. SPH4UI: Lecture 1 Course Info & Advice. Kinematics

Size: px

Start display at page:

Download "Fundamental Units. Then, watch out. Page 1. SPH4UI: Lecture 1 Course Info & Advice. Kinematics"

Calvin Pope
5 years ago
Views:

1 SPH4UI: Lecure 1 Course Info & Adice Course has seeral componens: Lecure: (me alkin, demos and ou askin quesions) Discussion secions (uorials, problem solin, quizzes) Homework Web based Labs: (roup eploraion of phsical phenomena) Wha happens if ou miss a lab or class es Read noes from online Wha if ou are ecused?? (Wha ou need o do.) Tha opic of our eam will be ealuaed as our es Kinemaics The firs few weeks of he course should be reiew, hence he pace is fas. I is imporan for ou o keep up! Then, wach ou. Fundamenal Unis How we measure hins! All hins in classical mechanics can be epressed in erms of he fundamenal unis: Lenh: L Mass : M Time : T For eample: Speed has unis of L / T (e.. miles per hour). Force has unis of ML / T ec... (as ou will learn). Pae 1

You should know where o look o coner back & forh. Eer heard of Goole Conerin beween differen ssems of unis Useful Conersion facors: 1 inch =.

2 Unis... SI (Ssème Inernaional) Unis: mks: L = meers (m), M = kilorams (k), T = seconds (s) cs: L = cenimeers (cm), M = rams (m), T = seconds (s) Briish Unis: Inches, fee, miles, pounds, slus... We will use mosl SI unis, bu ou ma run across some problems usin Briish unis. You should know where o look o coner back & forh. Eer heard of Goole Conerin beween differen ssems of unis Useful Conersion facors: 1 inch =.54 cm 1 m = 3.8 f 1 mile = 580 f 1 mile = 1.61 km Eample: coner miles per hour o meers per second: mi mi f 1 m 1 hr m hr hr mi 3.8 f 3600 s s Pae

3 Dimensional Analsis This is a er imporan ool o check our work I s also er eas! Eample: Doin a problem ou e he answer disance d = (eloci ime ) Unis on lef side = L Unis on rih side = L / T T = L T Lef unis and rih unis don mach, so answer mus be wron!! Dimensional Analsis The period P of a swinin pendulum depends onl on he lenh of he pendulum d and he acceleraion of rai. Which of he followin formulas for P could be correc? (a) P = (d) P d d (b) (c) P Gien: d has unis of lenh (L) and has unis of (L / T ). Soluion Soluion Realize ha he lef hand side P has unis of ime (T ) Tr he firs equaion Realize ha he lef hand side P has unis of ime (T ) Tr he second equaion (a) 4 L L L T 4 T T No Rih! (b) L L T T T No Rih! (a) P d (b) P d (c) P d (a) P d (b) P d (c) P d Pae 3

4 Soluion Realize ha he lef hand side P has unis of ime (T ) Tr he firs equaion (c) L L T T T Dude, his is i (a) P d (b) P d (c) d P Vecors A ecor is a quani ha inoles boh maniude and direcion. 55 km/h [N35E] A downward force of 3 Newons A scalar is a quani ha does no inole direcion. 55 km/h 18 cm lon Vecor Noaion Vecors are ofen idenified wih arrows in raphics and labeled as follows: Displacemen Displacemen is an objec s chane in posiion. Disance is he oal lenh of space raersed b an objec. We label a ecor wih a ariable. This ariable is idenified as a ecor eiher b an arrow aboe iself : A Or B he ariable bein BOLD: A 1m 3m Displacemen: Disance: 6.7m 5m 6m 3m 6.7m 5m 3m 1m 9m Sar Finish = 500 m Displacemen = 0 m Disance = 500m Pae 4

5 E R D Vecor Addiion A B C D R E B A C B A D R E A + B + C + D + E = Disance R = Resulan = Displacemen C R A B Recanular Componens - A opp sin R hp B adj cos R hp A opp an B adj Quadran II Rsin Quadran III A - Quadran I R B Rcos Quadran IV Vecors... The componens (in a paricular coordinae ssem) of r, he posiion ecor, are is (,,z) coordinaes in ha coordinae ssem r = (r,r,r z ) = (,,z) Consider his in -D (since i s easier o draw): r = = r cos where r = r r = = r sin (,) r arcan( / ) Vecors... The maniude (lenh) of r is found usin he Phaorean heorem: r r r The lenh of a ecor clearl does no depend on is direcion. Pae 5

6 Vecor A = (0,,1) Vecor B = (3,0,) Vecor C = (1,-4,) Vecor Eample Wha is he resulan ecor, D, from addin A+B+C? (a) (3,5,-1) (b) (4,-,5) (c) (5,-,4) Resulan of Two Forces force: acion of one bod on anoher; characerized b is poin of applicaion, maniude, line of acion, and sense. Eperimenal eidence shows ha he combined effec of wo forces ma be represened b a sinle resulan force. The resulan is equialen o he diaonal of a paralleloram which conains he wo forces in adjacen les. Force is a ecor quani. P Q P -P Vecors Vecor: parameers possessin maniude and direcion which add accordin o he paralleloram law. Eamples: displacemens, elociies, acceleraions. Scalar: parameers possessin maniude bu no direcion. Eamples: mass, olume, emperaure Vecor classificaions: - Fied or bound ecors hae well defined poins of applicaion ha canno be chaned wihou affecin an analsis. - Free ecors ma be freel moed in space wihou chanin heir effec on an analsis. - Slidin ecors ma be applied anwhere alon heir line of acion wihou affecin an analsis. Equal ecors hae he same maniude and direcion. Neaie ecor of a ien ecor has he same maniude and he opposie direcion. P P Q Q Q -Q P-Q Q Addiion of Vecors P P Trapezoid rule for ecor addiion Trianle rule for ecor addiion Law of cosines, R P Q PQcos B R P Q Law of sines, sin A sin B sinc Q R A Vecor addiion is commuaie, P Q Q P Vecor subracion P Q P Q Pae 6

P Q Addiion of Vecors S Addiion of hree or more ecors hrouh repeaed applicaion of he rianle rule Resulan of Seeral Concurren Forces Concurren forces: se of forces which all pass hrouh he same poin.

7 P Q Addiion of Vecors S Addiion of hree or more ecors hrouh repeaed applicaion of he rianle rule Resulan of Seeral Concurren Forces Concurren forces: se of forces which all pass hrouh he same poin. P P Q S P -1.5P The polon rule for he addiion of hree or more ecors. Vecor addiion is associaie, P Q S P Q S P Q S Muliplicaion of a ecor b a scalar increases is lenh b ha facor (if scalar is neaie, he direcion will also chane.) A se of concurren forces applied o a paricle ma be replaced b a sinle resulan force which is he ecor sum of he applied forces. Vecor force componens: wo or more force ecors which, oeher, hae he same effec as a sinle force ecor. Soluion Two forces ac on a bol a A. Deermine heir resulan. Graphical soluion - consruc a paralleloram wih sides in he same direcion as P and Q and lenhs in proporion. Graphicall ealuae he resulan which is equialen in direcion and proporional in maniude o he diaonal. Trionomeric soluion - use he rianle rule for ecor addiion in conjuncion wih he law of cosines and law of sines o find he resulan. Q P R 98 N 35 R Graphical soluion - consruc a paralleloram wih sides in he same direcion as P and Q and lenhs in proporion. Graphicall ealuae he resulan which is equialen in direcion and proporional in maniude o he diaonal. Pae 7

Soluion Trionomeric soluion From he Law of Cosines, R P Q PQcos B 40N 60N 40N60Ncos155 R 97.73N From he Law of Sines, sin A sin B Q R Q sin A sin B R 60N sin155 97.73N A 15.04 0 A 35.

8 Soluion Trionomeric soluion From he Law of Cosines, R P Q PQcos B 40N 60N 40N60Ncos155 R 97.73N From he Law of Sines, sin A sin B Q R Q sin A sin B R 60N sin N A A A bare is pulled b wo uboas. If he resulan of he forces eered b he uboas is 5000 N direced alon he ais of he bare, deermine a) he ension in each of he ropes for α = 45 o, Find a raphical soluion b applin he Paralleloram Rule for ecor addiion. The paralleloram has sides in he direcions of he wo ropes and a diaonal in he direcion of he bare ais and lenh proporional o 5000 N. Find a rionomeric soluion b applin he Trianle Rule for ecor addiion. Wih he maniude and direcion of he resulan known and he direcions of he oher wo sides parallel o he ropes ien, appl he Law of Sines o find he rope ensions. Graphical soluion - Paralleloram Rule wih known resulan direcion and maniude, known direcions for sides. Recanular Componens of a Force: Uni Vecors Ma resole a force ecor ino perpendicular componens so ha he resulin paralleloram is a recanle. F and F are referred o as recanular ecor componens T T 45 T 5000N 5000N T 3700 N T 600 N 1 Trionomeric soluion - Trianle Rule wih Law of Sines T T N sin 45 sin30 sin105 T 3700 N T 600 N 105 T1 1 Define perpendicular uni ecors are parallel o he and aes. Vecor componens ma be epressed as producs of he uni ecors wih he scalar maniudes of he ecor componens. F F iˆf ˆj F and F are referred o as he scalar componens of F and F iˆ and ˆj which F Pae 8

S j Addiion of Forces b Summin Componens S Si R j Q P QiPi Q j R Ri Pj Wish o find he resulan of 3 or more concurren forces, R P Q S Resole each force ino recanular componens R i R j P i P j Q i Q j

R P Q S R P Q S F F To find he resulan maniude and direcion, R 1 R R R an R Four forces ac on bol A as shown. Deermine he resulan of he force on he bol.

9 S j Addiion of Forces b Summin Componens S Si R j Q P QiPi Q j R Ri Pj Wish o find he resulan of 3 or more concurren forces, R P Q S Resole each force ino recanular componens R i R j P i P j Q i Q j S i S j P Q S i P Q S j The scalar componens of he resulan are equal o he sum of he correspondin scalar componens of he ien forces. R P Q S R P Q S F F To find he resulan maniude and direcion, R 1 R R R an R Four forces ac on bol A as shown. Deermine he resulan of he force on he bol. Plan: Resole each force ino recanular componens. Deermine he componens of he resulan b addin he correspondin force componens. Calculae he maniude and direcion of he resulan. Soluion Calculae he maniude and direcion. R Resole each force ino recanular componens. force ma comp comp F F F F 14.3N an 199.1N R R 14.3 Deermine he componens of he resulan b addin he correspondin force componens. R 199.6N 4.1 Equilibrium of a Paricle When he resulan of all forces acin on a paricle is zero, he paricle is in equilibrium. Newon s Firs Law: If he resulan force on a paricle is zero, he paricle will remain a res or will coninue a consan speed in a sraih line. Paricle aced upon b wo forces: - equal maniude - same line of acion - opposie sense Paricle aced upon b hree or more forces: - raphical soluion ields a closed polon - alebraic soluion R F 0 F 0 F 0 Pae 9

Free-Bod Diarams Space Diaram: A skech showin he phsical condiions of he problem. T AB 50 A 30 TAC 736N Free-Bod Diaram: A skech showin onl he forces on he seleced paricle.

10 Free-Bod Diarams Space Diaram: A skech showin he phsical condiions of he problem. T AB 50 A 30 TAC 736N Free-Bod Diaram: A skech showin onl he forces on he seleced paricle. In a ship-unloadin operaion, a 3500-lb auomobile is suppored b a cable. A rope is ied o he cable and pulled o cener he auomobile oer is inended posiion. Wha is he ension in he rope? Plan of Aack: Consruc a free-bod diaram for he paricle a he juncion of he rope and cable. Appl he condiions for equilibrium b creain a closed polon from he forces applied o he paricle. Appl rionomeric relaions o deermine he unknown force maniudes. T AB A 30 T AC SOLUTION: Consruc a free-bod diaram for he paricle a A. Appl he condiions for equilibrium in he horizonal and erical direcions. horizonal Verical TAB TAC 0 cos88tab cos30tac 0 T T T 0 AB car AC costab 3500lb sin30tac 0 T AB A 30 T cos88 TAB cos30 TAC 0 costab 3500lb sin30t AC 0 AC cos88t cos30 AB sin30tac 3500lb TAB cos T AC cos88 sin30tac 3500lb cos30 cos T lb T lb AC T 144lb AC AC 3500lb 3500lb Pae 10

T AC 58 10 T AB 3500lb Sole for he unknown force maniudes usin Sine Law. TAB TAC 3500lb sin10 sin sin58 TAB TAC 3570lb 144lb Someimes he Sine Law / Cosine Law is faser han componen ecors.

A model is placed in a es channel and hree cables are used o alin is bow on he channel cenerline. For a ien speed, he ension is 40 lb in cable AB and 60 lb in cable AE.

Epress he condiion for equilibrium for he hull b wriin ha he sum of all forces mus be zero. Resole he ecor equilibrium equaion ino wo componen equaions. Sole for he wo unknown cable ensions.

11 T AC T AB 3500lb Sole for he unknown force maniudes usin Sine Law. TAB TAC 3500lb sin10 sin sin58 TAB TAC 3570lb 144lb Someimes he Sine Law / Cosine Law is faser han componen ecors. Inuiion should ell ou which is bes. I is desired o deermine he dra force a a ien speed on a proope sailboa hull. A model is placed in a es channel and hree cables are used o alin is bow on he channel cenerline. For a ien speed, he ension is 40 lb in cable AB and 60 lb in cable AE. Deermine he dra force eered on he hull and he ension in cable AC. PLAN OF ATTACK: Choosin he hull as he free bod, draw a free-bod diaram. Epress he condiion for equilibrium for he hull b wriin ha he sum of all forces mus be zero. Resole he ecor equilibrium equaion ino wo componen equaions. Sole for he wo unknown cable ensions. T AB = A T AE =60 T AC F D SOLUTION: Choosin he hull as he free bod, draw a free-bod diaram. 7 f an f f an f 0.56 Epress he condiion for equilibrium for he hull b wriin ha he sum of all forces mus be zero. R T T T F 0 AB AC AE D T AB 40cos 60.6 TAC cos sin 60.6 A T AE =60 T AC TAC sin F D Resole he ecor equilibrium equaion ino wo componen equaions. Sole for he wo unknown cable ensions. T D AE AC F T cos sin T lb T sin cos 60.6 AC T T AC AC AC Pae 11

Recanular Componens in Space This equaion is saisfied onl if all ecors when combined, complee a closed loop. The ecor F is conained in he plane OBAC.

componens F Fhcos Fh Fsin F sin cos F Fcos F F sin h F sin sin Recanular Componens in Space Recanular Componens in Space Direcion of he force is defined b he locaion M,, z and N,, z of wo

12 Recanular Componens in Space This equaion is saisfied onl if all ecors when combined, complee a closed loop. The ecor F is conained in he plane OBAC. Resole F ino Resole F h ino horizonal and erical recanular componens. componens F Fhcos Fh Fsin F sin cos F Fcos F F sin h F sin sin Recanular Componens in Space Recanular Componens in Space Direcion of he force is defined b he locaion M,, z and N,, z of wo poins, Wih he anles beween F and he aes, F F cos F F cos Fz F cos z F F i F j F k z F cos i cos j cos k F z cos i cos j cos k z is a uni ecor alon he line of F acion of F and cos, cos, and cos z are he direcion cosines for F d is he lenh of he ecor F d ecor joinin M and N d i d j d k z d d d z z z F F 1 di d j d zk d Fd Fd Fdz F F Fz d d d Pae 1

13 The ension in he u wire is 500 N. Deermine: a) componens F, F, F z of he force acin on he bol a A, b) he anles,, z definin he direcion of he force PLAN of ATTACK: Based on he relaie locaions of he poins A and B, deermine he uni ecor poinin from A owards B. Appl he uni ecor o deermine he componens of he force acin on A. Noin ha he componens of he uni ecor are he direcion cosines for he ecor, calculae he correspondin anles. SOLUTION: Deermine he uni ecor poinin from A owards B. 40m 80m 30m AB i j k 40m 80m 30m AB 94.3 m i j k i j 0.318k Deermine he componens of he force. F F 500 N 0.44i j 0.318k 1060 N 10 N 795 N i j k Noin ha he componens of he uni ecor are he direcion cosines for he ecor, calculae he correspondin anles. cos i cos j cos k z 0.44i j 0.318k Moion in 1 dimension In 1-D, we usuall wrie posiion as (). Since i s in 1-D, all we need o indicae direcion is + or. Displacemen in a ime = - 1 is = ( ) - ( 1 ) = z 1 some paricle s rajecor in 1-D 1 Pae 13

14 1-D kinemaics 1-D kinemaics... Veloci is he rae of chane of posiion Aerae eloci a in he ime = - 1 is: 1 ( ) ( 1) a 1 1 rajecor V a = slope of line connecin 1 and. Consider limi 1 Insananeous eloci is defined as: d() () d so ( ) = slope of line anen o pah a D kinemaics... Acceleraion a is he rae of chane of eloci Aerae acceleraion a a in he ime = - 1 is: ( ) ( 1) aa 1 And insananeous acceleraion a is defined as: d a () d ( ) d ( ) d d() usin () d Calculus wa of sain es er er small Recap If he posiion is known as a funcion of ime, hen we can find boh eloci and acceleraion a as a funcion of ime! d a ( ) d d d d d Calculus (don worr ou will undersand his in ne ear.) a Pae 14

15 More 1-D kinemaics We saw ha = d / d In calculus lanuae we would wrie d = d, which we can inerae o obain: ( ) ( ) ( ) d 1 Graphicall, his is addin up los of small recanles: () = displacemen 1 Recap So for consan acceleraion we find: a 0 a a cons a Moion in One Dimension Quesion Soluion When hrowin a ball sraih up, which of he followin is rue abou is eloci and is acceleraion a a he hihes poin in is pah? (a) Boh = 0 and a = 0. (b) 0, bu a = 0. (c) = 0, bu a 0. Goin up he ball has posiie eloci, while comin down i has neaie eloci. A he op he eloci is momenaril zero. Since he eloci is coninuall chanin here mus be some acceleraion. In fac he acceleraion is caused b rai ( = 9.81 m/s ). (more on rai in a few lecures) The answer is (c) = 0, bu a 0. a Pae 15

For consan acceleraion: Recap: This is jus 1 0 0 a 0 a a for consan From which we know: acceleraion! 0 a( 0 ) 1 a (0 ) Recap: For consan acceleraion: 1 0 0 a Read Carefull!

16 For consan acceleraion: Recap: This is jus a 0 a a for consan From which we know: acceleraion! 0 a( 0 ) 1 a (0 ) Recap: For consan acceleraion: a Read Carefull! Problem Solin Tips: Before ou sar work on a problem, read he problem saemen horouhl. Make sure ou undersand wha informaion is ien, wha is asked for, and he meanin of all he erms used in sain he problem. Usin wha ou are ien, se up he alebra for he problem and sole for our answer alebraicall Inen smbols for quaniies ou know as needed Don plu in numbers unil he end Wach our unis! Alwas check he unis of our answer, and carr he unis alon wih our formula durin he calculaion. Undersand he limis! Man equaions we use are special cases of more eneral laws. Undersandin how he are deried will help ou reconize heir limiaions (for eample, consan acceleraion). 1-D Free-Fall This is a nice eample of consan acceleraion (rai): In his case, acceleraion is caused b he force of rai: Usuall pick -ais upward Acceleraion of rai is down : a = a = a Pae 16

17 Grai facs: does no depend on he naure of he maerial! Galileo ( ) fiured his ou wihou fanc clocks & rulers! On he surface of he earh, rai acs o ie a consan acceleraion demo - feaher & penn in acuum Nominall, = 9.81 m/s A he equaor = 9.78 m/s A he Norh pole = 9.83 m/s More on rai in a few lecures! Penn & feaher Grai facs: Acuall, rai is a fundamenal force. Oher fundamenal forces: elecric force, sron and weak forces I s a force beween wo objecs, like me and he earh. or earh and moon, or sun and Nepune, ec Graiaional Force is proporional o produc of masses: F(1 acin on ) proporional o M 1 imes M F( acin on 1) proporional o M 1 imes M oo! Proporional o 1/r r is he separaion of he masses For rai on surface of earh, r = radius of earh Eample of Gauss s Law (more on his laer) A he surface of earh raiaional force aracs m oward he cener of he earh, is approimael consan and equal o m. The number =9.81 m/s conains he effec of M earh and r earh. Quesion: Soluion: The pilo of a hoerin helicoper drops a lead brick from a heih of 1000 m. How lon does i ake o reach he round and how fas is i moin when i es here? (nelec air resisance) 1000 m Firs choose coordinae ssem. Oriin and -direcion. Ne wrie down posiion equaion: m Realize ha 0 = = 0 Pae 17

18 Soluion: 1D Free Fall Sole for ime when = 0 ien ha 0 = 1000 m m 14.3s 9.81m s 0 0 = 1000 m Alice and Bob are sandin a he op of a cliff of heih H. Boh hrow a ball wih iniial speed 0, Alice sraih down and Bob sraih up. The speed of he balls when he hi he round are A and B respeciel. Which of he followin is rue: Recall: Sole for : - a( - ) m/ s = 0 (a) A < B (b) A = B (c) A > B 0 A Alice 0 B Bob H 1D Free fall Since he moion up and back down is smmeric, inuiion should ell ou ha = 0 We can proe ha our inuiion is correc: Bob 0 Equaion: 0 ) H H ( 0 H = 0 This looks jus like Bill hrew he ball down wih speed 0, so he speed a he boom should be he same as Alice s ball. = 0 Does moion in one direcion affec moion in an orhoonal direcion? ball drop For eample, does moion in he -direcion affec moion in he -direcion? I depends. For simple forces, like raiaional and elecric forces, NO For more complicaed forces/siuaions, like maneism, YES In an case, ecors are he mahemaical objecs ha we need o use o describe he moion Vecors hae Maniude Unis (like meers, Newons, Vols/meer, meer/sec ) Direcion Pae 18

19 Vecors: In 1 dimension, we could specif direcion wih a + or - sin. For eample, in he preious problem a = - ec. In or 3 dimensions, we need more han a sin o specif he direcion of somehin: To illusrae his, consider he posiion ecor r in dimensions. Eample: Where is Waerloo? Choose oriin a Torono Choose coordinaes of disance (km), and direcion (N,S,E,W) In his case r is a ecor ha poins 10 km norh. Waerloo Torono A ecor is a quani wih a maniude and a direcion r -D Kinemaics Mos 3-D problems can be reduced o -D problems when acceleraion is consan: Choose ais o be alon direcion of acceleraion Choose ais o be alon he oher direcion of moion Eample: Throwin a baseball (nelecin air resisance) Acceleraion is consan (rai) Choose ais up: a = - Choose ais alon he round in he direcion of he hrow and componens of moion are independen. A man on a rain osses a ball sraih up in he air. View his from wo reference frames: Reference frame on he moin rain. Problem: Daid Ecksein clobbers a fasball oward cener-field. The ball is hi 1 m ( o ) aboe he plae, and is iniial eloci is 36.5 m/s ( ) a an anle of 30 o () aboe horizonal. The cener-field wall is 113 m (D) from he plae and is 3 m (h) hih. Wha ime does he ball reach he fence? Does Daid e a home run? Reference frame on he round. 0 h D Pae 19

20 Choose ais up. Problem... Choose ais alon he round in he direcion of he hi. Choose he oriin (0,0) o be a he plae. Sa ha he ball is hi a = 0, (0) = 0 = 0. (0) = 0 = 1m Equaions of moion are: = 0 = 0 - = = / Problem... Use eomer o fiure ou 0 and 0 : Find 0 = cos. and 0 = sin. remember, we were old ha = 30 de Problem... The ime o reach he wall is: = D / (eas!) We hae an equaion ha ell us () = a / So, we re done...now we jus plu in he numbers: a = - Find: = 36.5 cos(30) m/s = 31.6 m/s = 36.5 sin(30) m/s = 18.5 m/s = (113 m) / (31.6 m/s) = 3.58 s () = (1.0 m) + (18.5 m/s)(3.58 s) - (0.5)(9.8 m/s )(3.58 s) = ( ) m = 3.5 m Moion in D Two fooballs are hrown from he same poin on a fla field. Boh are hrown a an anle of 30 o aboe he horizonal. Ball has wice he iniial speed of ball 1. If ball 1 is cauh a disance D 1 from he hrower, how far awa from he hrower D will he receier of ball be when he caches i? Assume he receier and QB are he same heih (a) D = D 1 (b) D = 4D 1 (c) D = 8D 1 Since he wall is 3 m hih, Ecksein es he homer!! Thinkin deeper: Can ou fiure ou wha anle ies he lones fl ball? To keep hins simple, assume 0 = 0, and o from here Pae 0

21 Soluion Soluion The disance a ball will o is simpl = (horizonal speed) (ime in air) = 0 To fiure ou ime in air, consider he 1 equaion for he heih of he ball: 0 0 When he ball is cauh, = wo soluions (ime of cach) (ime of hrow) So he ime spen in he air is proporional o 0 : 0 Since he anles are he same, boh 0 and 0 for ball are wice hose of ball 1. 0, 0,1 ball 0, ball 1 0,1 0,1 0, Ball is in he air wice as lon as ball 1, bu i also has wice he horizonal speed, so i will o 4 imes as far!! Projecile Moion Moion in D Aain As ou can see, i can become difficul o sole problems ha inole moion in boh he and ais. Luck for ou, people from all oer he world hae had he same difficulies. Therefore a complee se of equaions hae been creaed ha will help sole hese problems. These are known as ballisic formulas. The assume launch heih and landin heih are he same. Two fooballs are hrown from he same poin on a fla field. Boh are hrown a an anle of 30 o aboe he horizonal. Ball has wice he iniial speed of ball 1. If ball 1 is cauh a disance D 1 from he hrower, how far awa from he hrower D will he receier of ball be when he caches i? Assume he receier and QB are he same heih Disance: Maimum: i sin d Rane i sin Trael Time: Heih sin i h Time o op: i sin (a) D = D 1 (b) D = 4D 1 (c) D = 8D 1 o sin d i sin d i sin 4 Pae 1

22 Eample A olfer his a olf ball so ha i leaes he club wih an iniial speed 0 =37.0 m/s a an iniial anle of 0 =53.1 o. a) Deermine he posiion of he ball when =.00s. b) Deermine when he ball reaches he hihes poin of is flih and find is heih, h, a his poin. c) Deermine is horizonal rane, R. a) 0 0cos m / scos m / s 0 0 sin m / ssin m/ s The -disance: 0. m/ s.00s 44.4m 1 1 The -disance: m / s.00s 9.80 m / s.00s 39. 6m Eample A olfer his a olf ball so ha i leaes he club wih an iniial speed 0 =37.0 m/s a an iniial anle of 0 =53.1 o. a) Deermine he posiion of he ball when =.00s. b) Deermine when he ball reaches he hihes poin of is flih and find is heih, h, a his poin. c) Deermine is horizonal rane, R. sin i h i sin d 37.0 m/ ssin m/ s sin m/ s 9.80 m/ s 44.7m 134m i sin 37.0 m/ ssin s 9.80 m/ s Shooin he Monke (ranquilizer un) Shooin he Monke... Where does he zookeeper aim if he wans o hi he monke? ( He knows he monke will le o as soon as he shoos! ) If here were no rai, simpl aim a he monke r = r 0 r = 0 Pae

23 Shooin he Monke... Wih rai, sill aim a he monke! r = r 0-1 / Recap: Shooin he monke... = 0 = - 1 / r = 0-1 / Dar his he monke! This ma be easier o hink abou. I s eacl he same idea!! The boh hae he same V () in his case = 0 = - 1 / Feedin he Monke Kinemaics Flash Reiew Pae 3

Fundamental Units. Then, watch out. Page 1. SPH4UI: Lecture 1 Course Info & Advice. Kinematics

Fundamental Units. Then, watch out. Page 1. SPH4UI: Lecture 1 Course Info & Advice. Kinematics SPH4UI: Lecure 1 Course Info & Adice Course has seeral componens: Lecure: (me alkin, demos and ou askin quesions) Discussion secions (uorials, problem solin, quizzes) Homework Web based Labs: (roup eploraion