WB_Solution_Ch02

WB_Solution_Ch02 - 2 Kinematics: The Mathematics of Motion...

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Unformatted text preview: 2 Kinematics: The Mathematics of Motion 2.1 Motion in One Dimension 1. Sketch position—versus-time graphs for the following motions. Include a numerical scale on both axes with units that are reasonable for this motion. Some numerical information is given in the problem, but for olher quantities make reasonable estimates. Note: A sketched graph simply means handsdrawn. rather than carefully measured and laid out with a ruler. But a sketch should still be neat and as accurate as is feasible by hand. It also should include labeled axes and. if appropriate, tick-marks and numerical scales along the axes. a. A student walks to the bus stop, waits for the bus. then rides to campus. Assume that all the motion is along a straight. street. x can)” Ila-s m 1: (MM) b. A student walks slowly to the bus stop, realizes he forgot his paper that is due, and quickly walks home to get it. X (M) 300 2.00 100 II a_ 3 r ttnia) c. The quarterback drops back 10 yards from the line of scrimmage, then throws a pass 20 yards to the tight end, who catches it and sprints 20 yards to the goal. Draw your graph for the football. Think carefully about what the slopes of the lines should be. K £65330 1.0 to O . V 3 t (5) ‘i 2-1 ‘10 2-2 CHA PIER 2 - Kinematics: The Mathematics of Motion 2. Interpret the renewing pnsition-versus—time graphs by writing a very short “story” of what is happening. Be creative! Have characters and situations! Simply saying that “a car mtwes 100 meters to the right" doesn’t qualify as a story. Your stories should make specific reference to information you obtain from the graphs. such as diatances moved or time elapsed. mm M on‘H-A + m- ‘h ‘ I-u 9* 60 “flu '1' prftaq as “in” PW Jul-M l3m- lOm I E r15“? AFC-:33" erg-92L. when]: codi'back DA ‘i‘k-L WM?- L0 mTv- \whl} I an; stoweaf‘i‘o éonpkbya a. Moving car 501 L°nyfm shirt-fl Q-eme. gar \owx‘un’ +k¢n . ____ __\fi _. gn'hi-‘k ¥w Mom I OM16 . 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Two fontbail players s“ PF.“ ‘ ‘ .r (yardill / Newman, Freoq mam 'H-i-b‘dl on Nloyv‘ : 1;,“ M52, M5 49fth adviser-«11'. 'bm’ruu “Mt-5 huh-n“: FMfiQiu 6m “Hi-n9" inflatable anlloh-t» “(55:5 0.5 'Fch LHSStb 4+»: 50 wth (.24.. B r-k‘hu Wink, +fi~es +0 (.OH'LK Us? ; '9“de Scores . 3°“ng l $rd-u (0-33an +0 run acF‘l-u- Foul has, 64de scoring. 0'4 8'12'16 Kinematics: The Mathematics ol‘Motion - ("mm 121% 2 2-3 3. Can you give an interpretation to this position—versus—time .t-tm'a graph? if so, then. do so. It not, why not? art-1 an Were is no sens'JoLa. ivfitrPr-ct‘ufiion D \Otcanit m fimpk rte-totter the. ' obi‘cf h be in +3” P [aces “fl-one!- . ..._.___._,_..._.-—'— Hag. 2.2 Uniform Motion 4. Sketch position—versus-time graphs for the following motions. Include appropriate numerical scales along both axes. A small amount of computation may be necessary. a. A parachutist opens her parachute at an altitude of 1500 m. She. then descends slowly to earth at a steady speed offi mfs. Start your graph as her parachute opens. too too 3” '6 ( 5) h. Trucker Bob starts the day 1'20 miles west of Denver. He drives east for 3 hours at a steady 6f) milcsr’hour before stopping for his coffee break. Let Denver be located at ..\‘ = 0 mi and assume that the x-axis points to the. east. (cam 7‘ 60 Last) Ptml" t rs) 4-60 was“ 4‘19 c. Quarterback Bill throws the hall to the right. at a speed of 15 mis. [t is intercepted 45 m away by Carlos. who is running to the left at 7.5 mis. Carlos carries the hall ()0 m to score. Let x = 0 m be the point where Bill throws the ball. Draw the graph for thelfootbalf. 4"" ‘W‘ZFLGPRJ. IS "- f (s) 2-4 mm s": 1—; R 2 - Kinermtics: The Mathematics of Motion 5. The. figure shows a position-versus—time graph for the motion of _r objects A and B that are moving along the same axis. a. At the instant I = 'l is the speed of A greater than, less than. or equal to the speed of 8‘? Explain. At {‘3- l 5., “Hun. Slope. er? ‘l'ktllm-gvrfl‘ (35:9? HM M For obfed' B. “Menu! Blue-ti" S Spufi is fire-den (Boil. are. with“ slopes.) b- DO C’blelilii A and B ever have the some speed? If so, at what time. or times? Explain. 140, Has. Spank m Awe-tho. Sunni... Eula has a cons-MM I... :- ¢ r 9924.31 (6.0 ash-«1‘ filo?!) OMNSL A 5 9941.3- ts “Luau-(5 fired-e _ (a. Interpret the following position-\-’ersus~tirne graphs by writing a short “story” about what is happening. Your stories should make specific references to the Speeds ol' the moving objects. which you can determine from the graphs. Assume that the motion takes place along a horizontal line. a. On a. «team‘s-q. “a... tuck—ch“, In»), olfl'vd wcs‘l' Pu.- ‘t'u-ro Mlle: «It (:0 Mela, «ha- busmud'fl ,m a“ 1..ng k1. Sine; ad" 9n returns GMT "k 60 M?“ Kinematics: The Mathematics of Motion 1 CHAPTER 2 2-5 2.3 Instantaneous Velocity 7". Draw both a position-versus-time graph and a velocity—versus—time graph for an object that is at rest at x =1 In. \ M 1'; 0 r o r Leash-4+ Oos'l'fib-fl arm “WW 8. The figure shows the position-versus-time graphs for two objects, A x B and B, that are moving along the same axis. a. At the instant I = i s, is the speed of A greater than, less than, or equal to the speed of B? Explain. N5 59494 is anneal-er at- t=l.s. Th Slope. 04 H _ rm 'H'W ‘l'RVI-fitfl'l' 'i‘b gas CMWE A‘l‘ “ta—l5 is Smallt“ ‘Hmm ‘HM. stspc o-P flc‘s line. I). Do objects A and B ever have the same speed? If so, at what time or times? Explain. A a.th E) heart. HM. 5AM Semi-7K 06* 39151" begot‘fi ’cs-Ss. Ai- VH/torl‘ Jam, “Hr-I- Slope. 0-? ‘HA 'lT‘MUQ-rfl' +0 Mean-c rcpt-{scuth E, 3 Mot-(M [, awat in. H... Slope at fie. ltwt. Permscn‘hnsfi, 9. Below are six position-versus-time graphs. For each, draw the corresponding velocity-versus- time graph directly below it. A vertical line drawn through both graphs should connect the velocity v, at time t with the position .v at the same time I. There are no numbers, but your graphs should correctly indicate the relative speeds. I I a. b. J J. 0 N I U l; l§—---——-—-n——-u--u— 0 ‘E—Wm I U 2-6 ("I-[A F'TF.R 2 - Kinematics: The Mathematics of Motion r“; 1.0. The figure shows a Imsition-versus-time graph for a moving object. At which lettered point or points: a. Is the object IBM the slowest? B b. is the object moving the fastest? D c. Is the object at rest'.J w d. Does the objeet have a constant B D nonzero veiocnv? _*__m e. Is the object moving to the left? D l l. The figure shows a position-versus-time graph for a moving object. At which lettered point or points: a. [s the object moving the fastest? D b. Is the t'thject moving to the left? C’ {D E c. Is the object speeding up? C d. Is the object slowing down? M 13 e. is the object turning around? Kinematics: The Mathematics at Motion ‘ C H A PTE a 2 2-7 2.2. For each of the following motions, draw - A motion diagram, - A position—versus—time graph, and - A velocity-versus—time graph. :1. A car starts from rest, steadily speeds up to 40 mph in 15 so moves at a constant speed for 30 s. then comes to a halt in 5 s. .qt U1. U: U“ Us. 9‘ U‘, U. Jr. '2 '3‘ 5:60 am a.“ an: out 5'” ‘E'g‘ 0a h. A rock is dropped from a bridge and steadily speeds up as it fails. it is moving at 30 nuts when it hits the ground 3 5 later. Think carefully about the signs. fr .. .. ii“ woo—— - v _, in» a l i 1 3 as) ..—, 1 Q Pl c. A pitcher winds up and throws a baseball with a speed ofriii) mfs. One-half second later the batter hiis a line drive with a speed oi'ofl mfs. The hall is caught 1 s al‘ter it. is hit. From where you are sitting, the batter is to the right of the pitcher. Draw your motion diagram and graph for the horizontal motion of the ball. 0 3—1,. A. “I. n =0 (.J—n—d up” HIE—r.é_—-—.‘__;fl a I; '0 V Rio 0 V Its) 2-8 CHA PTER 2 - Kinematics: The Mathematics of Motion 13. The figure shows six frames from the motion diagram 1 2 5 a . u ‘ A. o o o o of two moving cars, A and B. B. . . . . a. Draw both a position—versus—time graph and a l 6 velocity-versus-time graph. Show the motion of both cars on each graph. Label them A and B. x B l} 8 K EA 0 r u r b. Do the two cars ever have the same position at one instant of time? If so, in which frame number (or numbers)? 3 5 “l 3 Draw a vertical line through your graphs of part a to indicate this instant of time. c. Do the two cars ever have the same velocity at one instant of time? If 50, between which two frames? N" 14. The figure shows six frames from the motion 1 3 ‘1 5" . . A. O U I dlagram of two movmg cars, A and B. _ _ ‘ so a o a a. Draw both a posltton-versus-time graph and a 1 ‘1 S velocity—versus-time graph. Show both cars on each graph. Label them A and B. X 6 ii A l?) A b. Do the two cars ever have the same position at one instant of time? If so, in which frame number (or numbers)? M“ “It. L Draw a vertical line through your graphs of part a to indicate this instant of time. c. Do the two cars ever have the same velocity at one instant of time? If so, between which two frames? Z“; gm“ ‘f To 5 15. You’re driving along the highway at a steady speed of 60 mph when another car decides to pass you. At the moment when the front of his car is exactly even with the front of your car, and you turn your head to smile at him, do the two cars have equal velocities? Explain. N0) Thou-bk Yam haw-*- fi-A- Sc-mul- Posfiianotbvvs Han—romp, L“, ¢‘[,¢;+T ;, “EQ- his “black? uJ-CKQ. no 4? ‘5'an becawse- 1’“ u 't’“"‘"‘3 You“ at 44.. i’wn-t- o-Frcrr‘J' aft-shin {'Hm lat wuukok rim-1m em U" Kinematics: 'l‘hc Mathematica of Motion - 2.4 Finding Position from Velocity 'l (1. Below are Shoo-'11 four velocity—vermx—[ime graphs. For each: - Draw lhe C(H‘I‘ESIR‘HH'liIIg pq'ws'ilit'm-versus—time graph. - ("live a wrizleo description of the motion. , Assume {hm the motion takes: place alongr a horizontal line and than?1 : f). :1. ,. b. - Mm} %ru—mr& at? caught“ Sfud. .r—rkcfi'rs‘r '~""""‘°' i‘br ‘H" mt”. confirm-d S puck ‘H" 901*? L :- 'wu‘n h‘iufl" Rama °3 S ‘ 2L - Conshrdr {or-M 594.141 ho’i - Ayah: Canih'd' S?“J‘ badtuanfl> HM- l ""9 7M 3" 0&3“? I. H 2 2-9 .-‘~ I' - Cons'i‘hn't mtim‘fion _ clown nun-+21. l “L +KQA “(Ions “~“M; My? 3 ?¢¢J.:n,§ Mu? *9 rt‘hfi-r" +° "'" S‘OU’L‘ND &9""\ du‘t‘ “0;” fit- )flr‘hfla ?0"‘\TG+ 1. backurnl'd‘ km 2 +0 a, and stop p.43 ad'- 3. 240 (“1.1 -\?'TF R 1‘ - Kinematics: The Mathematics ol'Mouon 17. The figure. shows the. velocily-versumimc graph for a moving object whme initial position is In ; 20 111. Find the: objcci‘s m position graphically, using the geomeiry of the graph. at {he __ i// i \ \ {oliowing limcs‘ Finé‘knm up“ under-1““. cum. I = at. At 1:35. “0.9.444 rcc‘hmglc. MM 7/4 1 1 '1 X (35‘) = 7.0 + Vfio-s.) (‘55) -= 10m+ my; (353 = h. AIIIS S. +o+k¢ Pfluém «Mal-C1- ?“ are“ g TIM} area. Com in. hwé. bY «Ann-£43m rec-tuna“. "Pt-on 35 h: ‘15 an Q/q ‘04. 41‘,“- M... —ij Ham. ?9 r-‘h'dn 41-.“ 48 b 5.5. Or! e1QIWl¢W+krl x65): 50m+ was-mus) + Hm“ Wit-Ls») (h) -: 50M i~ [0'75 (is) +— 1; Liama-s'gfiu‘) -.~ S0n+l0n+Z$n 1' SENS-11 “a 35-“ C.Atr:7s. Add. h-HM. Pr-wgou unsun- *H-v. area oil-Hm +Hnngu. -Proa.. 55 {b 6.: («mtg Submcf -an. area M +1 5” c ) O‘r‘ 'HM. hl.fifl“-‘- ‘9’.“ (as +615,Tk¢5aar¢a5 owe- ‘t 5-305) 'z.(' '3) u - 5o. )4 (-1.) 2 51.5.... + 2.5.“ ~25.“ ; (L You should have found a simple reiaiionship between your answers to parts h and C. Can you expiain this"? What is the object doing? DW’MU 4“er 5‘5 41, ‘23, +|~L 9b;¢.9+ f5 Siou’ivss UL“?- Mo Q's-«3 in 44-1. ‘l‘X—durccrhan 3-m- lsefiafllfl} {ken 5P£¢¢tha “f LJ’Ai'c-fl. Maui-«3 "X o‘tI‘eCh'c-x {:31- 4’“ Secenéd-Suomp _ Eccam Han. acccla north“ is cons‘ha": «vi-é H~¢."flh¢$ are. 1%n0-\’ +1.4. HpfiOA ES SYMMIAV-mc . PA‘HK 'm rulers?“ Tim. tees us “W'fing van—=11 Kingtmutia‘u.‘ {'hc M;11i1:.‘1t'l;‘alit‘~ui'Mnlinm - {'EI \.'-‘. !- H 3 2-11 2.5 Motion with Constant Acceleration .18. .-'-\. car is trawling north. Can ils acceleration \cctor mar-point south? Explain. Yts. Macadam-.1104 W+Dr win Pan-d" .SOH.‘H'\ when +kn. car I5 5 icon in: (Loo-hm QKtK-I: fmdc'kima Aor'Ha . 10. (Jive :1 specific example for each (with: following siruatiom. For each. prm ide; ' A descripiinn. and ' A mmitm diagram-n. :‘t. u" : l'} but 11.95%}. Trnddm-J at“ CD‘S'hVfi‘ “leg \‘“1 :3; o. _ a, 5., auto ‘0 \'_.\:Uhu1ul+t0. Turn;n_3 «road Ugo—$0 _ . . & _... H , Tktb 'gurko'F-HM. w'hon d‘fimfl‘ "J7 ._ 5b....” ¢,:o‘bufl’ W40. é— . . / . .. a) A . - ‘ d ru'han. L. 1. Jidndn,_.. U. Steam“? ‘Lou’fl “LG”. “(IRA H m “taxi-W; t ‘I' \l‘} V U «Sh-- o (—L. ‘__'...... . . 5~ '61" .. A: 20. Hufnw 2111? three \‘L‘It'.3ClI)-'—\-'CI'5L15-llmi3 graphs. i-or each: ° Draw the Corresponding accelorutinn—versua—[time graph. - Draw a motion diagrer below the graphs. :1. ._._ h. .I L‘. to it t; (ox-7 ! Ea C no . I t C } ‘ . l ‘h‘f «a |.m)‘a----'-,.——--§pu_.——-}' HH:€_-fl_‘-—TOW arts- 23: onto i=1; Oat-t: 0~=o ta. 5—- \ A t (C) ‘tod’fiafl kl i ‘Jfi.H.H.—*9.—P’—|U I '4-"—-'(—-—"4———- .HOHL we hm‘fi fa :n't 2-12 CHAPTER 2 v Kinematics: The Mathematics omeion 21. Below are three acceleration-versus-time graphs. For each, draw the corresponding velocity versus~time graph. Assume that m, = O. a. N‘. b. N, C. a t | it. ———-- 5-; _ i 4 _ [I . I u 22. The figure below shows nine frames from the motion diagram of two cars. Both cars begin to accelerate. with constant acceleration, in frame 4. . . a“. 3 k 3 2 a B O O I O O I O I O I 4 3- b q, 3 9 :1. Which car has the largest initial velocity? The largest final velocity? 3— 13. Which car has the largest acceleration after frame 4’? How can you tell? B 33 accent-die v\ uub‘l' ho. aim-{tr- 1-0 aria-n A Small-fl" (“Hid «tout-L1 +0 05 \Al‘hflf‘filflwl Jet..ch , TL: div-wig): t'\ 5 pad-‘3 lug-tug «,4 sauce is We “Frail-Ht 15 greed-er?"- 2), c. Draw position, velocity, and acceleration graphs. showing the motion of both cars on each graph. { Label them A and B.) This is a total of three graphs with two curves on each. :«LL-m 3‘l 8 (3. Do the cars ever have the same position at one instant oftime? If so. in which frame? 3 a“; 8 t: 6. Do the two cars ever have the same velocity at one instant of time?1 1fso. identify the two frames between which this velocity occurs. 5 " ‘9 Identify this instant on your graphs by drawing a vertical line through the graphs. w. him Kinctnatics:'1‘hc Mathematics ofMotion - FHA PT ER 2 2-13 2.6 Free Fall 23. A hall is thrown straight up into the air. At each of the foliowing instants. is the hall’s acceleration s2. —;:, 0. < ‘9. or b g? a. .Iust alter leaving your hand? _-L b. At the. very top (maximum height}? __'3____ c. Just before hitting the ground? _"3_ 24. A rock is thrown (not dropped) straight down from a bridge into the river below. a. immediately after being released. is the magnitude of the rock’s acceleration greater than g. less than g. or equal to g“? Explain. Mamm rt": H‘ m‘ln'h'gn 3.3mm 'm Q...“ Ru 5 equaA h; a «1" Ali “haiku, tucker“.an 0+ Them-“kt «tact-t7, Tie. a.“ tint-'5“ “LI fills Roam Wb‘d‘] is . b. Immediately before hitting the water, is the magnitude of the rocks acceleration greater ’ " i o .. ' than. 5,. less than lg, or equal to g. Explain. Tl... Sad-huh— acufh‘m‘l 1‘" 15 S'i’til fibtflwii HM. ruck—i5 S‘hii In ‘Ql‘u. 9"“. TL:— Speufl u “\CPMSi‘uxa . ark ‘I'L; gm: rue («-d'xi-qhvd" flan-+13; b~( flu. sumo, AV excl-s seem-.69.. 25. Alicia throws a red ball straight up into the air, releasing it with velocity v“. As she is throwing it. you happen to pass by in an elevator that is rising with constant velocity 1-1,. At the exact instant Alicia releases her ball, you reach out of the eievator's window (this is a very fancy elevator!) and gently release a blue ball. Both halls are the same height above the ground at the moment they are released. a. Describe the motion of the two balls as Alicia sees them from the ground In what ways are the motion of the red bail and the blue. hall the same or different".I R‘Cfi‘kx 5&5 no di‘i‘Fcr-eneg in ‘HM—tamp'h‘m.~ 9+ ‘f'H—‘i'wo balls . tags». Slow which. mating mm, at taxman “match, my “.9 Meg». *1: $923.11 we 0mm:- wit-«.7 audit. 9' H4 dead-o:- o-F' ‘HM-I r€J~ ‘0‘“u. Ere acb‘ 'HMRI- “ft-“’4'- TLJL Mining“ has 'l'kg, lim'fioJ “PM attack-H o ka-h- its 'i‘kn. San-a. as 'HM. U3“ «\ Vibes 2-14 c H APTE R 2 ' Kinematics: The Mathematics of Motion b. Describe the motion of the two balls as you see them from the moving elevator. In what ways are the motion of the red ball and the blue ball the same or different? You 96¢ ‘H’LQU‘ Mo‘l’lbns 01s {70:45 rolva'HCal, (fio'l‘k balls (AN, ¥lellk5 Owl‘w’r “Prom you art \nCPefiSl‘n‘a S‘QQCGQ/ W~|+Ll Congl‘om’l aece lawman, ~35 . From your oarsPec‘hwe, loath smug Sim am west m; m :h med. 0. Alicia sees a well—defined “top” of the motion where her red ball reaches a maximum height and then starts to fall. Call the time of maximum height t1. As you watch from the elevator, do you see anything distinctive or different about the red ball’s motion at time Q? If so, what? 7/01» (Lo mil—Sate. Omy‘fklns (“weva “boats we, ‘0““3‘, wofio“ we “.65. from 7M rrwezdfve, ‘aofla we “dime +0 fall ma, W194 t/ch‘tmit‘ns speed. The. argzqé “News {to Maui «we? exva 7:» od— cw‘f velod‘hl. A‘t fikelhdan‘f ‘b" *er, \omUs Motown in be Walked «war Fem Yam aft-+3» samegfxceolasgari.“ (1. Does the red ball “stop” at time t1 when Alice sees it at the very top of its trajectory? As part of answering this question, define what you mean by the word “stop.” 1"?— 0w?— de‘lllnes m9???) m5 incubus A “\00957 5? {zero} «‘9 Only 16% cm ins‘tcm'l'y SHqu Alike. see: fig, reef ball CAMQ (I A \t W blue, $+o( Q‘l’h +99 o‘Q EZCQKSC Hm, “decry obServei oleemfli Won Webstmm (Aachen, \(o‘, 0X0 wo‘l.‘ see five. \omll; 5+0? 06% M7 “hm a-e'l'm' W7 Rain “‘95” “all” ' Kinematics: The Mathematics of Motion - C H A PT E R 2 2-1 5 2.7 Motion on an Inclined Plane 26. A ball. released from rest on an inclined plane accelerates down the plane at 2 W523. Complete the table below showing the hall’s velocities at the times indicated. Do not use a calculator for this; this is a reasoning question, not a calculation problem. Time (5;) Velocity (mis) h 0 0 (“111‘ wt. duel-mu welt“- le" a" 'l ——L‘—~ P o sH'iuc Add?!“ I M“ u “had-‘0 A at." Kean-[106 I) modem no. I “N 27. A bowling ball rolls along a level surface, then up a 30” slope, and finally exits onto another level surface at a much slower speed. 3. Draw position-. velocity-, and acceleration~versus-time graphs for the ball. i | | | ! I: ' I U .— _._.d v.“ .._ 1... r I I 1 r! I l i" I 1 i I I i I r I I I I I : [I —1—___-‘I m r i I 1 i I I E I I | I I I I Starts up Exits; dope slope 2-16 (“in PTER 2 - Kinematics: The Mathematics of Motion 21. Suppose that the hall’s initial speed is 5.0 mls and its final speed is 1.0 rm‘s. Draw a pictorial representation that. you would use to determine the height h of the slope. Establish a coordinate system, define all symbols, list known information, and identify desired unknowns. Note: Don’t actually solve the problem. Just draw the complete pictorial representation that you would use as a first step in solving the problem. A S : “Y5 "1.9 2.8 Instantaneous Acceleration '28. Below are two acceleration-versus—time curves. For each. draw the corresponding ve-locity~ versus—time curve. Assume that var = 0 a. a i. f i (1 _ ._ ..— —-I ...
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WB_Solution_Ch02 - 2 Kinematics: The Mathematics of Motion...

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