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Unformatted text preview: 2 Kinematics: The
Mathematics of Motion 2.1 Motion in One Dimension 1. Sketch position—versustime 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, tickmarks 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)” Ilas 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
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‘10 22 CHA PIER 2  Kinematics: The Mathematics of Motion 2. Interpret the renewing pnsitionversus—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‘HA +
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Foul has, 64de scoring. 0'4 8'12'16 Kinematics: The Mathematics ol‘Motion  ("mm 121% 2 23 3. Can you give an interpretation to this position—versus—time .ttm'a
graph? if so, then. do so. It not, why not? art1 an Were is no sens'JoLa. ivﬁtrPrct‘uﬁion D
\Otcanit m ﬁmpk rtetotter the. ' obi‘cf h be in +3” P [aces “flone! . ..._.___._,_..._.—'— Hag. 2.2 Uniform Motion 4. Sketch position—versustime 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 xaxis points to the. east. (cam 7‘ 60
Last) Ptml" t rs) 460 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) 24 mm s": 1—; R 2  Kinermtics: The Mathematics of Motion 5. The. ﬁgure shows a positionversus—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'ktllmgvrﬂ‘ (35:9?
HM M For obfed' B. “Menu! Blueti" S
Spuﬁ is ﬁreden (Boil. are. with“ slopes.) b DO C’blelilii A and B ever have the some speed? If so, at what time. or times? Explain.
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9924.31 (6.0 ash«1‘ ﬁlo?!) OMNSL A 5 9941.3 ts “Luau(5 ﬁrede _ (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‘sq. “a... tuck—ch“, In»), olﬂ'vd wcs‘l' Pu. ‘t'uro Mlle: «It (:0 Mela, «ha busmud'ﬂ ,m
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returns GMT "k 60 M?“ Kinematics: The Mathematics of Motion 1 CHAPTER 2 25 2.3 Instantaneous Velocity 7". Draw both a positionversustime graph and a velocity—versus—time graph for an object that is
at rest at x =1 In. \ M 1';
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8. The ﬁgure shows the positionversustime 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 annealer at t=l.s. Th Slope. 04 H _ rm
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‘Hmm ‘HM. stspc oP 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 Semi7K 06* 39151" begot‘ﬁ ’csSs.
Ai VH/torl‘ Jam, “HrI Slope. 0? ‘HA 'lT‘MUQrﬂ' +0 Meanc rcpt{scuth
E, 3 Mot(M [, awat in. H... Slope at ﬁe. ltwt. Permscn‘hnsﬁ, 9. Below are six positionversustime graphs. For each, draw the corresponding velocityversus
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——uu—
0 ‘E—Wm I U 26 ("I[A F'TF.R 2  Kinematics: The Mathematics of Motion r“; 1.0. The ﬁgure shows a Imsitionversustime 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 positionversustime 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 27 2.2. For each of the following motions, draw
 A motion diagram,
 A position—versus—time graph, and
 A velocityversus—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. Onehalf 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) 28 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
velocityversustime graph. Show the motion of both
cars on each graph. Label them A and B.
x B l} 8
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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 poslttonversustime graph and a 1 ‘1 S velocity—versustime 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) Thoubk Yam haw* ﬁA Scmul Posﬁianotbvvs Han—romp, L“, ¢‘[,¢;+T ;,
“EQ his “black? uJCKQ. no 4? ‘5'an becawse 1’“ u 't’“"‘"‘3 You“ at 44.. i’wnt oFrcrr‘J' aftshin {'Hm lat wuukok rim1m 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'mversus—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 '~""""‘°'
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and stop p.43 ad' 3. 240 (“1.1 \?'TF R 1‘  Kinematics: The Mathematics ol'Mouon 17. The figure. shows the. velocilyversumimc 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 \ \
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. . 5~ '61" .. A: 20. Hufnw 2111? three \‘L‘It'.3ClI)'—\'CI'5L15llmi3 graphs. ior each:
° Draw the Corresponding accelorutinn—versua—[time graph.  Draw a motion diagrer below the graphs. :1. ._._ h. .I L‘. to it t;
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I '4"—'(——"4——— .HOHL we hm‘ﬁ fa :n't 212 CHAPTER 2 v Kinematics: The Mathematics omeion 21. Below are three accelerationversustime 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 accentdie v\ uub‘l' ho. aim{tr 10 arian A Smallﬂ" (“Hid «toutL1 +0 05 \Al‘hﬂf‘ﬁlﬂwl Jet..ch , TL: divwig): t'\ 5 pad‘3 lugtug «,4 sauce is We “FrailHt 15 greeder?" 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. :«LLm
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 2 213 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
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than. 5,. less than lg, or equal to g. Explain. Tl... Sadhuh— acufh‘m‘l 1‘"
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it. you happen to pass by in an elevator that is rising with constant velocity 11,. 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‘Fcreneg in ‘HM—tamp'h‘m.~ 9+ ‘f'H—‘i'wo balls .
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kah its 'i‘kn. Sana. as 'HM. U3“ «\ Vibes 214 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
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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?
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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
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\(o‘, 0X0 wo‘l.‘ see ﬁve. \omll; 5+0? 06% M7 “hm ae'l'm' W7 Rain “‘95” “all” ' Kinematics: The Mathematics of Motion  C H A PT E R 2 21 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. duelmu welt“ le" a"
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no.
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“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~versustime graphs for the ball.
i



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Starts up Exits; dope slope 216 (“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 accelerationversus—time curves. For each. draw the corresponding velocity~
versus—time curve. Assume that var = 0 a. a i.
f
i (1 _ ._ ..— —I ...
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