24
CHAPTER
1
Representing Motion
SUMMARY
The goals of Chapter 1 have been to introduce the fundamental concepts of motion and to review
the related basic mathematical principles.
IMPORTANT CONCEPTS
Motion Diagrams
Describing Motion
The particle model represents a moving object as if all its mass
were co ncentrated at a si ngle point. Using thi s model, we can
represent motion with a motion diagram, where dots indi cate
the object's positions at success ive times. In a motion diagram,
the time interval between success ive dots is always the same.
Position locates an object with respect to a chosen coordin ate
system. It is described by a coordinate.
The C(lortlillllfe is the variable
1
Each dot rcprcsents thc position of the obJcct. Each
position i~ labeled with the time at which the dot wa,
• ... •
I =
Os
I ,
. •
2sJ
..,/
4,
3s
1
i
1
i
to deM'ribe the po\ition.
1
- 615 - 4 - 3 - 2- 1 0
ther~
5 s 6s
• • ••
111C time imerval betwcen
succes~ive position~ is the same.
This cow
i~
at
f
=
- 5 miles.
1
1
i
~
1
I 2 3( 4 5
~x(mi)
Thi\ car i~ at .( = + 4 mi les.
A change in position is called a displacement. For motion along
a line, a displacement is a signed quantity. The displacement
from Xi to X f is ilx = Xf - Xi'
Time iSJneasured from a particular in stant to which we ass ign
f = O. A time interval is the elapsed time between two spec ific
in stants Ii and If. It is given by ill = Ir - 'i'
Velocity is the ratio of the displacement of an object to the time
interval during which this displacement occurs:
Scalars and Vectors
Scalar quantities have only a magnitude and
ca n be represented by a single number.
Temperature, time, and mass
are scalars.
A~
A vector is a quantity
/
described by both a magnitude
and a direction. Velocity and
displacement are vectors.
Velocity vectors can be
drawn on a motion diagram
by connecting successive
points with a vector.
':t
u~ed
'---
Ax
AI
)I=~
~
Direction
~c lengt~
of a vc.ctor
1S propon10nai LO 1b;
magnitude.
Units
Every measureme nt of a quantity must include a unit.
Velocity vectors span
succe\sive points in _
a mOlion diagram.
Start
\
The standard system of units used in sc ience is the SI system.
Common SI units include:
_
':lI
......... .... The velocity
vec tors arc gelling longer,
so the object is ~peeding up.
Length: meters (m)
Time: seconds (s)
Mass: kilograms (kg)
APPLICATIONS
Working with Numbers
In scientific notation, a number is expressed as a decimal
number between I and 10 multiplied by a power of ten. In
sc ientifi c notation , the diameter of the earth is 1.27 X 107 m.
A prefix can be used before a unit to indicate a multiple of 10 or
1/ 10. Thus we ca n write the diameter of the earth as 12,700 km,
where the k in km denotes 1000.
We can perfonn a unit conversion to convert the diameter of the
earth to a different unit, such as miles. We do so by multiplying
by a co nvers ion factor equal to 1, such as I = I mill .6 1 km .
Significant figures are reliably known digits. The number of
signifi can t fi gures for:
Multiplication, division , and powers is set by the value with
the fewest significant figures.
Addition and subtraction is set by the value with the
small est number of dec imal places.
An order-of-magnitude estimate is an esti mate that has an
accuracy of about one sign ificant figure. Such estimates are usually made usin g rough numbers from everyday experience.
Question s
tMP)TM
;,:,.:,/
For homeWO~k assig~ed on MasteringPhysics, go t o
Problems labeled
BID
25
are of biolog ical o r medical inte rest.
www.mastenngp hyslcs.com
Problem difficulty is labeled as ! (straig htforward) to IJlIl (challenging).
Q UESTIONS
Conceptual Questions
I. a. Wr ite a paragraph describing the particle model. Wh at is it,
and w hy is it im port ant?
b. Give two examples o f situations, di fferent fro m those descri bed
in !.he text, for which the panicle model is appropriate.
c. G ive an example of a situation, di ffere nt from those described
in the text, for whic h it would be inappropri ate.
2. A softball player sli des in to second base. Use the particle model
to draw a motio n d iagram o f the pl ayer from the lime he beg in s
to sl ide unt il he reac hes th e base. Nu mber th e dots in order,
starting with zero.
3. A car travels to the left at a steady speed for a few seconds, then
brakes for a stop sign. Use the particl e model to draw a mot ion
di agram of the car for the entire moti on desc ri bed here. Number
the dots in order. start ing with zero.
4. A ball is dropped from the roo f o f a tall build in g an d
• 0
students in a ph ys ics class are as ked to sketch a motio n
di agram for thi s situation. A student submits the di a• I
gram show n in Fi gure QI A. Is the di agra m correct?
• 2
Ex pl ain .
FIGURE 01.4
.• 43
5. Write a sente nce or two descri b ing the d ifference between pos ition and di sp lacement. Gi ve one example of each.
6. G ive an example of a trip you mi ght take in your car for whi ch
7.
8.
9.
10.
11.
12.
the d istance traveled as measured on your car's odometer is not
equal to the d ispl acement between your initi al and fin al positi ons.
Wri te a sentence or two desc ri b in g the d iffere nce between
speed and veloc ity. G ive one exampl e of each.
The mot ion of a skateboard along a hori zontal ax is is obse rved
for 5 s. The initi al pos iti on o f the skateboard is negati ve w ith
respect to a chose n orig in, and its ve loc ity throughout the 5 s is
also negati ve. At the end of the observati o n time, is the s kateboard closer to or farther fro m the orig in th an initi all y? Ex pl ain .
Ca n the veloc ity of an object be pos iti ve duri ng a time interval
in whi ch its pos ition is always negat ive? Can its veloc ity be
pos itive during a time interval in whi ch its d isplacemen t is
negat ive?
Two Fri e nds watch a jogger compl ete a 400 m lap around the
track in 100 s. O ne of the fri ends states, "The jogger's veloc ity
was 4 mls d urin g thi s lap." The seco nd fri e nd objec ts, say in g,
"No, the jogger's speed was 4 m/s." Wh o is correc t? Ju sti fy
your an swer.
A softball pl ayer hits the ball and starts runnin g toward fir st
base. Draw a mot io n d iagram, using th e parti cle model , showing he r veloc ity vec tors durin g the firs t few seco nd s o f her
run.
A child is sledd ing o n a smooth, level patch o f snow. She
encoun ters a rocky patc h and s lows to a sto p. Draw a moti on
di agram, usin g the particle model, show ing her veloc ity vectors.
13. A s kydi ver jumps ou t o f an airpl ane. Her speed steadi ly
increases unt il she depl oys her parac hu te, at w hi ch po int her
speed quic kly decreases. S he subseq uentl y fa ll s to earth at a
constan t rate, stoppin g when s he lands o n the g roun d. Draw a
moti on diagram, using the particle mode l, that shows her pos iti on at successive times and incl udes veloc ity vectors.
14. Your roommate drops a tenni s ball fro m a third-story bak ony. It
hi ts the sidewalk and bo unces as hi gh as the seco nd story. Draw
a moti on d iagram, using the particle model , s howing the ball 's
veloc it y vectors from the lime it is released until it reaches the
max imum he ight o n its bounce.
15. A car is dri ving north at a steady speed . It makes a gradual 90°
left tu rn w ithout los ing speed, then conti nues driving to the
west. Draw a mo ti on d iagram, us in g the particle model , sho win g the car's veloc ity vectors as seen from a he licopter hovering
over the hi ghway.
16. A toy car ro ll s down a ramp, then across a smooth, horizontal
fl oor. Draw a mot ion d iagram, usin g the particle model. showin g the car 's veloc ity veclOrs.
17. Estimate the average speed w ith whi ch you go fro m ho me to
ca mpus (or anot her trip you common ly make) via whatever
mode of transport ati on you usc most common ly. G ive your
answer in both mph and m/s . Desc ribe how yo u arrived at thi s
estim ate.
18. Estimate the num ber of times you sneezed d uring the paSI year.
Desc ri be how you arrived at thi s est im ate.
19. Dens ity is the rati o o f an object's mass to its volume. Wou ld
you expect de nsity to be a vector or a scalar quantity? Ex plain .
Multiple-Choice Questions
20. I A studen t walk s 1.0 mi west and then 1.0 mi north . Afterward, how far is she from her start in g po in t?
A. 1.0 mi
8 . I A mi
C. 1.6 mi
D . 2.0 mi
2 1. I Whi ch of the follow ing mot ions is described by the moti on
di agram o f Fi gure Q 1.2 1?
A. An ice skater gli ding across the ice.
8 . A n airpl ane brakin g to a stop after landing.
C. A car pulli ng away from a stop s ign.
D. A pool ball bouncin g off a cushi on and revers in g di rection.
FIGURE 01 .2 1
o1
••
2
•
3
•
4
•
5
•
22. I A bird fli es 3.0 km due west and then 2.0 km due north . Wh at
is the mag nitude of the bird 's d ispl aceme nt?
A. 2.0 km
B. 3.0 km
C. 3.6 km
D. 5.0 km
26
Representing Motion
CHAPTER 1
23. II A bird fli es 3.0 km due west and the n 2.0 km due no rth.
Another bird fl ies 2.0 km due west and 3.0 km due north. Wh at
is the ang le betwee n the net displacement vec tors for the two
birds?
A. 23°
B. 34°
C. 56°
D. 90°
24. I A woman walks bri skly at 2.00 m/s . How much time will it
take her to walk one mile?
A. 8.30 min
B. 13.4 mi n
C. 2 1.7 mi n
D. 30.0 min
25. 1 Compute 3.24 m + 0.532 m to the correc t number of sign ificant fi gures.
A. 3.7 m
B. 3.77 m
C. 3.772 01
D. 3.7720 01
26. I A rectangle has length 3.24 m and he ight 0.532 m. To the correct number of signifi cant fi gures, what is its area?
A. 1.72 m'
B. 1.723 m'
C. 1.7236 m'
D. '1 .72368 01 '
27. I The earth fo rm ed 4.57 X 109 years ago. Wh at is thi s time in
seconds?
A. 1.67 X 10 12 s
B. 4.0 1 X 10 " s
C. 2.40 X 10" S
D. 1.44 X 10 17 s
28. III An object's ave rage density p is defin ed as th e rati o of its
mass to its vol ume: p = M /Y. The earth's mass is 5.94 X 1024 kg,
and its vo lume is 1.08 X J0 12 km 3. What is the eart h's average
density?
A. 5.50 X 10 3 kg/m3
B. 5.50 X 10' kg!m3
3
D. 5.50 X 10 12 kg/m3
C. 5.50 X 10' kg/m
PROBLEMS
Section _1.3 Velocity
Section 1.1 Motion: A First Look
1. I You' ve made a video of a car as it skids to a halt to avoid hi tti ng an obj ec t in the road . Use the images from th e video to
draw a motio n di agram of the car from the time the skid beg ins
until the car is stopped.
2. I A man rides a bi ke along a straight road for 5 mi n, then has a
nat ti re. He stops fo r 5 mi n to repair the flat , but then realizes he
cann ot fi x it. He conti nues hi s journey by walkin g the rest of the
way, which takes him another 10 min . Use the parti cle model to
draw a moti on diagram of the man for the e ntire mot ion
described here. Number the dots in order, starti ng with zero.
3. I A jogger running east at a steady pace suddenl y develops a
cramp . He is lucky: A westbound bus is sitting at a bus stop just
ahead. He gets on the bus and enjoys a qu ick ride home. Use the
part icle model to draw a moti on diagram of the jogger for the
entire motion descri bed here. Number the dots in order, starti ng
with zero.
8. I A security guard walks 110 m in one trip around the perimeter of the bu il ding. It takes him 240 s 10 make thi s trip. Wh at is
hi s speed?
9. II Li st the follow ing ite ms in order of decreas ing speed, from
greatcst to least: (i) A wind-up toy car that moves 0. 15 m in 2.5 s.
(ii ) A soccer ball th at .roll s 2.3 m in 0.55 s. (iii ) A bicycl e th at
travels 0.60 m in 0.075 s. (iv) A cat that runs 8.0 m in 2.0 s.
10. II Figure PI . IO shows the mot ion diagram for a horse gall opi ng
in onc di rection along a straight path. Not every dot is labeled,
but the dots are at equall y spaced in stanls of time. What is the
horse's veloc ity
a. Duri ng the fi rst len seco nds of its galJop?
b. During the in terva l from 30 s to 40 s?
c. Duri ng the interva l from 50 s to 70 s?
70s
•
50s 30s
•
••••
lOs
•
•
x(m)
50 150 250 350 450 550 650
Section 1.2 Position a nd Time: Puttin g Numbers on Nature
4. I Figure PI .4 shows Sue between he r home and the cinema.
Wh at is Sue's posit ion x if
a. Her home is the ori gin?
b. The c inema is the origin?
Home
Sue
g
FIGURE P1.4
Ci nema
B
2 mi
3 mi
5. I Ke ira starts at position x = 23 m along a coordin ate axis. She
then un dergoes a disp lace ment of -45 m. What is her final
position?
6. I A car trave ls along a stra ight east-west road. A coord inate
system is establ ished on the road, with x increasing to the east.
The car e nds up 14 mi west of the inte rsec ti on with Mu lberry
Road. If its disp laceme nt was -23 mi , how far from and on
whi ch side of Mul berry Road did it start?
7. I Foragin g bees ofte n move in straight li nes away fro m and
BID toward their hi ves. Suppose a bee starts at its hi ve and fli es 500 m
due east, th en fli es 400 m west, the n 700 m east. How far is the
bee from the hi ve?
FIGURE Pl .l0
II . II It takes Harry 35 s to walk from x = - 12 m to x = - 47 m.
Wh at is hi s ve locity?
12. I A dog trots from x = - 12 m to x = 3 m in 10 s. Wh at is its
veloc ity?
13. I A ball rol lin g alo ng a stra ight line with veloc it y 0.35 mls
goes from x = 2. 1 m to x = 7.3 m. How muc h time does thi s
take?
Section 1.4 A Sense of Scale: Significant Figures, Scientific
Notation, a nd Units
Convert the follow ing to SI uni ts:
a. 9. 12/Ls
b. 3.42 km
c. 44 cm/ms
d. 80 kmfhour
IS. Convert the follow ing 10 SI unit s:
a. 8.0 in
b. 66 ftls
c. 60 mph
16.
Convert the follow ing to Siuni ts:
a. l.0 hour
b. 1.0 day
c. 1.0 year
17. Li st the follow ing three speeds in order, fro m smaLl es l to
largest: I mm per /-Ls , 1 km per ks, I cm per ms.
14 .
Problems
18. I How many sig nifi cant figures does eac h of the following
numbers ha ve?
a. 6.21
b. 62. 1
c. 0.620
d. 0.062
19. I How many sig nificant figures does each of the following
numbers have?
a. 0.62 1
b. 0.006200
c. 1.0621
d. 6.2 1 X 103
20. I Compute the following numbers to 3 significant figures.
a. 33.3 X 25 .4
b. 33 .3 - 25.4
d. 333.3';- 25.4
c. V3D
2 1. JlIII The Empi re State Buildin g has a height of 1250 ft. Express
thi s he ight in meters, giving your resu lt in sc ie ntific notation
with three significant figures.
22. I Estimate (don' t measure !) the lengt h of a typi cal car. Give
yo ur answer in both feet and meters. Briefl y desc ribe how YOll
arrived at this est im ate.
23. III Blades of grass grow from the botto m, so, as growth occurs,
BlO the top of the blade mo ves upward. During the summer, when
yo ur lawn is grow in g quickly, estimate thi s speed in m/s.
Explain how you made thi s est imate, and ex press your resu lt in
scientific notation.
24. II Estimate the average speed with wh ich the hair o n your head
810 grows. Give your answer in both mls and ,umlh. Briefly
describe how you arri ved at thi s est imate.
25. II Estimate the average speed at which your fin ge rnail s grow, in
BlO both mls and ,um/h. Brieny describe how yo u arrived at thi s
estimate.
Section ].5 Vectors and Motion: A First Look
26. I Carol and Robin share a house. To get to work, Carol walks
north 2.0 km while Robin drives west 7.5 km . How far apart are
their workplaces?
27. 1 Joe and Max shake han ds and say goodbye. Joe walks east
0.55 km to a coffee sho p, and Max n ags a cab and rides north
3.25 km to a bookstore. How far apart are the ir dest inat ions?
28. II A city has streets laid out in a square grid, with eac h bloc k
135 m lon g. IF yo u dri ve no rth for three bloc ks, then west for
two blocks, how far are you from your starting po int ?
29. II A butterfl y nies from the top of a tree in the ce nter of a garden to rest on top of a red fl ower at the garden 's edge. The tree
is 8.0 m taller than the flow er, and the garden is 12 m wide.
Determine the magn itude of the butterfly 's di splace ment.
30. III A garde n has a c ircular path of radiu s 50 m. John starts at the
easternmos t po int o n thi s path , then wa lk s counterclockwi se
around the path until he is at its southernmost point. What is
John 's displacement ? Use the (mag nitude, d irecti on) notat ion
for your answer.
3 1. I Mi grating geese tend to travel in straight-line path s at approx BIO imately constant speed. A goose fli es 32 km south , then turn s 10
fly 20 km west. How far is the goose from its origin al position ?
32. Ilill A baUon a porch roUs 60 cm to the porch' s edge , drops 40 em ,
continues rollin g on the grass, and eventually stops 80 cm fro m
the porch' s edge. What is the magnitude of the ball 's net di splaceme nt, in centimeters?
33. II A ki c ker punts a footba U from the very center of the fi e ld to
the sideline 43 yards down field . What is the net displacement of
the ball ? (A football field is 53 yards wide. )
27
General Problems
Proble ms 34 throug h 40 are motio n probl ems similar to tho se you
will learn 10 sol ve in C hapter 2. For now, simply interpret the problem by drawin g a motion diagra m showing the object's position
and its veloc ity vec tors. Do not solve these problems or do any
mathemati cs.
34. Il In a typical greyhound race , a dog accelerates to a speed of
BID 20 mls over a di stance of 30 m. It then maintain s thi s speed.
What would be a greyhound ' s lime in the 100 m dash?
35. Il BiUy drops a wate rmelon from the top of a three-slory building, 10 In above the sidewalk. How fast is the watermelon go ing
when il hits?
36. Il Sam is reckl ess ly dri ving 60 mph in a 30 mph speed zo ne
when he suddenly sees the police. He steps on the brakes and
slows to 30 mph in th ree secon ds, looking nonchal ant as he
passes the offi cer. How far does he Lravel whil e brakin g?
37. II A spee d skate r mo ving across fricti o nl ess ice at 8. 0 mls
hit s a 5.0-m-wide patch of rough ice. Sh e slow s steadil y,
Ihen co nlinues o n at 6.0 m/s. What is her accele rali o n on the
rough ice?
38. Il The g iant eland, an African ante lope, is an exceptional
BID jumper, able 10 leap 1.5 m off the ground. To jump thi s hi gh, with
what speed must the eland leave the ground?
39. II A ball roll s along a smooth horizontal floo r at 10 mis, then
starts lip a 20° ramp . How hi gh doe..... it go before rolling back
down ?
40. Il A motori st is travelin g at 20 m/s. He is 60 m from a stop li ght
when he sees illurn yellow. Hi s react ion time, before stepp ing
on the brake , is 0.50 s. What steady dece leration whil e braki ng
will bring him to a SLOp right at the li ght?
Problems 41 through 46 show a motion diag ram. For each of these
problems, write a o ne or two sentence "slory" about a real object that
has thi s motion diagram. Your slories should talk about people or
objects by name and say what they are doin g. Problem s 34 through
40 are examples of motion short stories.
41. I
Stop
FIGURE Pl .41
-rO.-.."MO.-_.'----..-..............~
42. I
II •
..
.
II . ..
Til! '
II •
Stop
FIGURE Pl .42
43.
•
Top view of motion
horizontal plane
:1
i)
jj -----................
FIGURE P1 .43
Circular arc
II •
II •
28
CHAPTER 1
Repres enting Motion
44. I
SIan
Stop
The two pans of Ihe motion
diagram <Ire displaced for
clarity. but the motion <lctually
occurs along a single line.
Same
point
FIGU RE P1 .44
45. I
/
Side view or mot ion
"
Circular arc
i"'"'"i7 - - . . (
FIGURE P1 .45
-, ~
55. I Yo ur bra in co mmuni cates with yo ur body usin g lieI've
BID impulses. electri cal sig nals pro pagated along axo ns. Axo ns
come in two variet ies: in su lated axons with a sheat h made of
mye lin , and unin stll ated axons with no stic h sheath. Myel inated (s heat hed) axon s co nd uc t ne rve imp ul ses muc h fas te r
than unm ye li nated ( un sheathed) axons . The impul se speed
de pends o n the di amete r of the axo ns and the sheat h, but a
typi cal myel in ated axo n tra nsmi ts ne rve impu lses at a speed
o f aboul 25 mis, muc h fas ler th an the Iypi cal 2 .0 mls for an
unmye li na ted axo n. Fi g ure PI. 55 shows three eq ual.le ng th
nerve fibers co nsisting of e ig ht axo ns in a row. Nerve
impulses enter at the left side simultaneous ly and travel to the
ri g ht.
a. Draw motion di agrams for the nerve impu lses travel ing
alo ng fibers A, B, and C.
b. Which nerve impul se arri ves at the ri ght side fi rst?
c. Which will be last?
Unmyelinated fiber A:
46. I
""- Individual
Panly myelinated fiber Bj axons
Fully myelinated fiber C:
FIGURE P1.46
47. 11111How man y in ches does li ght travel in o ne nanoseco nd? T he
speed ofli ghl is 3.0 X lOll m/s.
48. I Joseph watches the roadside mile markers d uring a lo ng car
trip on an interslate hi ghway. He not ices that at 10:45 A.M. they
are pass ing a marker labe led 10 1. and at II :00 A.M. the car
reaches marker J 19. Wh at is the car's speed, in mph ?
49 . II Alberta is go ing to have din ner at her grandmo th er's house,
but she is run ning a bit behind sc hed ule. As she gets onto the
hi ghway, she knows that she must ex it the hi ghway withi n 45
min if she is not goin g to arri ve late. Her ex it is 32 mi away.
What is the slowest speed at whi ch she could drive and still
ill-ri ve in time? Ex press your answer in mi les per hour.
50. II Th e end of Hubbard Glac ier in Al aska advances by an average of 105 fee t per year. What is the speed of advance of the
glacier in mls?
5 1. I The earth co mpletes a circular orbit around the sun in o ne
year. The orbit has a radius of 93,000,000 mil es. W hat is the
speed of the earth aro und the sun in m/s? Report your res ult
using scientifi c notat ion.
52. III Shan no n dec ides to chec k the accuracy of her speedo meter.
She adj usts her speed to read exactly 70 mph on her speedometer and ho lds thi s steady, measuri ng th e ti me between successive mile markers separated by exactly 1.00 mile . If she
measures a time of 54 s, is her speedometer accurate? If not, is
the speed it shows too hi gh or too low?
53. II Motor neurons in mamm als transmi t signals from the brain to
BID skeletal muscles at approx im ately 25 m/s. Estimate how much
time in ms (10- 3 s) it will take for a signal to ge t fro m yo ur
bra in to your hand .
54. III Sate llite data take n several times per ho ur o n a pani cular
BID albatross showed travel of 1200 km over a time of 1.4 days.
a. Gi ve n these data, what was the bird's average speed in mph ?
b. Data on the bird 's pos itio n were recorded on ly intermitten tly. Explain how this means that the bi rd's actual average
speed was hi gher than what you calcul ated in part a.
FIGUREP1 .55
==
= =
56. II The bacteriu m E:icherichia coli (or E. coli) is a single-ce ll ed
BID organi sm that li ves in the gut of healthy humans and anim als.
Its body shape can be modeled as a 2-,um- long cy lin der with a
I t-Lm d iameter, and it has a mass of I X 10- 12 g. Its chro moso me cons ists of a sing le dou bl e-stran ded chain of DNA 700
times longer than its body length . The bacterium moves at a
cons ta nt speed of 20,um/s, thoug h not always in th e same
di rection. Answer the fo Ll ow ing questi ons about E. coli using
SI uni ts (unless spec ifi cally
req uested othe rwise) and
correct sign ifica nt fi gures.
a. Wh at is its length ?
b. Di ameter?
c . Mass?
d. What is the length of its
DNA, in mill imeters?
e. If th e o rgan ism were to
move along a straight path , how many meters would it travel
in one day?
57 . II The bacterium Escherichia
BID coli (or E. coli) is a sing lece lled organ ism that lives in
the gut of healthy humans and
an imals. When grown in a uni form medi um ri ch in sallS and
ami no ac ids, it sw ims along
A
zig-zag paths at a constant
E
speed. Fi gure PI .57 shows
the pos iti ons of an £. coli as it FIGURE Pl .57
moves from po int A to po int
J. Eac h segmen t of the mot ion can be identifi ed by two letters,
such as segmen t Be. During whi ch segment s, if any, does the
bacterium have the same
b. Speed?
c. Veloc ity?
a . Di splace ment?
Problems
58. III In 2003, the population of the United States was 291 million
people. The per-capita income was $3 1,459. What was the total
income of everyone in the United States? Express your answer in
sc ientific notation, with the correct number of significant figure s.
59. 111 The sun is 30° above the hori zon. It makes a 52-m-long
shadow of a tall tree. How high is the tree ?
60. II A large passenger aircraft accelerates down the runway for a
di stance of 3000 m before leav ing the ground. It then climbs at
a steady 3.0° angle. Arter the plane has traveled 3000 m along
this new trajectory, (a) how hi gh is it , and (b) how far horizontally is it, from its initial position ?
61. III Start in g from its nes t, an eag le flies at consta nt speed fo r
3.0 min du e eas t, then 4.0 min due north. From there the eag le
fli es directly to its nest at the same speed. How long is the eagle
in the air?
62. 111 John walks 1.00 km north, then turn s right and walks 1.00 km
east. His speed is 1.50 m/s during the entire stroll .
a. What is the mag nitude or hi s d isplacement , rrom beginning
to end?
b. Ir Jane start s at the same time and place as John , but walks
in a straight line to the end point of John 's strolJ , at what
speed should she walk to arrive at the endpoint just when
John does?
29
Passage Problems
Growth Speed
The images or trees in Figure P I.63 come from a catalog adverti sing
fast-growing trees. Ifwe mark the position of the top of the tree in the
sllccess ive years, as shown in the graph in the figure, we obtain a
motion diagram much like o nes we have seen for other kinds of
motion . The motion isn't steady, of course. In some months the tree
grows rapidly; in other month s, quite slowly. We can see, though, that
the average speed of growth is fairly consta nt for the first few years.
v( ft )
30' e t = 3 years
21 e t=2years
FIGURE Pl .63
•
Is! 2nd
year year
12 e t = I year
o
3<d
year
63. I What is the tree's speed of growth , in feet per year, from
(= I yrto{ =3 yr?
A. Ilftlyr
B. 9 ftlyr
C. 6 ftlyr
D. 3 ft/yr
64. I What is thi s speed in m/s?
A. 9 X 10- 11 m/s
B. 3 X 10- 9 m/s
C. 5 X 10 6 m/s
D. 2 X 10 6 m/s
65. I At the end of year 3, a rope is tied to the very top of the tree
to steady it Thi s rope is staked int o the ground 15 feet away
from the tree. What angle does the rope make with the gro und?
A. 63 °
B. 60°
C. 30°
D. 27°
01)
Stop to Think 1.1: B. The images or Bare rarther apart, so B travels
a greater distance than does A during the same intervals of time.
Stop to Think 1.2: A. Dropped ball. B. Dust particle. C. Descendin g
rocket.
Stop to Think 1.3: C. Depending on her initial positi ve position and
how far she moves in the negative direction , she could end up on
either side or the origin.
Stop to Think 1.4: D > C > B = A.
Stop to Think 1.5: B.The vector sum is found by placing the tailor
one vector at the head of the other.