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Summary
281
SUMMARY
The goals of Chapter 9 have been to introduce the ideas of impulse. momentum, and angular
c
moment um and to learn a new prob lem-solving strategy b ased on conservation laws .
GENERAL PRINCIPLES
Law of Conservation of Momentum
T he total mo me ntum P = P I + P2+ ... o f an isolated system is
a con stant. Thu s
Solving Momentum Conservation Problems
PREPARE Choose an iso lated syste m or a system that is isolated
d uring at least part of the probl em. Draw a v isual ove rview o f the
system before and after the interact ion .
SOLVE Write the law of co nservati on of mo mentum in te rms of
vector compo nents:
Conservation of Angular M omentum
(p I.,)f + (P 2>:)r + ... = (p Ix), + (P2x)i
T he angular momentum L of a rotatin g object subject to zero
ex tern al torque does not chan ge. Thus
(P ly)f
+ (P 2y )r + ... =
+.
(P ly), + (P 2y); + .
In term s of masses and ve loc ities, thi s is
Thi s can be wrilten in term s o f the mome nt o f inert ia and
angular ve loc ity as
m I( Vlx)r
+ m 2(v2~)r +
m I(V I)f + f1I 2( V2y)f +
ASSESS
= III I(V tx) i + J11 2( V2<) i +
= m I( VI.J i + 11I 2(V 2)i
+
Is the result reasonable?
IMPORTANT CONCEPTS
Momentum
p=
11/V
/II
Impulse i t = area under force curve ....
lmpu lse and momentum are
related by the impulsemomentum theorem
System A group of in teracting particles .
Intcma l forces
Isolated system A system o n which the
net ex ternal force is zero.
<~
P
tk
I
;,
/ '-1
Before-and-after visual overview
I
Defi ne the system.
T hi s is an alternati ve state ment of Newton's
seco nd law.
Use two draw ings to show the system
befo re and af ter the interactio n.
Angular momentum L = Iw is the rotatio nal
ana log of linear momentum p = mv.
Li st known informat ion and ident ify
what you are try in g to find.
APPLICATIONS
Collisions Two or more particles come
together. In a perfectly inelastic
colli sio n, they stic k together and move
with a common fi nal ve locity.
Explosions Two or more particles
move away from each other.
Two dimensions Both the x- and y-components of
the total momentum P must be conserved, giving two
s imultaneous equati o ns.
282
CHAPTER 9
tMP)TM
;.:;./
Momentum
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Problem difficulty is labeled as ! (straightforward) to nUl (challenging).
Problems labeled
Worksheet;
INT
rli can be done on a Workbook Momentum
integrate significant material from earlier chapters;
BID are of biological or medical interest.
QUESTIONS
Conceptual Questions
I. Rank in order, from largest to sma Ues t, the momenta PI.
through P 5x of the objects presented in Figure Q9.1. Explain.
20g
20g
87:;
U I-- 2-m
-I'- "
10 g
109
[2]1--m/s
2
~
D
OO g
5
0.1 m/s
FIGURE 09 .1
2. Starting from rest, object 1 is subject to a 12 N force for 2.0 s.
Object 2, with twice the mass, is subject to a 15 N force for 3.0 s.
Which object has the greater final speed? Ex plain.
3. A 0.2 kg pl ast ic cart and a 20 kg lead cart can roll without friction on a hor izontal surface. Equal forces are used to push both
carts forward for a time of 1 s, starting from rest. After the force
is removed at t = I s, is the momentum of the plast ic cart
greater than, le ss than, or eq ual to the momentum of the lead
cart? Explain.
4. Two pucks, of mass m and 4m, li e on a frictionless table. Equal
forces are used to pu sh both pucks forward a distance of I m.
a. Which puck takes longer to travel the distance ? Explain.
b. Which puck has the greater momen tum upon completing the
distance? Ex plain.
5. A stationary firecracker ex plodes into
N
three pieces. One piece travels off to the
A.
east; a second trave ls to the north. Which
E
of the vectors of Figure Q9.5 could be the W
veloc ity of the third piece? Explain.
D.
6. Two students stand at rest, facing eac h
S C.
other on frict ion less skates. They then start
.
7f
tossing a heavy ball back and forth FIGURE 0 9 .5
between them. Describe their su bsequent
motion.
7. Two particles collide, one of which was initiaUy moving and
the other initially at rest.
a. Is it possible for both particles to be at rest after the colli sion ? Give an exa mpl e in which this happens, or exp lain
why it can't happen .
b. Is it possible for olle particle to be at rest after the coll ision?
Give an example in which thi s happen s, or explain why it
can' t happen.
8. Automobiles are designed with "c rumpl e zones" intended to
co llapse in a collision. Why would a manufacturer design part
of a car so that it collapses in a colli sion ?
9. You probably know that it feels beller to catch a baseball if you
are wearing a padded glove. Explain why thi s is so, using the
ideas of momentum and impul se.
10. In the early days of roc ketry, some
people claimed that rockets couldn' t fly in outer space as there was
no air for the rockets to push
against. Suppose you were an
early in vestigator in the field of
rocketry and met someone who
made this argument. How would
you co nvince the person that
rockets could travel in space?
11. Two ice skaters, Megan and Jaso n, push off from each other on
frictionless ice. Jason's mass is twice that of Megan.
a. Which skater, if e ither, experiences the greater impul se during the pu sh? Ex plain.
b. Which skater, if either, has the greater speed after the pushoff? Explain.
12. Suppose a rubber ball and a steel ball co llide. Which, if e ither,
receives the larger impulse? Explain.
13. While stand in g st ill on a basketball court, you throw the baU to
a teammate. Why do you not move backward as a result? Is the
law of conservation of momentum violated?
14. To win a pri ze at the county fair, you ' re trying to knock down a
heavy bowling pin by hitting it with a thrown object. Should
you choose to throw a rubber ball or a beanbag of equal size and
weight? Explain.
IS. Rank in order, from largest to smallest, the angu lar momen ta L I
through L5 of the balls shown in Figure Q9.15. Explain.
G
0
UUU
3m/, kg
~ kg~ kg d
I
2
I
4m
m/'2kg
2m/
4m ',_kg
3
4
5
FIGURE 09.15
16. Figure Q9.16 shows two
masses held together by a
thread on a rod that is rotating about its center with
angular velocity w. If the
thread breaks, the masses
will slide out to the ends of FIGURE 09 .16
the rod. If that happe ns, will
the rod 's angular velocity increase, decrease, or remain
unchanged? Explain.
17. If the earth warms signi ficantly, the polar ice caps will melt.
Water wiU move from the poles, near the earth 's rotation axis,
and will spread out around the globe. In principle, this wiLl
change the len gth of the day. Why? Will the length of the day
increase or decrease?
Questions
18. The disks shown in Figure Q9.IS have eq ual mass . Is the angular momentum o f disk 2, on the ri ght , larger th an, smaller than ,
or eq ual to the angular momen tum of disk I ? Explain.
FIGURE 09.18
Multiple-Choice Questions
the veloc ity of the boat as a fun ct ion of time? Pos itive ve loc ity
is forward, negative veloc ity is backward. Neglect any drag
force on the canoe from the water.
23. II Two blocks, with m asses 1111 = 2.5 kg and 111 2 = 14 kg ,
approach each other along a horizonta l, fri cti o nless track. The
initial veloc ities o f the blocks are VI = 12 .0 m/s to the right and
V2 = 3.4 m/s to the lefL Th e two blocks then colli de and sti ck
toge ther. Which o f the graphs could represe nt the force of block
1 on block 2 during the coll ision?
30.00:;~
(N)
A.
3~~
(N)
B.
20,000
10,000
20
10
o
19. 1 Curling is a sport played w ith 20 kg sto nes that s lide across
an ice surface. Suppose a curl in g sto ne slidi ng at 1 mls strikes
another stone and comes 10 rest in 2 ms. Approx im ately how
much force is there on the stone during the impact?
A. 200 N
B. 1000 N C. 2000 N D. 10,000 N
20. 1 Two balls are hung from cords. The fir st ball , of mass 1.0 kg,
is pulled to the side and released, reachin g a speed of 2.0 m/s at
the bottom o f its arc. Then, as
shown in Figure Q9.20, it hits
and sticks to anOlher ball. The
speed of the pair just after the
colli sion is 1.2 m/s. What is
the mass of the second ball?
A. 0.67 kg
8. 2.0 kg
C. 1.7 kg
FIGURE 09 .20
D. 1.0 kg
2 1. I Fi gure Q9.2 1 shows two blocks s liding on a frictionless surface. Even tuall y the smaller block overtakes the larger one, co llides with it, and sticks. What is the speed of the two blocks after
thecolli sion?
B. 4vJ5
c. Vj D. 5v,l4
A. vJ2
~
FIGURE 09 .21
~
22. I Two fri e nds are si ttin g in a stat ionary canoe. At I = 3.0 s the
person at the fro nt lOsses a sac k to the person in the rear, who
catches the sac k 0.2 s later. Which plot in Figure Q9.22 shows
A.
B.
't=="" "C==",
3.0
3.2
3.0
C.
D.
+-~""
3.0
3.2
3.2
FIGURE 09 .22
+-==,.
3.0
3.2
283
o
t(ms)
10 20 30 40 50
o
o
t(ms)
10 10 30 40 50
:~
( N)
c.
300:;~
(N)
D.
100
2000
1000
o
o
/ (ms)
10 20 30 40 50
o
o
t (ms)
10 20 30 40 50
FIGURE 09 .23
24. I A small puck is s liding to the ri ght w ith
momen tum Pi on a hori zon tal, fri cti o nless
surface, as show n in Figure Q9.24. A force is
applied to the puck for a short time and its
momen tum afterward is Pf. Whic h lettered
arrow shows the directi on of the impul se that
was deli vered to the puck ?
FIGURE 09 .24
25. 1 A red ball , initiall y at rcst, is s imultan eous ly hit by a blue ball traveling from weslto east at 3 m/s and a
green ball travel in g east to west at 3 m/s. A ll three balls have
equal mass. A ft erward, the red ball is trave li ng south and the
gree n ball is moving to the east. In whic h directi on is the b lue
ball travel in g?
A. West
B. North
C. Between north and west
D. Between north and east
E. Between south and west
26. 1 A 24 g, 3-cm -d iameter thin , holl ow sphere rotates at 30 rpm
about a vert ical, fri ct ionless ax is thro ugh it s center. A 4 g bug
stands at the top of the sphere. He then walks along the surface
o f the s phere until he reaches its "eq uator." When he reaches
the equator, the sphere is rotating at
A. 15 rpm
B. 24 rpm
C. 30 rpm
D. 37 rpm
E. 45 rpm
27. I A 5.0 kg solid cy linde r o f radius 12 cm rotates w ith
Wi = 3.7 radls about an ax is through its ce nter. A torque of
0 .040 N · m is applied to the cy linder for 5.0 s. By how Illuc h
does the cyl inder 's angular momen tum change?
A. 0.12 kg · ml/s
B. 0.20 kg · m 2/s
C. 0.38 kg · mlls
D. 0.52 kg · m' !s
E. 0.S8 kg · m 2/s
- ~ --:--
284
CHAPTER 9
Momentum
PROBLEMS
9. II A ch ild is slidin g o n a sled at 1.5 mls to the ri ght. YOll stop
Section 9. t Impulse
rI the sled by pushing on it for 0.50 s in a directi on oppos ite to its
Section 9.2 Momentum and the Impulse-Momentum Theorem
I. ! At what speed do a bi cycle and its rider, with a combined
mass of 100 kg, have the same momentum as a 1500 kg car
u'aveling at 5.0 m/s?
2. I A 57 g tenni s ball is served at 45 m/s. Ir the ball started from
rest, what impul se was appli ed to the ball by the racket?
3. II A student throws a 120 g snowball at 7.5 mls at the side of
the schoolhouse, where it hits and sticks. What is the magn itude
of the average force on the wall if the durati on of the colli sion is
0. 15 s?
4. 1111 In Fi gure P9.4, what value of Fn....~ g ives a n impul se of
6.0 N · s?
¥-~-~~--I-~ I (ms)
246
10
FIGURE P9 .4
rI
II
rI
/
F,
motion . If the mass of the child and sled is 35 kg, what average
force do you need to apply to stop the sled? Use the concepts o f
impul se and momentum .
10. III A n ice hock ey puck slides alo ng the ice at 12 mls. A hockey
stic k del ivers an impul se of 4.0 kg' mIs, causing th e puck to
move off in the opposite d irection with the same speed. What is
the mass of the puc k?
II. I As part of a safe ty in vesti gatio n, two 1400 kg cars traveling
at 20 mls are crashed into different barri ers. Find the average
forces exerted on (a) the car that hits a line of water barrel s and
takes 1.5 s to stop, and (b) the car that hi ts a concrete barrier and
takes 0. 1 s to stop.
12. II In a Littl e League baseball game, the 145 g ball ente rs the
strike zo ne with a speed of 15.0 m/s. The batter hits the ball ,
and it leaves hi s bat with a speed o f 20.0 mls in exac tl y the
opposite direct ion.
a. What is the mag nitude o f the impulse delivered by the bat to
the baB ?
b. If the bat is in contact with the ball for 1.5 ms, what is the
magnitude of the average force exerted by the bat on the ball ?
Section 9.4 Conservation of Momentum
5. 1 A sled and rider, g lidin g over horizon tal, fri ctionl ess ice at
4.0 mIs, have a combined mass of SO kg. The sled then slides over
a rough spot in the ice, slowing down to 3.0 m/s. What impulse
was deli vered to the sled by the friction force from the rough spot?
13. 1111 A small, 100 g cart is moving at 1.20 mls o n an air track
II when it co llides with a larger, 1.00 kg cart at rest. Aft er the col14.
II
Section 9.3 Solving Impulse and Momentum Problems
6. II Use the impul se- momentum theore m to find how long a
II stone fallin g strai ght down takes to in crease it s speed from
5.5 mls to 10.4 m/s.
7. II a. A 2.0 kg o bject is mo vin g to the rig ht wi th a speed of
1.0 mls when it ex periences the force show n in Fi gure
P9.7a. What are the object's speed and direction after the
force ends?
b. Answer th is question for the fo rce shown in Figure P9.7b.
15.
rI
II
(a)
16.
II
(b)
F, (N)
F~ (N)
2
2
17.
0
f-----1
I
(s)
0
I
(s)
1s
-2
II
18.
-2
11
FIGURE P9 .7
8. 1111 A 60 g tenni s ball with an
ini tial s peed of 32 mls hit s a
wa ll and rebounds with the
same speed. Fig ure P9.S
s hows the force of th e wall
o
o n the bail during the collision . Wh at is the va lue of FIGURE P9 .8
F m ax , the maximum value o f
the co ntact force durin g the co lli sion?
19.
iI
rI
I (ms)
20.
rJ
li sio n, the small cart reco il s at 0.850 m/s. What is the speed of
the large cart after the collision?
II A man standin g o n very sl ic k ice fires a rifle hori zo ntall y.
The mass of the man together with the rifle is 70 kg , and the
mass of the bullet is 10 g. If the bullet leaves the muzz le at a
speed o f 500 mis, what is the final speed of the man ?
III A 2.7 kg bloc k of wood sits on a table. A 3.0 g bullet, fired
horizontall y at a speed of 500 mIs, goes completely through the
bl ock, emerg ing at a speed of 220 m/s. Wh at is the speed of the
block immed iately after the bu llet ex its?
I A strong man is compre ss in g a li ghtwe ig ht sprin g bet wee n
two weights. One weight has a mass of2.3 kg, the other a mass
o f 5.3 kg. He is holdin g the weights stationary, but then he loses
hi s grip and the weigh ts fly off in oppos ite directions. The
li ghter of the two is sho t out at a speed of 6.0 m/s. What is the
speed of the heav ier weight?
II A 10,000 kg railroad car is roLlin g at 2.00 m/s when a 4000
kg load o f g rave l is s udden ly dropped in . What is the car's
speed j ust after the gravel is loaded?
I A 5000 kg open train car is roll ing o n fricti on less rail s at
22.0 mls w hen it starts po uring rain. A few mi nutes later, the
car's speed is 20.0 mls. What mass of water has coll ected in the
car?
II A 50.0 kg archer, standing o n fri cti onless ice, shoo ts a 40 g
arrow at a speed o f 60 m/s. What is th e recoil speed o f the
arc her?
II A 9.5 kg dog lakes a nap in a canoe and wakes up to find the
canoe has drifted out onto the lake but now is stationary. He
walks alo ng the length of the canoe at 0.50 mIs, relat ive to the
water, and the canoe simultaneously moves in the opposite direc~
tion at 0. 15 mls. What is the mass o f the canoe?
Problems
Section 9.5 Inelastic Collisions
2 1. II A 300 g bird flying along at 6.0 mJs sees a 10 9 in sec t heading straight toward it with a speed of 30 m/s. The bird ope ns its
mouth wide and enjoys a nice lunch. What is the bi rd's speed
immed iately after swallow in g?
22. 11 A 7 1 kg baseball player j umps straight up to catch a line
drive. If the 140 g ball is movi ng hori zo ntall y at 28 m/s. and the
catch is made whe n the ballplayer is at the hi ghest point of hi s
leap, what is hi s speed immed iately after stoppin g the ball ?
23. !II A kid at the j unior hi gh cafeteria wants to prope l an e mpty
miLk carton along a lu nc h tab le by hitti ng it wilh a 3.0 g spit
ball. If he wants the speed of the 20 g carton just after the spit
ball hits it to be 0.30 mis, at what speed shou ld hi s spit ball hit
the carton?
24. 1 The parking brake on a 2000 kg Cadillac has failed, and it is
roll in g slow ly, at I mph. toward a group of small childre n. Seeing the situation, you realize you have just enough time 10 drive
your 1000 kg Vol kswagen head-on into the Cad ill ac an d save
the children. With what speed should you impact the Cad ill ac to
bring it to a halt?
25. J A 2.0 kg block slides along a frictionless surface at 1.0 mls. A
secon d block, slidin g at a faster 4.0 mIs, collides with the first
fro m behind and sticks to it. The fin al veloc ity of the combined
blocks is 2.0 m/s. What was the mass of the second block?
285
32. II What is the ang ular momentum about the ax le of the 500 g
rotating bar in Fi g ure P9.32?
g
120 rpm
ra
rJ
Section 9.6 Momentum and Collisions in Two Dimensions
26. 1111 A 20 g ball of c lay traveling eas t at 3.0 m/s coll ides with a
30 g ball of clay travel in g north at 2.0 m/s. What are the speed
and the directi on of the resulting 50 g ball of clay?
27. II Two part icles coll ide and
Py (k g 'm/s)
bounce apart. Figure P9.27
4
3
shows the ini tial momenta
of both an d the final
mome ntum of particl e 2.
What is the final nlO mentum of particl e I? Show
-2
your answer by copying the
-3
fi gure and drawing the final
-4
momentum vector on the
FIGURE P9 .27
fi gure.
28. 11 A 20 g ball of c lay travel in g eas t at 2.0 mls co llides with a
30 g ball of clay travelin g 30° south of west at 1.0 mls. What
are the speed and direction of the resulting 50 g blob of clay?
29. 11 A firecracker in a coconut blows the coconut into three pieces.
Two pieces of equal mass fl y off south and west , perpendic ul ar
to eac h other, at 20 m/s. The third pi ece has twi ce the mass as
the ot her two. What are th e speed an d direct ion of the third
piece?
PI
~
A,I,
\ Axle
ra
ra
FIGURE P9 .32
FIGURE P9 .33
33. 1111 Wh at is the angular momen tum about the axle of the 2.0 kg,
4.0-cm-diameter rotating di sk in Figure P9.33 ?
34. 1 Di vers change their body positi o n in mi da ir while rotat ing
BID abou t their center of mass . In one di ve, the di ver leaves the
board with her body nearly straig ht, then tucks into a somersault position. If the momen t of inertia of the diver in a strai ght
position is 14 kg· m2 and in a tucked position is 4.0 kg· m2 , by
what factor is her angul ar veloci ty when tucked greater than
when straight?
35. II Ice skaters oft en end the ir perform ances with spin turn s,
BID where th ey spin very fast aboutlheir center of mass with their
arms fo lded in and legs together. Upon e nding, their arms
ex te nd outward, procla imin g their finish. Not quite as not iceably, one leg goes out as we ll. Suppose that the moment of inerti a of a skater with arms out and o ne leg extended is 3.2 kg· m2
and for arm s and legs in is 0.80 kg' m2 . If she starts out spinning at 5.0 revIs, what is her angular speed (in rev/s) whe n her
arms and o ne leg open outward?
General Problems
36. III What is the impul se on a 3.0 kg particle that experie nces the
force described by the graph in Figure P9.36?
F, (N)
1500
D
1000
PI
- 500
Section 9.7 Angular Momentum
30. III What is the ang ular momentum of the moo n around the
INT eanh? The moon's mass is 7.4 X 1022 kg a nd it orb its
3.8 X 10 8 m from the earth .
3 1. III A little girl is go ing on the merry-go-round for the first time,
an d wan ts her 47 kg mother to stand nex t to her on the ride, 2.6 m
from the merry-go-round 's center. If her mother 's speed is
4.2 mls when the ride is in motion , what is her angu lar momentum around the cen ter of the merry-go-round ?
~
2.0m
-
500
I'
o +-+-~+-~~,
FIGURE P9.36
2
4
8 10
(m"
- 1000
37. 111 A 600 g a ir-trac k gli der coll ides with a spri ng at one end of
l i the track . Figure P9. 37 shows the glider's velocity and the force
exerted on the gl ider by the spring. How lo ng is the gli der in
con tact with the spring?
F, (N)
'\ (m/s)
3
0
-3
FIGURE P9 .37
36 j
I
(s)
0
/\.,
f--------<
I
(s)
286
Momentum
CHAPTER 9
38. II Far in space, where grav ity is negligible, a 425 kg rocket
traveling at 75.0 m1s in the positive x-direction fires its engines.
Figme P9.38 shows the thru st force as a function of time. The
mass losl by the rocket during these 30.0 s is neg li gible.
a. What impulse does the engine impart to the rocket?
b. At what time does the rocket reach its maximum speed?
What is the maximum speed?
II
1
l~r ~
O~'(')
o
to
20
30
FIGURE P9 .3 8
39.
11111
rJ
O L----+-~--- t (ms)
FIGURE P9 .39
A 200 g baH is dropped from a height of 2.0 m, bounces on a
II hard floor, and rebounds to a height of 1.5 m. Figure P9.39
INT show s the impul se recei ved from the floOl". What maximum
force does the floor exert on the ball?
11111 A 200 g baU is dropped from a height of 2.0 m and bounces
on a hard floor. The force on the baB from the floor is shown in
INT Figllre P9.40. How high does the baU rebound?
40.
rJ
/\
l~r
o~'(m')
o
~;~kL
(N)
FIGURE P9 .40
FIGURE P9 .41
2
4
6
8
1200
~
o
I(S)
0.2
0.4
0.6
41. III Figure P9.4l is a graph of the force exerted by the floor on a
woman making a vertical jump. At what speed doe s she leave
INT the ground?
Hint: The force of the floor is not the on ly force acting on the
woman.
42. II A sled sli des along a horizontal surface for which the coeffic ient of kinetic friction is 0.25. Its ve locity at point A is 8.0 m/s
INT and at point B is 5.0 m/s. Use the impul se-momentum theorem
to find how long the sled takes to travel from A to 8.
43. III A 140 g baseball is moving horizontally to the right at35 m/s
when it is hi t by the bat. The ball flies off to the left at 55 mIs, at
an angle of 25° above the horizontal. What are the magnitude
and direction of the impul se that the bat delivers to the baH?
44. II Sq uids rely o n jet propulsion, a versatile technique to move
around in water. A 1.5 kg squid at rest suddenly expels 0.10 kg of
BID water backward to quickly get itself movi ng forward at 3.0 m/s.
If other forces (such as the drag force on the sq uid) are ignored,
what is the speed with which the sq uid expels the water?
45. III The flowers of the bu nchbe rry plant open with asto ni shing
BID force and speed, causing the pollen grai ns to be ejected out of
the flower in a mere 0.30 ms at an accelerat ion of
2.5 X 104 m/s 2• If the acceleration is constant, what impul se is
deli vered to a pollen grain with a mass of 1.0 X 10- 7 g?
46. II a. With what speed arc poUen grains ejected from a bunchberry fl ower? See Problem 45 for information.
BID b. Suppose that 1000 ejected pollen gra in s slam into the
abdomen of a 5.0 g bee that is hovering just above the
flower. If the collision is perfectly inelastic, what is the bee's
speed immediately afterward? Is the bee likely to nOli ce?
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47. III A ten ni s player sw in gs her 1000 g racket with a speed of
11 10 m/s. She hits a 60 g tenni s ball that was approach ing her at a
speed of 20 tn/s oThe ball rebounds at 40 m/s.
a. How fast is her racket moving immediately a fter the impact?
You can ignore the interacti on of the racket with her hand
for the brief duration of the collis ion.
b. Jfthe tenni s ball and racket are in con taci for 10 ms, what is
the average force that the racke t exerts on the ball ?
48 .
A 20 g ball of clay is thrown horizo nt all y at 30 m/s toward a
1.0 kg block sitti ng at rest on a fri ctionle ss surface. The clay
hits and Slicks to the block.
a. What is the speed of the block and clay right after the
coll ision?
b. Use the bloc k's initial and final speeds to calcu late the
impulse the clay exerts on the block.
c. Use the clay's ini tia l and final speeds to calc ulate the
impul se the block exerts on the clay.
d. Docs 1blockonctay = - l clayonbtock?
49. II Dan is glid ing on hi s skateboard at4.0 m/s. He suddenly jumps
backward off the skateboard , kicking the skateboard forward at
8.0 m/s. How fast is Dan going as hi s feet hit the ground? Dan 's
mass is 50 kg and the skateboard's mass is 5.0 kg.
50. I James and Sarah stand on a stationary cart with frictionless
wheels. The total mass of the cart and riders is 130 kg. At the
same in stant, Jame s throws a 1.0 kg balJ to Sarah at 4.5 mIs,
while Sarah throws a 0.50 kg ball to James at 1.0 m/s. James's
throw is to the right and Sarah 's is to the left.
a. While the two balls are in the air, what are the speed and
direction of the cart and its riders?
b. After the balls are caught, what are the speed and direction
of the cart and riders?
5 1. III Ethan, whose mass is 80 kg, stands ar one end of a very long,
statio nary Whee led cart that has a mass of 500 kg. He then
starts sprinting toward the ot her end of the cart. He soon
reaches hi s to p speed of 8.0 mIs, measured relative to the carl.
What is the speed of the cart whe n Et han has reached hi s top
speed ?
52. II The cars of a long coal train are filled by pulling them under
a hopper, from whi ch coa l fall s into the cars at a rate of
10,000 kg/so Ignoring friction due to the rail s, what is the average force that the engine mu st exert on the coal train to keep it
moving under the hopper at a speed of 0.50 m/s?
53. II Three identi cal train cars, coupled together, are rolling east at
2.0 m/s. A fourth car trave lin g east at 4.0 m/s catches up with
the three and coup les to make a four-car train. A moment later,
the train cars hit a fifth car that was at rest on the track s, and it
coup les to make a five-car train . What is the speed of the ri vecar train?
54. I A liD kg li nebacker ru nnin g at 2.0 m1s and an 82 kg quarlerback running at 3.0 mls have a head-on colli sion in midair. The
linebacker grabs and holds o nto the quarterback. Who e nd s up
moving forward after they hi t?
55 . II Most geologi sts believe th at the dinosaurs became ex tinct
65 million years ago when a large comet or asteroid struck the
INT earth, throwing up so much dust that the sun was blocked o ut
for a period of many month s. Suppose an astero id with a diameter of 2.0 km and a mass of 1.0 X I Ot3 kg hits the earth with an
impact speed of 4 .0 X 104]n/s.
a. What is the earth' s recoil speed after such a co lli sion? (Use a
reference frame in wh ich the earth was initially at rest.)
b. What percentage is thi s of the earth 's speed around the sun?
(Use the astronom ical data in side the back cover.)
II
II
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Problems
56. III At the cente r of a 50-m-diameter c irc ular ice rink , a 75 kg
~ skater tra ve lin g north at 2.5 mls co llides with and holds onto a
INT 60 kg skater who had been heading west at 3.5 m/s.
a. How long will it take them to glide to the edge of the rink?
b. Where will they reach it ? Give your answer as an angle
north of west.
57. II Two ice skaters, with masses of 50 kg and 75 kg, are at the
center of a 60-m-d iameter circular rink . Th e skaters push off
IN! against eac h other and glide to opposite edges of the rink. If the
heavier skater reaches the edge in 20 s, how long does the
li ghter skater take to reach the edge?
58. III One billiard baJJ is shot east at 2.00 m/s. A second, ident ical
biUiard ball is shot west at 1.00 m/s. The balls have a glanc in g
colli sion, not a head-on collis ion, deflecting the second ball by
900 and send ing it north at 1.41 m/s . What are the speed and
direct ion of the first ball after the collision?
59. 1111 A \0 g bullet is fired into a 10 kg wood block that is at rest
on a wood table. The block, with th e bullet e mbedded, slides
INT 5.0 cm across the table. What was the speed of the bullet?
P;\Clwge
60. III You are part of a searc h~~
an d-resc ue mi ssion that has
",""
INT been called out to look for a
lost exp lorer. You ' ve found
the mi ss ing ex pl orer, but
you·re separated from him by
200m
a 200-m-high cliff and a 30m-wide raging ri ver, as shown
in Figure P9.60. To save hi s
Explorer
life, you need to get a 5.0 kg
30m
pac kage of emergency suppli es across the ri ver. Unfor[Unately, you can ' t throw the FIGURE P9 .60
package hard enough to make
it ac ross. Fortunately, you happe n to have a 1.0 kg rocket
intended for launching flares. Improvisin g quickly, you auach a
sharpe ned stic k to the fro nt of the rocket. so that it will impale
itself into the package of supplies, then fire the rocket at ground
leve l toward the supplies. Wh at minimum speed must the roc ket
have just before impact in order to save the explorer's life?
61. 1111 A 1500 kg weather rocket accelerates upward at 10.0 mls 2 .1t
ex plodes 2.00 s after liftoff and breaks into two fragme nt s, one
tNT twice as mass ive as the ot her. Photos revea l Ihat Ihe li ghter
fragment traveled straight up and reached a maximum hei ght of
530 m. What were the speed and direction of the heav ier fra gment just after the explosion?
62. 1I Two 500 g blocks of wood are 2.0 m apart on a frictionless
table. A to g bullet is fired at 400 m/s toward the blocks. It
passes all the way through the first bloc k, then embeds itsel f in
the second block. The speed of the fi rst block immed iately
aft erward is 6.0 m/s. What is the speed of the seco nd block after
the builet stops?
63. I A 500 kg cannon fires a 10 kg can nonball with a speed of
200 m/s re lative to the mu zz le. The cann on is on wheels that
roll without friction. When the can non fires, what is the speed
of the cannon ball relati ve to the earth ?
64. III Laura, whose mass is 35 kg, j umps hori zonta ll y otf a 55 kg
canoe at 1.5 m/s rel at ive to the canoe. What is the canoe 's speed
just after she jumps?
65. II A spaces hip of mass 2.0 X 106 kg is crui sing at a speed of
5.0 X 106 m/s when the antimatter reactor fails, blow in g the
ship into three pieces. One section, havin g a mass of
5.0 X 105 kg, is blown straight backward with a speed of
II
2.0 X lOt) m/s. A seco nd piece, with mass 8.0 X 105 kg, con tin ues forw ard at 1.0 X 106 m/s. What are th e direction and speed
of the third piece?
66. III A proton is shot at 5.0 X \0 7 m/s toward a go ld target. The
nucleus of a gold atom, with a mass 197 times th at of the proton , repe ls the proton an d denects it strai ght back with 90% of
its initial speed. What is the recoil speed of the gold nucl eus?
67. III Figure P9.67 shows a coLli sion between three balls of clay.
The three hit simultaneously and stic k together. What are the
speed and direction of Ihe res ultin g blob of clay?
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PI
4.0m/~
3.0m/s
- - o 30g
+2.0m/s
PI
PI
PI
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287
FIGURE P9 .67
0
20g
6B. III The carbon isotope 14C is used for carbon dating of archeological artifacts. 14C (mass 2.34 X 10- 26 kg) decays by the process
kn own as beta decay in whi ch the nucleus emits an electron (the
beta particle) and a subatomi c particle called a neutrino. In one
such decay, the electron and the neutrino are emitted at right
angles to eac h other. The electron (mass 9. 11 X 10- 31 kg) has a
speed of 5.00 X 107 mls and the neutrino has a mome ntum of
8.00 X 10- 24 kg ' m/s . What is the recoil speed of the nucleus?
69. III A I.O-m-Jong mass less rod is pivoted at one end and swings
around in a circle on a frictionless table. A block with a hole
through the center can sl ide in and out along the rod . Initi aUy, a
small piece of wax holds the block 30 cm from the pi vot. The
block is spun at SO rpm, then the temperature of the rod is slowly
increased. When the wax melts, the block slides out to the end of
the roo. What is the final angular speed? Gi ve your answer in rpm.
70. II A 200 g puck revolves in a circle on a fri ctionless table at the
end of a 50.0-cm-long strin g. The puck's angular moment um
INT about th e center of the circl e is 3.00 kg· m2/s. What is the te nsion in the string?
100,
7 1. III Fi g ure P9.7 1 shows a 100 g
puc k revo lvin g in a 20-c m20cm
INT radiu s c ircle on a frictionless
table. The string passes through
a hole in the center of the table
200 g
and is tied to two 200 g weights.
200 g
a. What speed does the puck
Pin-i
need to support the two
FIGURE P9 .71
weights?
b. The lower weight is a lighl bag filled with san d. Suppose a
pin pokes a hole in the bag and the sand slow ly leaks out
while Ihe puck is revo lvin g. What will be th e puc k' s speed
and the radius of its trajectory after all of the sand is gone?
72. II A 2.0 kg, 20-c m-diameter turntable rotates at 100 rpm on
fri cti onless bearings. Two 500 g bl ocks fall from above, hit the
turntable simultaneously at opposite ends o f a diago nal , and
st ick. What is th e turntable's angular speed, in rpm, j ust after
thi s event ?
73. II Joey, from Example 9.10, stands at rest at the outer edge of
the fri ctionless merry-go-round of Figure 9.28. The merry-goround is also at rest. Joey then beg in s to run around the perimeter of the m erry~go-ro und , finally reac hin g a constant speed,
measured relalive to the ground, of 5 .0 m/s. What is the final
angular speed of the merry·go-round?
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II
II
288
CHAPTER
9
Momentum
74. III A 3.0-m-d iameter me rry-go-ro und with a mass of 250 kg is
spi nnin g at 20 rpm . John runs aro und the merry-go-round at
5.0 mIs, in the same direct ion that it is turnin g, and j umps onto
the outer edge. John 's mass is 30 kg. What is the merry-goround's angul ar speed, in rpm , aft er John jumps on?
75. III Disk A, with a mass of 2.0 kg and a radiu s of 40 em, rotates
ciock-wise about a fri ctionless vertical axle at 30 rev/s. Disk 8 , also
2.0 kg but wilh a rad ius of 20 em, rotates counterclockw ise about
that same ax le, bUi at a greater he ight than disk A, at 30 rev/s. Disk
8 slides down the axJe unt il it lands on top of disk A, after which
they rotate together. ~fte r (he colli sion, what is the ir common
angul ar speed (in rev/s) and in whi ch directi on do they rotate?
Passage Problems
Hitting a Golf Ball
Cons ider a go lf cl ub hiui ng a go lf ball. To a good approx imat ion, we
can model thi s as a co lli sion between (he rapid ly movin g head of the
golf clu b and the stat ionary golf ba U, ignoring the shaft of the cl ub
and the go lfer.
Stop to T hink 9. t: F. The cart is initi a ll y movi ng in the negat ive
x-d irect ion, so (Px); = -20 kg ' m/s. After it bounces, (Px)f =
10 kg ' mls. Thus ilp ~ ( 10 kg ' mls) - (-20 kg ' mls) ~ 30 kg ' mls.
Stop to Think 9.2: B. The clay ball goes fro m (vx ); = v to (V.I.)f = 0,
so J day = ~ Px = -//lV. The ru bber ball rebounds, go ing from (Vt)i = v
to (Vx)f = -v (sa me speed, oppos ite directi on). Thu s J rubher = ~ Px =
-2mv. T he rubber ba ll has a greater momentum change, an d thi s
requires a greater impul se.
Stop to Think 9.3 : Ri ght end. Th e ball s started at rest, so the total
momentum of the system is zero. It 's an isolated syste m, so the total
momen tum after the expl os ion is still zero. The 6 g ball has momentum 6v. The 4 g ball, with velocity - 2v, has momen tum - Sv. T he
combined momentum of these two balls is -2v. in order for P to be
zero, the third ball must have apos;r;ve momen tum (+2v) and thus a
positive veloc it y.
A golf ball has a mass of 46 g. Suppose a 200 g club head is movin g at a speed of 40 mls j ust before strikin g the gol f bal l. After the
coll ision, the golf ball 's speed is 60 m/s.
76. I Wh at is the momentum of the clu b + ball system ri ght before
the collision?
A. 1.8 kg ' mls
B. 8.0 kg ' mls
C. 3220 kg' mls
D. 8000 kg ' mls
77. I immed iately after the colli sion, the momentum of the club +
ball system will be
A. Less than before the colli sion.
B. The same as before the co lli sion.
C. More than be fore the co lli sion.
78. I A man ufacturer makes a golf ball that compresses more than
a traditi onal go lf balJ whe n struck by a cl ub. How wiU thi s
affect (he average fo rce durin g the co lli sion?
A. T he force wi ll dec rease.
B. The force wi ll not be affected.
C. The force will inc rease.
79. I By approx im ate ly how much does the cl ub head slow down
as a result of hilli ng the ball ?
A. 4 mls
B. 6 mls
C. 14 mls
D. 26 mls
Stop to Think 9.4: B. !fhe momen tum of parti cle 1 is
(DAD kg)(2.5 mls) ~ 1.0 kg ' ml<. whil e that of panic le 2 is
(0.80 kg)( - 1.5 m/s) = - 1.2 kg ' m/s . The total mome ntum is then
1.0 kg ' m/s - 1.2 kg ' m/s = - 0.2 kg ' m/s. Beca use it 's negati ve,
the total momentum, and he nce the fi nal veloc ity of the parti cles, is
directed to the left.
Stop to Think 9.5: B. Angular momentum L = Iw is conserved.
Both boys have mass m and initi all y stan d di stance R12 fro m
the ax is. Thus the initi a l moment of inerti a is Ii =
2
I disk + 2 X I11(RI2f = I di$k + "ir1lR • The Fi nal moment of inerti a is
I f = Idi.~'.:. + 0 + mR2. because the boy stand ing at the ax is co ntributes
not hi ng to the mome nt of inerti a. Because I f> I ; we must have
wr<wi '