Download Impulse and Momentum AP Physics 1 packet answers

Name:
Unit 4: Linear Momentum
Big Idea I: Objects and systems have properties such as mass and charge. Systems may have
internal structure.
Essential Knowledge I.A.I: A system is an object or a collection of objects. Objects are trcated as
having no internal structure.
a. A collection of particles in which internal interactions change little or not at all, or in which
changes in these interactions are irrelevant to the question addressed, can be treated as an object.
Jlj" Idea 3: The illtl'raetiolls of an obieet with other obieets call be lleseribed bv forces.
Learning Objective (3.D.I.I):
Essential Knowledge 3.D.I:
The
student is able to justily the selection of data needed to
The ehange in momentum of an
determine the relationship between the direetion of the foree
object is a vee tor in the
acting on an objeet and the change in momentum caused by that
direction of the net foree
exe11ed on the object.
foree.
Essential Knowledge 3.D.2:
Learning Ob.iective (3.D.2.1):
The ehange in momentum of an The student is able to justify the seleetion of routines lor the
ealculation of the relationships between ehanges in momentum of
objeet oceurs over a time
an objeet, average loree, impulse. and time of interaction.
interval.
a. The l(lI"Cethat one object
Learning Objective (3.1).2.2):
exerts on a second object
The student is able to predict the change in momentum of an
object from the average force exerted on the object and the interval
changes the momentum of the
second object (in the absence of of time durin" which the force is exerted.
other lorces on the second
Learning Objective (3.D.2.3):
object).
The student is able to analyze data to characterize the change in
b. The change in momentum of
momentum of an object from the average lorce exerted on the
that object depends on the
object and the interval of time during which the lorce is exerted.
impulse. which is the product of Leaming Objective (3.D.2.4):
the average force and the time
The student is able to design a plan for eollectiug data to
interval during which the
investigate the relationship between changes in momentum and the
interaction occurred.
average force exerted on an object over time.
Bin Idc~l4: Il1tcnlctions
_
I'roficicnt
hetween s\'stems can result in challuC's in those svstcms.
Lcaming Ob.jeetivc (.t.B.I.I):
Thc studcnt is able to calculate the change in linear momcntum of
a two-object system with constant mass in linear motion from a
representation of the system (data. granhs, etc.).
Leaming Objcctivc (.t.B.1.2):
The student is able to analyze data to lind the change in linear
momentum for a constant-mass system using the product of the
mass and the change in veloeity of the center of mass.
Essential Knowlcdgc .t.B.2: The change in Learning Ob.jcctive (.t.B.2.1):
The student is able to apply mathematical routines to
linear momentum of the system is given by
calculate the change in momentum of a system by
the product of the average l(lree on that
analyzing
the average loree exerted over a certain time
system and the time interval during whieh
on the system.
the I(lree is exerted.
a. The units f(lr momentum are the same as
Learning Objcctive (.t.B.2.2):
the units of the area under the curve of a
The student is able to per/arm analysis on data
presented as a I(lree-time graph and predict the change
force versus time graph.
b. The changes in linear momentum and
in momentum of a system.
II(lITe are both veetors in the same direetion.
Esscntial Knowledgc .t.B.I:
The change in linear
momentum lor a constant-mass
system is the product of the
mass of the system and the
changc in velocity of the center
of mass.
1
Unit 4: linear Momentum
Name:
_
Bil! Idea 5: Changes that occur as a result of interactions are constrained by conservation laws.
Lenning Objective (5.A.2.1):
Essential Knowledge 5.A.2: For all systems under
The
student is able to de/inc open and closed
all circumstances, energy, charge, linear momentum,
systems
for everyday situations and apply
and angular momentum are conserved. For an
conservation concepts for energy, chargc,
isolated or a closed system. conserved quantities are
and linear momentum to those situations.
constant. An open system is one that exchanges any
conserved quantity with its surroundinl.(s.
Learning Ohjective (5.0.1.4):
Essential Knowledge
The student is able to design an cxperimental test of an application of the
5. D.I: In a collision
principle ufthe conservation of linear momentum, predict an outcomc of
between objects, linear
the experiment using the principle, analyze data generated by that
momcntum is
experiment whose uncertainties are expressed numcrically. and evaluate
conscrvcd.
the match between the prediction and the outcome.
a. In a closed system,
the linear momentum is Lea,'ning Objective (5.D.2.4):
constant throughout the The student is able to analyze data that veri fy conservation of momentum
in collisions with and without an cxternal friction force.
collision.
Learning Objective (5.D.3.1):
Essential Knowledge 5.0.3: The velocity of the
The
student is able to predict thc vclocity of
center of mass of the system cannot be changed by an
the center of mass of a system when there is
interaction within the system.
no interaction outside of the system but there
a. The center of mass of a system depcnds upon the
is an interaction within the system (i.e., the
masses and positions of the objects in the system. In
an isolated system (a system with no external forces), student simply recognizes that interactions
within a system do not affect the center of
the vclocity of the center of mass does not change.
mass motion of the system and is able to
b. Whcn objects in a systcm collide, the velocity of
dctcrmine that there is no external force).
the center of mass of the system will not change
unless an extcrnal force is exerted on the system.
2
Impulse and Momentum Reading Assignment
Directions: Read Chapter 9 sections 9.1-9.6. As you read answer all Stop to Think questions (Check your
nswers on page282) and work through all example problems. Below is a list of what you need to take
away from your reading.
1. Define/Know
a.
Impulsive force
b. Impulse: formula, symbol, units
c.
momentum: formula, symbol, units
d. total momentum
e. the law of conservation of momentum
f.
explosion
g. perfectly inelastic collision
2. Explain:
a. how the magnitude of an impulsive force changes with time.
b. impulse and momentum theorem
c.
how to use the impulse momentum theorem to decrease the force of impact in a collision
d. how conservation of momentum is used to solve collision problems
e. when momentum is conserved (and when it is not)
3. Be able to:
a. Calculate impulse in terms of time and force.
b. Interpret a Force vs. time graph, and what the area under the curve represents.
c.
Calculate momentum if mass and velocity is known.
d. Calculate the change in momentum and relate that to the magnitude of the impulse.
e. calculate an initial (or final) velocity using conservation of momentum (see example 9.5)
3
Impulse and Momentum Problems
Momentum
1. What is the momentum for a 0.05 kg paintball traveling 150 m/s?
V:: vv-.V
::. C. CCO)UCSU) :=""\. S ~.
~
2. What momentum does a 1400 kg automobile traveling 0.9 mls (a few miles per hour) have?
? -:.'('('."
::(\~CD)
CCl ~)
7::
I '2 (PO \L<j' ~
3. A pair of 100,000 kg railroad cars move east; the first moving at 12.5 mis, the second at 11.2 m/s.
What is the magnitude of the total momentum of the system?
p -::yY\
V,;- V?- l
(
-=
'l-~"1O,ODD
'f-~. ~
iFlt;
4. 6.50 kg block slides down a ramp that is elevated at 42.0. a distance
of 1.50 m. The coefficient of kinetic friction is 0.235.
a. What is the acceleration of the block down the ramp?
fa-
'z,.(:.'j. -::
'I -
t~
ClS"'- :: lla.t;)l q.V)<;ri~4'2-
V,c,o.. = 4'2.. (q It.t;,o.. -:: ~\.4g
\ \.
C".l.~S)((".S)(q.~
\'1.
\:
0\,:;
Co~42.)
4.«
~'J.
\
b. What is the speed of the block when it reacnes the bottom?
\if 7..
:: V i '2. +
Yf?-::
c.
D'J. -f
\ v.~.-= ~.~ ~
LlA.. 6..,.,
'2.(Lt.~)(VO) -)Yf1..
=14.Y.
'------
What is the momentum of the block at the bottom of the ramp?
5. Rank in order, from largest to smallest. the momenta (P.,)I 10 (Px)s.
:!Og
r-;-L~
i
20g
[2]
.•
?1 > r\ -=
Order:
Explanation:
10 g
J(Jg
2 mls.
p,!>
· CD
2
/ll/~
= Pc; >?If
•
r:4L~
,,2ooS:
• I
I ••. ~•.
~:; 5::
;-f~>:~~.~
'O.lm/s
..•• •~~.- ....v
.
J
Impulse and Momentum Problems
The pasition-versus-timc
graph is shown far a 500 g oQject. Draw the corresponding
tUIll-vcrsus-lime graph. Include an appropriate vertical scale.
v ,;'.'
10 -
~.-, _~o :
5
~\ -=(. 5)(~) -= '1...5
I
_2SS~
Lt
(I'
'~l)
5l'V'
~
I (s)
~ 'l-
,
01'2
momen-
,
iiI
-=
t 5)(-2..5)
-::-,.115'
(,.,
3456
-I. f:l
-'L.S'
7.
The momenlum-versus-lime
graph is shown for a 500 g object. Draw thc corresponding
acceleralion-versus.timc
graph. Include an appropriatc vertical scale.
~\o~::. 0..
p(k~mhJ
_!9-
IO~Y""
5
o
/71., ,
I
2
3
4
\0
1.-
-
[5)
"'-'1
~
o
a (ml,,')
5
I (S)
(I
6
=- _r-"""';)
().."2.. = If
?z-10
--0~)
8.
+-.--.....-....,...-,..-,-,
I
-S-
r
3
4
J (.s)
6
1..'
_
I
In each of Ihe following. wliere a rubber b,,1l bounces with no loss of speed. is the change in
mOlllcntum !lp positive (+). negative (-) • or zcro (O)? Explain.
I.-@
a.
c.
b.
.-li_
~
,11' .• =
-t
l1l'.\=_
\J:::' \1\ \1.5; -
'fY\
V;
:?::. t'v\ \I.e:- - m V:
? =. \'(\
{Y\
(-v; J
!'( -=- '(Y\\lf
'\if -
~'J.f-6)
Impulse
9.
-r, -
5
m(+V: )
,='(V\lff-(rn<
'(Y\\J'\
-
..
-
b.
P(N)
5.
II
:::. '(v\
f-~
0.2
-5.
1
(s)
IJ
':i
-5
&\/'.
Yf-
e. F(N)
F(:<)
5
5
Z
'.
'("'0
What impulse is delivered by eaeh of these forces?
'l.
-- -
? -=- {'(\ "+ - lY\ V \
? ::. 'J~ - I'Y\ c.-v: )
I O.4~ I
I
I
0h
I
(s)
II
I
Is)
fiJ..lS1
.5
.L
J'-::
\(f
"2..
::.
\
(.4](5)
tV
'SJ
5
Impulse and Momentum
10.
Problems
A 2 kg object is moving to the right with a speed of 1 mls when it experiences an impulse due
to [he force shown in the graph. What is the object's speed and direction after the impulse?
a.
b.
FIN)
2.
~-t ~ t'o" Vl. - "'" VI
l'l)(\) -= 7.
2.
"2 -
2
(J.
1«)
~-t -::('1\ 111..• my:
1 (!i)
(J'
I 0.5 , 1
~'j,.'(:;J"=2v'l.- ~
\ -=
F (N)
I
\
2. \I'l. - '2.
'2. - 2..
"2~
.L.
2.
"2-
\V2 ~o ~ J
I. 5 ~ I
__
-=
--J
"bi1.1I~~cks A and B. both initially at rest, are pushed to the right continuously by identical eonSHUltforces. Block B is more massive than Block A. Which block crosses the finish line with
morc momcntum? Or do they finish with equall11omcnta? Explain.
St::.n
Finish
i
~
I
~
!
12. Blocks A and B arc pushcd to thc right continuously by identical constant forces for exactly
1.0 s. sl:uting from rest. Block B is more massive than Block A. After 1.0 s, which hlock has
morc momentum'! Or do thcy havc equal momenta'? Explain.
SUrt
-'-G:I
~
I
13. A participant in the World's Strongest Man competition exerts a force of 650 N on a 1,800 kg cable car
for 25 seconds. Ignore friction between the wheels and the track.
(a) Calculate the Impulse exerted on the cable car.
T::: f"-\:
-=
ltoS0 )l2
5)
-::. ~-\-I,.,-/l..-S-o-rv-.~'\
(b) If the cable car started from rest, what was its speed after the strongman pushed for 25 seconds?
.r-+ ~
~~
r-\:;
-::
fY\V
v.
h'\( Vi. -
V,)
2 - tv\
Y¥\lV'2. -
D)
\ loP, "l50
-=
"1. =.
'~DO'V1.
9~
S
6
Impulse and Momentum
Problems
14. It takes you all of 0.018 s to initially touch and then catch a 0.600 kg football travelling at 16.0 m/s.
(a) What is the change in momentum for the football?
1
(b) What is the impulse?
(c) What is the force that must be exerted to stop the ball?
q. t&>
-
.CiB
15. A horizontal force of 25 N acts for 5 seconds on a 50 kg object that starts from rest on a frictionless
horizontal surface.
(a) Calculate the impulse on the object?
-= (2S 'JlS)
(f:= ~-t
-::\\'2.S N . ~ \
(b) How fast is the object moving at the end of the 5 second interval?
ft-.:: P
~-\:- :; lY\
\ '2.~-::
~O~
\Vf :: L.. S ~
Yf
L Vf)
1
16. The front of 1400 kg car is designed to absorb the shock of a colliSion y having a "crumple zone" in
which the front 1.20 m of the car collapses in absorbing the shock of the collision. If a car traveling 25.0 m/s
stops uniformly in 1.20-m
(a) what is the acceleration of the car?
Xf::' V L -\:
V 7..._ V; '2._ 20.. Ay..,
+
o'l--;:.
~s) .•...
- '2. ~L\.'2.-)
(b) how long does the collision last?
Xf ::. Vi -\:;~ ~ 0.1:"72. - br-
\.7. -:. "2'St +l-I'Jo :2.')-\::.
(c) Calculate the initial momentum
?i :::\'Y\V;
=
v.+- -= Ve + o-.t
o = "2-r; - 2~o.t.tL-\;)
of the car?
01100)(25)
- f"SPO() ~. 9
(d) What is the magnitude of the average force on the car?
5<:>OOD
.0'1~
7
Impulse and Momentum Problems'
17.
•
i\ carnival g.amc requircs you to knock ovcr a wood post by throwing
a ball at it. You're offered a very bouncy rubber ball and II very sticky
clay ball of equal mass. Assume that you can thmw them with equal
speed and cqual accuracy. You only get one throw.
a.
r;~
f\-:: t.p J
~~YOu~~s~!
V\a.A
0-. C¥\'\{clkf ~
iV\.0Y' 0.0 ~
'\M.f vJ,K til ~
e'fcr
\1'\
'o()1Mlt(lj
().
VJ aJ.)
VV\~
'ooJi ~ ',1- WI\\
J6y..ifJc 0-
\~
~.
b. Let's think about the situation more carefully. Uoth balls have the same initialmomenlulll
(P..li just before hitting the post. The clay ball sticks, the rubber ball bounces off wilh
essentially no loss of speed. What is the final momentum of each ball'?
Clay ball: (px)r = . '()
Rubber ball: (Px)' =
Hint: r.,lomentum has a sign. Did you take the sign into account'!
c. What is the
chllll~e
Clay ball: !J.P., =
-(p)[)\
in the momentum of each ball?
l?iJ l
Rubber ball: !!J.p, =
2.(P)LY\.
d. Which ball experiences a larger impulse during the collision? Explain.
e. From Newton's third law, the impulse thallhe ball exerts on the post is equal in magnitude,
although opposite in direction, to the impulse that the post exens on the ball. Which ball
exerts the larger impulse on the post?
f. Don't change your answer to pan a, but are you slillhappy
with thai answer'! If not, how
would you change your answer? Why?
18.
A small, light ball S and a large, heavy ball L move tow,lrd each other: colliec. and bounce
apart.
-0
a. Compare the force Ihat S exerts on L to the force that L exerts on S. That is, is Fs "" I. larger,
smaller, or equal to Fl.u" s'! Explain. (Hint: One of Newton's laws is especially relevant.)
8
Impulse and Momentum Problems
18 (continued).
b. Compare the lime inlerval during which S experiences a force to the lime interval during
which L experiences a force. Are they equal. or is one longer than the other'!
c. Sketch a graph showing a pial/sible Fs on L as a function of time and another graph showing
a plausihle FL on S as a function of time. Be sure to think about the .\'igll of each force.
,..
~ ••• I.
-I
d. Compare the impulse delivered to S 10 the impulse delivered to L. Arc they equal. or is one
larger than the other?
e. Compare the momentum change of S to the momentum change of L.
f. Compare the velocity change of S to the velocity change of L.
VL <. VS
WlL'>
('oeC~
Y\'\s ')
Conservation of Momentum
19.
As you release a hall. it falls-gaining
speed and momenlum. Is momentum conserved'!
t
a. Answer this question from the perspective of choosing the ball alone as the system,
r
Y\O{
~C~
'1
W
~
..).6
QcA1Vt}
--t-,(~Y\-t&
{U-rL-(.
0etLe
Em.- ~
b. Answer Ihis question from the perspective of choosing ball -I- eanh as the system.
if' \of
C ouA+l
-i-htX't
(U'i
VlO
.{,)Lj-erno1
~(LS .
9
Impulse and Momentum
20.
Two panicles
Problems
collide, one of which was initially moving and the olher initially at rcsl.
a. Is i( possible for bolll particlcs 10 be al rest after thc collision'!
Ihis happens. or explain why it can't happen.
'{\O
.leD De
'o~C~
~
~eJ
'(Y\O~.
-ttNJ
<A.{\eIfV\-'Clf'dS
'ZS\OvQ\
I
Lt51A.A6-Ot-
'Ou--;\-
~
b. Is il possible r r olle panicle 10 bc at rest after Ihe collision'!
happens. or cxplain why il can't happen.
r~
-tw pOJ'h
~
VV\evft.
~
~
.
CZ£-
Givc an example
Give an example
pc;v.1J1 c.{~
\OJ"C( ?c:urt\c\e:' Pf -=
V\.()JJf.
0-.
~ ~
~
-tce[)..lJ.
~
~
'o€. t\V'\c0l. l?).
in which this
•
~'5jf~
to WI olDd
LOi.j..Vv
D.
~
V\I'{)
I.r00
VV\\)V\1\3 V~
V'v\lAlc
c.t.e iY\oviVVj.. ~
+\Ao.-Vl
in which
vi V'j
J> ~tl
21. At the lak out on a float is a water slide. You have been sliding down-It and
landing in the water with a velocity of Vw all morning. Then a large, very fast
boat comes by that is making very large waves with causes the rope that
anchors the floating slide to break so now it is free to move in the water. After
the waves dissipate and the water is smooth you decide to slide down the slide
again. If the water provides no resistance force to the motion of the floating
slide, how does your velocity when you hit the water compare to Vw?
Explain this result in terms of the law of conservation of momentum.
will
f.,1l ~
IMV<' l?f' oJ- Y'2*
r~' =
10
Impulse and Momentum Problems
22.
PhyzJob: Conservation of Momentum
II )
(
n the Rail Yard
"
'''___
Ii
SIGN CONVENTION:
1. A 58RN woman stands atop a SOOkgflatcar; both are
initially at res!. The woman then nms to the right and
jumps off the right end with a speed of7m/s.
a. What is the initial (and tinal)\momentum
~~<;~ ~~\~fia
~h:
f-~
+
.•...• ~ .... i!!t _.. _
of the system?
\7 ::;;0 _
fi(~~;IL~)II~
\~;OJ~?f'\IS
-
:~
...............-. ~.
_ •• <;!!5." :. • ••
••
)
c. What thcn, is\11e final momentum of' the l1atcar1
~
- L-11..0 ~
«POl'\J'J
= [ 625
\J ~ -~o
'f
2. 100kg Digby exerts a 300N force on a ?5,000N b-ox-c-a-r--'~
(initially at rest) for lOs.
a. What is the impulse imparted to the boxcar?
b_ What is the speed orthe boxcar when Digby lets go?
r¥\::
\~'
J:: 6\/:::
q .'6 ~
\<;J""o'v
3Q1:5O:::
\j_-
1'i">'30.lt>/V)
1°0;'
16
.V
~
c. If the boxcar then couples with a 2,OOOkgcoal car.
(initially at rest), with what speed will the two
subscqucnllymove?
?i:::f(:
"'bbo
0
\If=:'
-= ~
'S'3D.\0 -1 'Z.O()()] Vf
=)
""SOD 0
:, 5'60. ~
d. Felix thcn grabs a ropc dangling frolll the boxcar and
SlowsYle two railroad cars to rest in 15s. How much force
does Felix exert whlie stoppmg the cars'!
(f -
Dr
/~
.
~-\:; =. Pf - 'P~
-Fi. -= 0 -
f
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11
Impulse and Momentum Problems
22 (continued)
3.
Il1A.~
~
.......
.::::
fig J.
50,OOOkg
rTlB ~
10,OOOkg
'I"
~'=D
~. .=: .. ...~
fAl
.,",."...8. "'.~.•.,'~
~
fig.
2.
•.••.••
rTle =
.._...
30,OOOkg
Illo ~
III
f!l...
'.
~.n
•.•
5,OOOkg
0
~~1!IS
~~n~
'i'=?
:•••• 1••'•••.
~~~i#i1'::~~JiJ{PJ~.*
ri
,
. .rrmmrrm
_____
.-
-
~
- .- -
-
-
..•..•......• -..._.
_
•• » »
NO..
.__ ....•.
m-.
fig. 4.
oJjigurcsjin' exercise 3
I. Boxcar A moves wilh speed VA Imt'urd empty coal car B initially al rest.
2. The 111'0 collide elastically,. the coal car moves with speed v'JJwh;Ie boxcar A moves "vith .\peed v'A3. Coal car B then couples with hnxcar C oneljlalear D; thc)' all move lvilh speed v'BCI)
4. Since v~.lis greater t/zan v'BCI> hoxcar A catche,'}'up ...d,1l cars B, C. and D. This lime boxcar A couples
with cars B, C, and D: aI/move at v:1BCf.}
E\lJ/anllliol1
a. If VA = IOm/s, what is the momentum
of the system in figure I?
VA:: r<I~ VA :: ~~OO)()o)
b. What is the total momentum
t3
00,
000
= \SOO,~oo
~
~
J
-= '(Y\p.,V" T ('(\-e"g
'DOCPOO -::: ~\),OtX»lll.~;) +~OjXX»\rS
?e,f -=
=
VfA
I
the speed v's of the coal car?
d. What is the momentum
\
ufthe system in figures 2, 3 and 4'1
e. If the speed of the boxcar aftcr the collision shown in figure 2 is
500, DOC>
'f-{j ~
6.67m/s, what is
19 l.P'5 00 = lC}QOOv'.s
10, 000
\ V5f=
ID~
\v.LPtQ
~]
~
\
of coal car B after the collision?
l'f\& Y\?f
-=
CIt>/~OO) (\",.LPS)
-
[I V>lP SOO
~
e. What is the spced of cars B, C. and D after coal car B couples to cars C and 0'1
\ LP (y500
--=
(N\
fb T \'"Y' c. -I- fY\c.I)
V-r
)
\1.9v,5ClO
-== [Lo,oOJ)+( ~O,OOO)t-( '5,OuO)J Vi='
C What is the final speed of cars A, B, C, and 0 in figure 4'1
£
I VI
jUlJ>5o.:> = ~
=
l
3./ ~
_.---
-'J
12
2\1TI~~OOO,OOC .6£
The Book of Phyz@Dean
Baird. All rights reserved.
5'quo
4b~()
4{;000
db
'-f
Impulse and Momentum
Problems
23. During practice, a quarterback throws a 0.43 kgJootball with a velocity of 8.2 m/s to the south into a
.]&Jsg.trash can laying on its side. The trash can travels with the ball after the collision. What is the final
velocity of the combined mass?
ft\~.Lj~
4
?',::: 'Pf
rfbJlvin,) -to'~lVi~)~
0'.f-b .•.vY\.0V-F
(4:' )l<{". 2.') + 00-..~-::(tn +. Iv) 'if
3. S'Z...lJ> -== (1C\) V L
Vift>-= ~.2-0?;
f'l\.r ::: • \ \.t \L-g
V; r-":: 0 ~
Vf- -=- ?
. '5" "I
•
?5:"l
24. A 2.15 kg ball (m,) moving at 5.00 m/s to the right hits a 1.15 kg ball (m2) head-on that is traveling at
3.50 m/s to the left. The second ball (m2) ends up going to the right with a velocity of 3.00 m/s. What is the
~
~@\.15\'9
VI f?
G>---7
I/.M'
(Y\\ V,
=-
y:
'":eflocity of the first ba~ after the collision?
~.\
Y,;"S'5
VJ,j. :0 3.'55
7
-t"
p~
1"\1. VZ- ~
'::
-.I, r
(Y\,
rY'
-= 3 -:;
1'<'12.. v' 2.f
2..16 'If -;-(1.\5)('3)
(?.IS)\'5)r(t.\5)(-3.S):::
10.,5" -
-+
y.b2S
-= 2.I~Vf--t- ~.'-t5
U>.,2 5' - -.,.'1 C;; = 2.15V.p
G~-~S-~
\!+-- \. ::,
I
25. A 3.50 kg object, object A, moving at 5.50 m/s to the right collides head on with a 5.00 kg object, object
, that is a rest. The 3.50 kg object ends up with a speed of 0.50 m/s in the opposite direction. What is the
.elocity of the two objects after the collision?
';SV-C3
"'.esc;;
0,----,7
L-@
.s ~
l?J.'S')CS'CS') -\-~)lO)
'>'f-<;l
@
-==
1:>.6(-.'5}
t-
Slv~)
O~
S
\
-r CO"4:-
''I, ') :: - \.l5
.2.,1. '2..
"5
-
'5
Yf -
11.213
'?
26. Dan (50 kg) is gliding on his skateboard (5 kg) at 4 nfj5."""ReSuddent<,rjumps off the skateboard, kicking
the skateboard backward at 8 m/s. How fast is Dan going as his feet hit the ground?
C'50 +SX 4)
2.1.X))
-:: 'S"D'If"-=:-
S L-«j
'5 0 v+- - L-t 0
e~!-1
wO-:=: SOVt=
M-------;
13
Impulse and Momentum
Problems
27. You find yourself stranded on this very slippery sheet of ice. There is s.9 little friction that you can't walk
at all. No worries, you've got this lovely 2.5 kg physics book. You throw it away from yourself giving it a
speed of 8.4 m/s. Figure your mass at 42.0 kg.
~1.r'3
z.'<;¥"j
V,,"Z 8',Lt~
(a) What is your speed after you let go of the book?
0 -r
c;
p; -::Pf
v-:::oTD.
s
Vp~
0:; (j.'S)l<?lf)
-'})
\_VP-=
-(~2)"'p
::: - /.-f2 Vp
-.S~"
_
~
~
0
_
(b) How much time does it take for you to reach the other side of the ice 15.5
Fnetiw\- W ve,\a 0:
\1\-0
V:::
.9-t
:::i'
t~AV
IS,E,
01
away?
Q.OY\-.n-~-t
-
\50
6\~
.5
28. Determine whether each of the following graphs represents an inelastic collision between
Ohject A (solid line) and Object B (dashed line). The objects in an inel,lstic collision Illust
stick together. and the collision must conserve momentum. Part of your explanation should
consider Ihe relative masses of A and B.
NOle that part a is a posilion-versus-lime graph, but pmts b-d are velocity-versus-time graphs.
r"
\
'
•••
;
~
b
0
o
0
DIe A c.. 0' ~
i(\
UjJ
I===-=--=--=-i-~\~\~\\l'. V\9.icl
I
-\\u SC0'YU- p\o.-c.e.
Y'l'\~o--lY'-
tw
'f ~
'0'0 \-\t\ \;V}.J'lf. SCUYU
Ofu(".
~()..,VvU
DO~ h M '6\) w\ \i~(jY\."is ,
c.
1
J
A
'{'vQ;~
\.
lm~~-~-
c..
d.
A
0
1
---ii---
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d-o
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tcu'l n£.l- ~ ~~
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~f~I
yV\.~
~ct
Impulse and Momentum Problems
2.0 m/,
250 kg ~,
~,
---15
III
200 kg
---I
29. Several students are riding in bumper cars at an amusement park. The combined mass of car A and its
occupants is 250 kg. The combined mass of car B and its occupants is 200 kg. Car A is 15 m away from car B
and moving to the right at 2.0 mis, as shown, when the driver decides to bump into car B, which is at rest.
(a) Car A accelerates at 1.5 m/s2 to a speed of 5.0 mls and then continues at constant velocity until it strikes
car B. Calculate the total lime tor car A to travel the 15 m.
Q 'If =\J i.
0
+ ()...t:,-t
V"l. -:::~
S -:::\~
S~ 2 1- V:; f:::,'c
~
~
:; bt'
=
1--5
.
\. -J
l:f -;(~)l1-}-!' 1(1' '5~
-) ~IXf :: 1\'\'\
.
-1:;:;0
\.Ii
(b) 'Alier the collision, car B moves to the right at a speed of 4.8 m/s.
1. Calculate the speed of car A alier the collision.
V'I t'f
-:>
(2'50)('5) -= ('2.'5b)\If
t(UJdJ '-t.~)
\ 'l.'50 -9(00 -:: 2. '50 \if
ii. Indicate the direction 9f motion of car A after the collision.
__
To the left _.../__ 1'0 the right __
None; car A is at rest.
Explain your answer
15
Impulse and Momentum Problems
~ ,.0
F.llJ',ine
,wl.
;~i,..
E,.,~•.••sh\ll> 00'00.
___
0"",""
M•• lm"",to:~"':
p.n<;h<Me
Iiq:ok>,..
30. A model rocket of mass 0.250 kg is launched vertically with an engine that is ignited at time t = 0, as
shown above. The enginc provides an impulse of20.0 N.s by firing for 2.0 s. Upon reaching its maximum
height. thc rocket deploys a parachute, and then desccnds vertically to the ground.
(a)
On the figurcs below, draw and label a free-body diagram for the rockct during each of the following
intervals.
(b)
Determine the magnitude of the average acceleration of the rocket during the 2 s Jiring of the enginc .
.J"~ f. -c
'::.'; f =r
L~
-.t.1"" ~ yY\ ()... ::: -r
-
20N'l.
7..g
-=
\oN
f"?l -::.";> • 2-';
-
What maximum height will the rocket reach? (caution....
O~
-=
'l~"" ?
l"
W't\
'L--z.
"l
0..::,,0,..
t-=- 1.S
_.....,
b'f. - ~
(d)
X+ -=
~lOO _"L)l '-I)
2.
lPo.4rn
t
=
0-::lPO.tf
-t&q.&')t
-~3.---=--~
...
____
-9.«
-\:;~
-
tp.\\.QS
"f "l-=- Vi
Vf -=- 0 :/~rYl
0= (uo-"i)'1. -tt-I't,Io)b)C
-:: - .,,'
-\:;-:;:.
~
-b
s
L
"77.
~
C!willthe maximum height be reached?
-+
-~lt~~.\1.L"
-~\p
- _}'\.v
- ?
1:>'1-- ..
b t~-t'-b~'2.
V.f::\J~ .f(}...t
~O.'l. """/,'are wo I crent phases of motion)
2- •
vi:: vf~:I:
(11.,
\II' -:::V-I + o..:t
_
r -=- (~O,'2- "Ie. '2.') = G,c ,&.oj • '15
(f\il~'i.')
time aft~
l' z.~JLcd')
V_~
X.f ~ kfv_~;b\.o--t:'1.
-:: .L l"bO,'2..')(2)'-
\'VI.Cl.r.<:c.
\0 -
0.. -
0..=
(c)
\fo:
iii. After the parachute is
deployed
ii. Afrer the engine stops.
but before the parachute
is deployed
i. While the engine
is firing
t>'to -=- W LF•\
•
"l(-'1.Y)b)(.
_I~"
.,
':)
~
l"~\
•.••.
J
x-= \~.\~+uO.l.I
-=-""\
__
::: 1.4t,. 531"1\.
~
-
-\:;::: 1.- + LJ, Iv
~ -\::;-:. C6•
I It 5
'---16
J
Impulse and Momentum Problems
31. A 70 kg woman and her 35 kg son are standing at rest on an ice rink, as shown above. They push against
each other for a time of 0.60 s, causing them to glide apart. The speed of the woman immediately
separate is 0.55
m/s.
after they
Assume that during the push, friction is negligible compared with the forces the people
exert on each other.
(a) Calculate the initial speed of the son after the push.
"?~-=
Yf
0:; (/U)~5)-C3S)VI?
-3~,5 -:: 05 Va
(b) Calculate the magnitude
of the average force exerted on the son by the mother during the push .
.f:::
£:J!-
..,0('~5)- a
1:;
.It>
-
(c) How do the magnitude and direction of the average force exerted on the mother by the son during the
push compare with those of the average force exerted on the son by the mother? Justify your answer.
f!1w& tuA- c9f100Si I-€
I
fS -= (3CS)(
f~) - Q
.tp
-
\
-fs =
+..,...
(d) After the initial push, the friction that the ice exerts cannot be considered negligible, and the mother
comes to rest after moving a distance of 7.0 m across the ice. If their coefficients of friction are the same,
how far does the son move after the push?
Wt -= () v..
JAI'N -:
1
0 - 2- I'Yw
L.
jAv-tMl~\.~)l1)-::-i~('55)'l.
••
~ \'--:; (-. S5)_
2("\.g)l1)
-
. t:>C)1- 1-
17
Imp
6 Momentum and Collisions in Two Dimensions
32.. An object initially at rest explodes into three fragments.
The momentum vectors of two of the fragments are shown.
Draw the momentum vector P3 of the third fragment.
? /' "5)( :'-\
- 2-
~ \.. ~3~-=
-2
J _
1'. (kg m1s)
33 .. An object initially at rest explodes into three fragments.
The momentum vectors of two of the fragments are shown.
Draw the momentum vector P3 of the third fragment.
I'ULcl
to
-t '3 t'l'
G(..VY\ c~
('?sO - "3py:)
p. (kg m/s)
-2
,
~~~~
,
A 500 g ball traveling to the right at 4.0 m/s collides with
and bounces off another ball. The figure shows the
momentum vector P I of one ball after the collision.
Draw the momentum vector P2 of the second ball.
C: £)(4)
P i.,f
=
\\0... ':l
35.
-2
1', (kg
._--,
m/s)
.. 2
Pi
2 Y,
p .• (kg m!s)
2
-2
.• 2'/.
CCA.N\c-V, ~
l,o'i~
I
..-' _-.1
CGU'\.c..u.wl. .
34.
,i
'il'I:l
~
PZ-'1 -~.
A 500 g ball traveling to the right at 4.0 m/s collides with
and bounces off another ball. The figure shows the
momentum vector P I of one ball after the collision.
Draw the momentum vector P2 of the other ball.
It;;)L Li) -;: 2- p;
() - -lvl-"2.~
f",f~ J
P,. (kg m/s)
p. (kg m/s)
~
18
Impulse and Momentum Problems
36. A 23,000 kg cannon mounted on a railway car fires at 45 kg shell at a velocity of 650
'4° above the horizontal. What is the recoil velocity of cannon/railroa~car
//0
45kg
aher the cannon is fired?
?:: Z.qI 'LSo ¥-~'1
i:;l~t,O
v=o~Om/s
mls at an angle of
~~
- -"..t" -
q,
recoil
230.000 kg
fi = ()
'b'o
~
'tic. "+. -= me V.fc
Rsl'-t
>(-\= -::
Z4}.'-t'l
"'7
23,OCCV~ (. -::.2'"\. "2-lj'l
=CJ.-q,1..So)
c..o~3lf
:: 2.11 ,2*<1
~9s
~C~
=
Yf-L
I"h
'2l..ls '2.4Q
2 ~,000
\ '\ffc. -:: \ .DS
I
-f \
mlsec to the right collides with an identical object B
which was initially at rest. Aher the collision, object A moves with a velocity of 20 mls at an angle of 3r
37. Object A of mass 1 kg traveling with a speed of 25
above the horizontal while object B moves at an angle of 53° degrees below the horizontal. Determine the
speed of object B aher collision.
-
'37'~+
-,
I ) ('2.'5 ~,,~
_
B' -= \'f\ \I
- l\
~
J -
2c:;u. ~
~. J S
A\
A
. VAl
... '
()
t"jI.
+~1-,'1re.fx
:: Q)( '2.0) lCP~31}
2.S = I C:;.q1 -t Vrf,y.
Vs
I After
;'~3
(')
t'Mi -=
I
~Bf
"
'I
_
+ (I) VSFl<.
1
",o3'f
OJ.O~~S
t>X -
'Vh,&,:: ~p.':}f- r~
o = Q) (:20) 'ls\Y\31 ::;0 h'6'jf
"
~
V -= J[?.
\JfMf :::. 12..01>\P~
1'2..0
~~t
-t!L'2..
all';
OoLD
f -::
\c;) .\j-tl ~
mls to the right collides obliquely with a 3.0 kg puck initially at rest. After
the collision the 2.0 kg puck travels at 2.5 mls at an angle of 34° above the horizontal. What is the velocity
38. A 2.0 kg puck travellthg at 5.0
(speed & direction) of th
.0 kg puck after collision?
?\ i. " ?fl'( ,..."
l?-':l(C;) -;; l'}.
')
7-
2.\CL3@ S"~ ~-;}'
34°
or-
~.-<C
,? ) '-"-'"
-:; O~~'7
S
•
I I
\0-'"1"
I' L 5 "
r. "3)1;
'6Y
1"?~)(.
'?Wfsrr.
,-
(7,,?" 1>
I
Ed :: D -:: PA~+ (J~
l2.) ('2.., 6') 'iri 1>l..t :: 1J cl
V\
2- . ~ ~~
6
~ Y'Q.'1
fr;
~J\2'~C:S+tz..~)-z..-= (P.Lt~Yj~
?Q.:::
'2,.. '{
'('<1 Q, \l1bf
U .4'l -: ~ Ve.f
~,,::: '2. .\\;
l'
-I>v'lo= S-~
--- ~
19
Impulse and Momentum
Problems
Answers
18) a) equal b) same c) same graph just mirror
1) 7.5 kg m/s
images (second graph is negative of first graph)
2) 1260 kg m/s
d) equal e) equal f) VL < Vs
3) 2,370,000 kg m/s
4) a) 4.86 mis'
19) a) no b) yes
b) 3.8 m/s
c) 24.7 kg m/s
5)2>1=3=5>4
20) a) no b) yes
21) Less
6)
~~
,\
",
-
....,t. ..••
22) (answers at bottom of page ...use a mirror)
23) 5.98-m/s
"
24) + 1.52 m/s
.
"
u
25) +4.2-m/s
26) 5.2 m/s
27) a) 0.5-m/s
b) 31-s
28) a) yes b) yes c) no d) no
29) a) 3.6 s b) i) 1.2 m/s ii) right
8)a)+
b)-
9a) 1 Ns
c) +
30) b) 30.2 m/s2 c) 244 m d) 8.1 s
b) -2 Ns c) 1 Ns
10) a) 1.5 m/s
31) a) 1.1 m/s b) 64.2 N c) equal d) 27.5 m
b) 0
32) missing vector needs -2 py and -1 px
11) Block B
33) missing vector needs -3px
12) Equal
34) vector needs -3py
13) a) 16,250 kg b) 9 m/s
35)vectorneeds+3pxand+2py
14) a) 9.6 kg m/s b) 9.6 kg m/s c) 533-N
36) 1.05 m/s
15) a) 125 Ns b) 2.5-m/s
16) a) -260.4 mis'
b) 0.096-5
37)15 m/s
c) 35,000-kg m/s
d) 364,600-N
17) a) Bouncy b) CB = 0 RB = -Pxi c) CB= Pxi RB =
2(Pxi) d) Rubber e) Rubber f) totally stoked!
38) 2.2 m/s at 30° SofE