themotionis given.Remember
thatv:2nlT,wherc r is theradiusof thecircularorbit. You can
usethis expressionto eliminatev in favor ofr or r in favor ofv in the expressionfor the
centripetalaicceleration.
All relativemotion problemsareessentiallyvectoradditionproblems.Identifi the two reference
framesof interest.Identifr the velocitiesin the equationi pa= i ps+ i sn, write the equationin
componentform and solvefor the unkno*rr quantities.Sometimesthe equationmustbe
rewritten in tenns of mag,nitudes
an6anglesratherthan in terrnsof comp-onents.
Questionsand ExampleProblemsfrom Chapter 4
QuestionI
The figure below showsthreesituationsin which identicalprojectilesarelaunchedfrom the
ground(at the samelevel) at identicalinitial speeds3a6 engles.The projectilesdo not land on
the sarnetenain, however.Rankthe situationsaccordingto the final speedsof the projectilesjust
beforethey land, greatestfirst.
(b)
(c)
q>t>c
Question2
An airplaneflying horizontallyat a constantspeedof350 lcn,tr over level groundreleasesa
bundleof food supplies.Ipore the effect of the ah on the bundle.What arethe bundle'sinitial
(a) vertical and (b) horizontalcomponentsofvelocity? (c) Wlat is its horizontalcomponentof
velocityjust beforehitting the ground?(d) Ifthe airplane'sspeedwere,instead,450 kmlh, would
the time of fall be larger,smaller,or the same?
(q)
(b)
35o Ko/t,
I Cl
Z9o(nf'n
(,J)
tVv
aor,'-
Problemt
Thevelocityi ofaparticlemovinginthexyplaneisgivenby
n=6.0t-4.U\i+8.0j,with
t in
meten per secondandt Q 0) in seconds.(a) What is the accelerationwhent:3.0 s?(b) When (if ever)
is the accelerationzero?(c) When(if ever)is the velocity zero?(d) When (if ever)doesthe speedequal
10m/s?
'""""
-- | 1a"o,7r')t-( q"onfs')
to ] t * (t.o,fs)!
V-+
--t
r-\
|
/^\
Tt
(o)
= f'(t"-o^/s,)q- = dt a ,
(s"oef
sr)t7 t
/at
(- ta"/s) I
(U) .^* u-",J[ ar = o ---)
'" /
t = (6"on/e)
( u-o,/t>)' 6t.o.lsz)t --o
/(r.o*ls')
-----)t= o"c75s
(c)
rd)
vx - r b.on/s
,l
((""Dnk)t - Lq.o^/s,)7\= -tn"or1r
---'a (-tt.o.fs)t\+ t "o,')/s^)
tf 6"0n/5=o
f =g.r.
Problem 2
positionis givenby r = 1f.OO4i- 1+.OOl)j+ Z.OO;0,
An elecfton"s
wlTl
andr' in
meters.(a) In unit-vectornotatio& what is the elechon'svelocity t(t) ? At t : 2.00 s, what is i
(b) in rmit-vectornotationandas(c) a magnitudeand (d) an anglerelativeto the +x-axis.
^g
J--
f'&) A^A
b l'l*Aon "u '.
+ (a.,".)p
t - LU-oo,1t)t'J3
C(*) = l1z.oo,/s)t1
(o) i=J?/tt=
(:.oo"7,;i
')zl i
-lG*"./'
(a.uos)i
(b) 7(*=e.oor)= (a"oo'/s)i -Q"oo',,1')
( a . o o o / . ) ? - ( I t " . o , o / ii )
(c) lv'l
/\\
\o/
-,(y;_\
n _
+__t /-tt .o-lr,\
- )F
=
d
n
v
\ v,(/
)a!\ t;.r,r/
lIo.5 n/<
O =-??-t
Problem 3
A rifle is aimedhorizonlally at a target30 m away.The bullet hits the target 1.9cm below
the
TT'le noint. what are(a) the bullet'stime of flight and(b) its speedasit-emergesfrom the
rifle?
-ro
ri = -O, Dl9rq
/= T;
,I
^
Y4v
I
VoY = u r"/,5
a.,,=-1.tonfs^
t=
---o
)
='/*qrl'
1(- o"otQ-
.I-.-.
./
\ --l
'--o
y-y" = V"lt*//r4)l'
(q)
V^ = O r.)
o,062.s
L-l
/t-\
X^= om
X = gcj,
\ v/
oX
:
X = Vort
X-X"=Vo*t
\/ o x ,- x../, "
r)
30-./
/ o.o6ts
/ C
V,* = 4"4^ 1o"no/s
t= o,oae.s
Problern 4
A small.ballrolls horimyally offthe edgeof a tabletopthatis 1.20m high.It strikesthefloor at
a point 1.52m horizontallyfromthetableedge.(a)Howlongis thebdl L theak? whar is
O)
its speedattheinstantit leavesthetable?
-Y^ o .'r
V...7= o,o/s
Gy = - 7,w-)s
c-- I
x = /,5er",
Y"r=?
t=
4 o
-- Vryt* )/larfr
= t/sa1t^
)
t
-'-
v)
Xo = O r''
y-y"
= - l'?on-.'
)
b)
..o
[{
Yay
(-l"ro'"
- q"tor.,/Sa
O"Ll75s
o"tll5s
\/
\,/
A-
Ao
--
\/
iF
L
Vox
\ , / --- \' JO ' . e +
,(
\/
VoX
=
Y^
/
,/t
=
| rl
1.2<
/ t
I
r I
- 3"b'to/s
vox
I,-------_________
I
l,
I
n
t/)
rrq R<
\V ,
Problem 5
The punteron a football teamtries to kick a football sothat it staysin the air for a long "hang
time". (a) If the ball is kicked with an initial velocity of 25.0 m/s at an angleof 60.00abovethe
ground,what is the "hangtime"? (b) How far doesthe ball travel beforeit hits the ground?
f' "yl *'z"-zl-i,
)
V,y = ( )9.o,,l)
fu;ak fuXAtf pa"i'J'+J^^W ^/
) &L;* '<Dtu'"r'-k
zt/
= V"r +art --- O = V,.y+ 4y t
V..r
-lt
ltiq $o.s
- Ar"to/s
-t = -Y "j/ot
o,rls ) 9'-1M WA
t j-
- )l'\,ls
:
n
/\-
o(. 4r
- ? ' t o t . t1 s
hbf"^t- r\ a;4"- e (r..1,) :FQ
,-t
-|1"zonls^
t=
(>
/\ "_/
v,)
Xo
X- X" -- V"xt
= 6 ttl
x= ?
X - Voxt =(tt.so1'1G"1)
Vr" = (a5"o,4i, edrLo"
:
lA"5 n/s
X -- 55"oo
4"1s
t=
Problem6
A rocket is fued at a speedof 75.0 m/s ftom groundlevel, at an angleof 60,0oabovethe
horizontal,The rocket is fued toward an 1I .0 m high wall, which is localed 27.0 m away. By
how much doesthe rocket clearthe top of the wall?
,t.
"*
,/
l9?:
-i,,'x w*"J[ y ,u{sr r= At.or.)
)
u
i' -8rid t ulx\s,Lx- S?,orq",,uunt b -8'}"Jy
L>
o\ /. <-rrq
tro
X - Xo--V,*d ---+ X = Vo*f ---+ t= x.*
Xr--or',
X = !?.on"t
t =o.tas
Voy = (ls.or1rS qr: bo'
_
3.t-5n/5
t=?
't/sa1t"
-
y ={rf
*lsayt'
V^:Or')
v= ?
V,) - L?5'onls)lvmbo"
-b5'os1t
/a-
/\
-)
^
= -4ra*/.f,
t.ov.
7 _-o.7),s
/
/>
\
-
_,y = (aao,ls)(s"tas),t/s(-1.zo,lt')@.rar)^
,,1. 41-1,3
m
h-- 4'1.3"!,-ll.oq =33.3m
/
'==-
Problem7
A golfball is struckat groundlevel. The speedofthe golfball asa function ofthe time is shown
in the figuni below, wheret: 0 at the instantthe ball is stuck. (a) How far doesthe golf ball
travel horizontallybeforereturningto gound level?(b) What is the maximumheightabove
grotrndfevel auainedby the ball?
9 ;ly" r,vga,cfu u c ^.nrrnl/yyurft d tg, t"-{Ard
p"rrt "rt ^" Vy = o-/s 3 J b" )u7A""I [riJ,
t1
9A
c9R
- 9e,
--)lS,-,n'Vuy'44, Vox=l9o/s*"^)r"tre!-Ln' G
Jfu "-r'tLa s"os
9l
l9
(q)
01234b
, (s)
x=?
*.'-"1
-
x-X"=r,/)*f ---; X = v " x f
- ---+lX=95nr
X =(lr"1s)1s.os7
"o
V,x = \1..,/s (
I
t= S.os
Xo=orq
(U)
\-.
Vo = 3)"r1s -*
t M
t.A.s?+
/o=or.r
) =?
rc^n {.^ird V,y = ?.?"5'n/s
.ao
w"n ""N
V'y - 11.5"r/5
q) - -9,t0'r/sl
jAarl \
y - ). = V"yl*
y = V".,t*'{ay*r
v=(r?.s"/.) (?.s:) *'/r(-t,z'/s) (r-sr) * :l t,"n.,,
Problen8
A plane,diving with a constantspeedat an angleof 53.0owith the vertical, releasesa projectile
at an altitudeof730 m. The projectilehits the ground5.00 s after beingreleased.(a) What is the
speedofthe aircraft?(b) How far did the pmjectile travel horizontallyduring its flight? What
werethe (c) horizontaland (d) vertical componentsof its velocityjust beforestriking the
ground?
z4v
r/ = - V^ t-ac53-o'
/i>.Qylr
"
*
{: J" V"yt
)Aa')t\
O: yqtVuyt t
o)
r/" = ?3or"'r
\
I s'.o"\
V,
= or"'
!
t= E,oo,
O
A,j='1'noqf sr
= JZo r, - \/oLz)53.o"(s.ao5) - 1-1:t-r,(s,a)\
vo--a,.li1=':",i'-'''::lil.;l
'o'qs,.o'(s'"ot)
V'=o."t
--J
[U) X-X"=v,"t
X=Vovt =(ral./')(s-oo';
x=7
=
Vo" Llurq/r) ax"Eze
= I b Irr1s
(J )
(.)
V* = Vo*
Vy = Voy+ay f -= (-lozel)
-- t\tnlsl
IV.7
a-.-.--t
I
(t'zo-/s') (rXj
ooc53.oo1
r
Problem 9
In the figure below, a baseballis hit at a height h : 1.00m andthen caughtat the sameheight.It
travelsalonlside a wall, moving up pastthe top of the wall 1.00s after it is hit andthen down
pastthe top of the wall 4.00 s later, at distanceD : 50.0 m farfheralongthe wall. (a) Whal
horizontaldistanceis traveledby the ball from hit to catch?What arethe (b) magnitudeand (c)
angle(relativeto the horizontal)ofthe ball's velocityjust after beinghit? (d) How high is the
walf?
t=Z.oos
it\\s
t=,1'b0t
aaaat--
-
rT-'
AX= 5D.Orn
Vtu9r, = AX =
At
d =Soos
At= q. oos
,',
5O.Orr
_
| ),.5efg
+ . o os
I
I
Nir/t
/y .^a e '1,,.^a'6'))
V1=V""=
t=3.ooe f. Xfi fr,'-4j.*
&ffi'rq
Vu,x" -- I e"s '!/s
p.,J ua^" Vy =on1s
/,o
%r=?
Vy = %ytarf
'i::y:l1#
---
Vo, = -ayt
- - L-q.7oe/s')G"oos;
V"y'=X'ttrrt
O.y= -$,6o'e/9r
= lr"So,/s
(") x"=:* V"" (o-Oo
x=(
f=
(u)q=
s
X-X"=Y"jt-
y =(re.Eol5)(ra""r)
\= f5.on
3r.1,ls
\--l (J /'r*1-,*}''r^4r*'t
f=l.oo5
|
)
( l"os)r
y - I'orr+(e i.q,)/ s)( t..os) * /t e9 ;o "ls)
t/yaytr
Y Yu":V"y*+
'
I'ofl
= )"5.5-
a
Problem 10
A sprinterrunning on a circulartrack hasa velocity of
constantmagnifude9.2 m/s anda
:ftxi*3|"1".;}"**n
ofmagnitude
3.8nr./s2.
wt'ut*" 1u;tn taci *ai* *J OirL"p"rr"a
"r
V = l"Ark
. )/
A=V-,/r- a
(a)
!
q: 3.znfst
:/
- Y>/
,| - --'./^
/.
t- - (z"l n)s)^ __-_z"8nfs^
i- = ?
l-= ?
T=
lr
( aa,-)
9.a '/s
Problem11
(a) wbat is the magnitudeof the centripetaracceleration
ofaa-objecton
Earth,sequatorovdngto
tle rotationof Earrh?
(b)whatwouldiheperiodof .otuti* orEurtn
il,
* ,n"
h";;i;
equatorto-bavea centripetalaccelerationwith a magnitude
"1.i".*
of 9,g nr,/s,?
= 6-3?x;66.
[-
T= bx (Ybr*)(W)
a ) V= '4I_r=
-r
\^
= r6,eoos
?,n (A"*
* l06n)
V - 463nf 5
86)qoos
r \
b)
t-
.7t
u1r
Q=
4.?r
a
-)
v: \rr/r
/'
),
q
T= s.l" lo3s
x 6q ^wi\
q
/-\-.,
' \
/tl
\
-
?rlf
-f1
4 r'(a.:?" ion)
?.so,r/s^