Ifup is definedaspositive,thenar: -9.8m/s2.If downis definedaspositive,then
av:9.8 rnls',If the initiai speedandlaunchangleof theprojectilearegiven,thenvox: v6cos(o)
and vqy= v0 sin(0), assuming0 is given with respectto the +x axis.
Oneof the key things to realizewhen dealingwith two dimensionalmotion is that you cantreat
eachdimensionseparately.The x part of the motion andthe y part of the motion arecompletely
independentof eachother;eachbehavesas if the othet did not exist. However,the two motions
me connectedby the variablet.
Questions and Example Problems from Chapter 3
QuestionI
A tennisball is hit upwardinto the air andmovesalongan arc. Neglectingair tesistance,where
along the arc is the speedof the ball (a) a minimum and (b) a maximum?Justiff your answers.
lu A&Aat-Q"{*rn o} o *u)'r;,*"*
J$" i.pr!.t 4 u! PJ-J.Q
p^t^t
Vy -- O-/5 . tlfc ^.;"" a+!"d ; V"*^ = V* = V,*
(a)
(b) frt ,\+r'AL ^
'.1
^^4rI
-r
^
wA9\
tf
n
/a'at',.
a
Question2
The figure showsthreepathsfor a football kicked from groundlevel. Ignoring the effectsof air
resistance,rank the pathsaccordingto (a) time of flight, (b) initial vertical velocity component,
(c) initial horizontalvelocity component,and (d) initial speed.
(o)
o.-!! J.^1
(b)l,loEJ
( c)Il;;L w^r"r)
(r)lEcT"'@)
\W:
4r^'t, ilua ^!9^r, h]tu a'""^oA'&d,
pv azvr^tkl
,o-^n\ )"frtuA
^,t^^9\ Pr:.:i guz.4o.tAtPa-.Jt
6!9 b"'pv
p* )v*'
2an"u
go, L
"8
b"ah
''--/
Problem I
The punteron a football teamtries to kick a football so that it staysin the air for a long "hang
time". (a) lf ifrc Uattis kicked with an initial velocity of 25.0 m/s at an angleof 60.0oabovethe
ground,what is the "hang time"? (b) How far doesthe ball travel beforeit hits the ground?
N>J- '. ,Dv aa^a^aot-a h. W )Q
h -S..^})fi, .f^a x")d^r}t$"j
(/
=' X"l"ls
(w'u'4!
P-.i
O = V'Yr-AYt
+
-t
t
- l\,'lr"ls
-"")/at=
n r.
= na,ele
t=
ffi6,
bo'u^ ait = 1[l.ar,) =[E^e
\=E=
.
V*))
-0y - q,xo n)5J
1-=7
.:
Vy =V'yta:t
Vo, = (Ie .o./r) arn tno.o"
Vy = Omle (t^4\"f
b)
/'n "u)r 'r-a
-- onfe) *A a^"e4eX
.-D
a)
F\e
o
X-X.:Voxt
X = ()1.5 ,ls)('/'qts)
Vo.,,= (rs.o'/r; c.r-a[oo,oo.
I0.5q/s
X, -9r-, t= q.q^,
X= <
:
X
#
55,3n"r
Problem 2
A diver runs horizontally with a speed of I .20 m/s off a platform that is 10.0 m above the water.
Whatis the speedjust beforestrikingthewater?
--t.a-,o.f
s
?-{*
/1..
-+-^
"
t,ho pW\Y1
| ," ^
lo.u tn \,
J
-d-
=) 4/*
ru" /r*J t$r.^J ]r', fu<"tE,)'*
-
\
&
Yo=lo"orr
!=oVo, = D^/s
lr_;t't","
;
c)D
z:',
r
= ,/r.tt* '{ o,.7
t
Y )"
4
-
-
----->t =
=
)o /a.a.rtt
-AY.
-r,l
----t= t",t3s
t= ,f6,
\ _9,ton19*
#
l Ll.l, fs
Problem3
A golfball iq struck al groundlevel. The speedofthe golfball as a firnction ofthe time is shown
in the figure below, whlre t = 0 at the instantthe ball is struck.(a) How far doesthe golfball
travel horizontallybeforeretumingto groundievel? (b) What is the maximumheight above
groundlevelattainedby theball?
't'A .1,tJ€
*"'.d.) 'a''J #a b;o
+''z""\Jt\" ?^
-UE:
^
1lt
d96
.uu'.'9*'^z't't:P"i
a'A
,,^ ol4 i-,u t--s'os
90
\--/
t^^
V.wJ,r : V* = Vr"
)
4.o
19
2.u
X = (le,lr)(E.os)
X = QSr"
5
= Oro
A!"A)
"dn ^'\^d
b)" ft
x-x" = V.,(t
= l9 '"/s
6)
Ea .$.w., r VoY
-^)Je arlA
= -d'r t
,..
Vy - O"T/5 "f -!"2q"* Wd
Q
-----t
Vy= V-* t Qn t
- L-1.1n/E)Q'rs)
V
\/-r, = 0"1.6^)s
Vo.7= ?
+=".-----''
t =r,Ee,")"ArJ WJ
a ) = - 1 . s os 1 t x
y = /l,Vyt
= '/\(tq'5'/5)(''ut)
= r-:-:-l,.J
[3
Problem 4
A car drives straightoff the edgeof a cliff that is 54 m high. The police at the sceneof the
accidentnotethat the point of impact is 130m ftom the baseofthe cliff. How fast was the car
travelingwhen it went over the cliff?
-> -B;t Ac*!^.e-l" ar" #s\
t
.,t_=
-.
/]Je/J).@!
vt.
tr()lr-)
-) m^*
%tL^ir,4. t'^\
t=?
j'40'"/'e!
,1ogfitdul
uox
--b
X- X"= V o Y
V," = \,/t
=
@
L-
l?on
;t---
Vo" = 39't.75
:-=:
t'
t=. 3.3ts
L
,. /.'
f = 3.3\s
-y" = ' [ a 1 t ]
V'y=?
G) = - 1-7o-l5t
P
-_
y-)" =v;-%.a,*'
= 59)Y=oV.y = Oo15
\,,
I
t\
\/
_
'f 4{1ah
D
l--.
t'*
Xo=On
X = l3DFr
b'!
7.3^ e
--'-'
rc.o4'o ztgnA vna
3?.ois
Problem 5
A golfer, standingon a fairway, hits a shotto a greenthat is elevated5.50m abovethe point
where sheis standing.If the ball leavesher club with a velocity of 46.0mls at an angleof 35.00
abovethe ground,find the time the ball is in the air beforeit hits the grcen.
q u Po l s
r'--/a!5'"-
if\
J
5,5o t ,
4o
y-)-
-= ).6.f
V,, = (lt .o"/.)-a^,"13s.o'
"/s
y"=o
= 5.so)
Or = - 7.86nfsa
I/ta1t^*Vort
l=o
s.5DD =D
-\,.)
(-u.1o"1t')t^+(\G"4-l)t5=-'l.Qon/sr
B=a6.1"r/s
t=?
- 5J '.r5- . o
rl
)
C
\_ =
= V"f t *'/aatta
-&{:" a ry"d^-fa'zV-frd'
\
t=_gyfb!6.
A
4 'A
t
-
J _ _L
n
\!u.
t/o6r t2
ot 5'tlr
t = -2G.1n/g J (la-q.l;)"-'t(''t,lo,1r'1(-s'lo^) '-4
L
^l-q.9o,t/t)
Problem 6
A diver springsupwardfrom a boardthat is threemetersabovethe water.At the instffiEhe
contactsthe waterher speedis 8.90 m/s andher body makesan angleof75.0o with respectto the
horizontal surfaceof the water. Determineher initial velocity, both magnitudeaad direction;
t-l'
V" = (r.tonlr)rc,o"l5.o'=
9..
T
lll 11
Vt = - (z.lo'/.) -a.^^n
?5'oo ='
1-om
)
.t
\/
vo)
8"1"o-fs
h----
I
Vy = V'x =+
)"3onf s
"'/r, = 0,"3onlS
'=-::::
tu4 ^M'A .b, {,^J
V", aruJ V"t
Vy'=V,.i* AoyC,{-i;,>
-- -" -/
r,/, - -/l"or,1t
!o
= 3'o"r
Y'oQ)
t=
= -9'gor/1
"
"
VD
tt,5A*ls
-
(ut-;=t^,r'(m)
9 =F,n-'
:
F1
Problem7
The highestbpnier that a projectile canclear is 13.5m whenthe projectile is launchedat an
angleof 15.0 abovethehorizontal.Whatis theprojectile'slaunchspeed?
.3fi"n*
)3-5"D
J*4^*'^^
W\
"
Y"=o'
= 13's'"
)
V'y=?
Voy , t-.rx,4\ F
V" &\-"\
Vq=!"9
(i- u";* Aayf v -r'"i ---- o --v,l *l'.')l
Vv = D r,/s -* )"40-n
l6'3 m
v-a
= -Q.gonfrt
16.3 ro/s
A.1
A)v\ \9. o'
b
Vry =V-r,v".,
Vo= 6A"9nfs
Problem8
ln thejavelin throw at a track-and-fieidevent,thejavelin is launchedat a speedof29 m/s at an
angleof36" abovethe horizonta.l.As thejavelin travelsupward,its velocity points abovethe
astime passes.How muchtime is requiredfor the angleto
horizontaiat an anglethat decreases
be reducedfrom 36oat launchto 18"?
*
r\ = -r -r/VY-\
O )o'"1 la
/ u,
^
^a quvq )\
-fr" a"gA .L $! Nel4;hd
4Dz
\f,*= (lf./s)
-t. o =Luyu_)
coa3lo'=A3'L/a^fs
= l?.o5*ls
V"1= (af"/r) -a,v\3G"
Jr^ A Vt/v* n
I\
^
,l
-r
t
r at',
i \
V) = (A3.tlb,lt) Fn
V^..= /"7,05n|.
'
tr
Vt
--
a_t r
/\
|
t.t"d.t\f5
a , , = - 1 . Ao , / < .
#=?
t
18" -= 1.t"1.-/s --a
::=-
,l
( -r-^* l-'-,
Vy = Vx i"''ne)
v* = {or=
\3"ti6"/t)
fua'u {',1uA*n
O-ts"
V"yt0yt
t=
vr-4
t= o"lGs
rt.G2.'
ls- (rf.os./s)
- 1 , r o n /s ^