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Dynamics of Machines, Mechanical Engineering, Ingenieria Mecanica, Dinamica de Particulas, Kinematica, Kinematics
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2 KINEMATICS OF PARTICLES
zj; 1 s:::; .: t.J - 2t ~ (,..3
V= S = tZ-4t::. t(t -4)
a..=- v=2t-4Cl = - 4 /1/S~c.2
o0=4-4==0-<-2 --Q = 4- .rI/sec Z
'#
2/2 I .s = 4t3 of- /5 t-z-l.Pt f- 6 YYl
1/= S ::IZ&z.;-3ot: -/8 = 6(zt -I J(t «s )zr=o t-Ohen z:'= '/2 5 (or r~-3 5)
a = V = Z4t 1-30 )' for t =isJ a.. = Z4(~) +3D=: 42 rn/52
2/3 1 s == - '? T- 4t" - t7/3 mmzr = ~; = 4- - :2.i/3 ry,,.., /sa ~ if = - ~/3 WlW//~ 2 c-pn.51',
t=CJ5J
V=4-y(<;)
= -2 ""''''''/$'12
5=-9 +4(9)- 3"= 0
Frol'Yl t = 3 10 t== 9 5,
J)= +3+/-3/:: 6 ;nl'>'?
2/6 I v2= V;z-t- 2a..LJ5,,'(/20 44 )~o+ 2(0.&J)530
c- (120)<4-4/30)2v - ;; Roz. :/lz (0.([,)(32. e )
V= v-; t-at' t =)
v-z= Z/2f-Za.45/
o z: [100 Zr2(-32,2)h
h:::; 993fl /f
v:::: v+at· -Iloo= f?oo-3Z,2CI )
t-;:: 49.7 sec
1ZOx 44-/30
O. Gx3Z. Z" ~./ / .sec
2/71 All"" fadt -14(-S1Jdt • -4(5)("1.81)
C> • - 1"'.2 ,.,/$v/;a -/00 =-/96.2, ";,2;:: -9'.2 IYI/:l
2/'11 a= dv . zr-=Jodt =.ot -= 6(ZO)=1201"H/sdt J ...,..,
CorrespondlJo1y h = .!.-ot 2 == .!..(G )(20/':: 1200 n?2 2
Dur,".-,Q u..pl<-'uy4 coost J 0 f LlhJ / .. vd-v = -9 dy
-2. vI?? 0v;,.,z=2:}tJh) Llh=~ e, 7341'??
2.(q.~/)Maj(. h=IZoor73sf= /934177 or" h=/.'l34 kl?7
2/8 1 30/3.6 61>0
fVdl/ = fa. c/s /' f Vdv = aJd's2.00 km /h 200/J.6 0
=200/3.6117/.5 / 2. ( ../02_2002) = 600aZ(3.6)
30 irn-?/h a.. == - 2.51 rn/52
= 30/3, G rn/.5
2/10 I2./~ 1 if 2=, if Z f- Z a .LJ 5' zr z:: 42 i- .2 9;?/(5')') Q
V2,;;: 32.4) 1/= S. G'1 /"h/sIFd,;- :: ad."
a",..,. d.r_ /SSS-40d~ - /5
= /5 ,"/s"
2/;/50~~--'--'--,--r--.--r--4O~~~~4--+--r--r~--~~/
v; km /h .]0 +-"'~",--+--+--+_f----+-/---TtV_1
20 ~ /
r-, ./o oL--L_2..-I---L--"'4~~---;6:-...J....-;!8
10
2//2 I80L-L-L-~~-~~~~~/~;~~~*~~4060 1-"""'+i'-...~-+---lI/~-+-+-t--t--t--t-----j 30
""/x in. /,I'~ 1/= )G-' 40 1--I----+44---+.--+-+--If----t-+-+-+--i 20 ;". / Sec.
/1/ ""'I'-...
f7 r':o ~L....J___.L__L-----L--'-_'__-:---'~~r--j L 0o 1 2 3 4 ,,----".-5+--+--1"
t sec 't.at t z: 2 .sec I I \ -10
x.:::::; 72/3.5 = 2/ 1'n./.5eC -~
i::::; -36/4.7 =: -7.5 in.)5cc2 -20-I-
C n7ln)
v- = v: = /5" kl'Y7 = 4s km/ho e 2/60 h
Vz:= 1{ = 0.33(60) =. 20/rrn/h
.2//5/ Y=Yo s/n 217nt I if= j = Z17n.Yo cos 27Tnt2 2. . ~a.. = v= - 417 »r Yo .5/1'7 27717
When C = 1/(41'7) V= v,;. =. 277nY,.COS-zTr= 0) 4 0 __
:z 2. • ITa: =: Cl'14 = - 4/T n y" .5/17 Z
- 4 z 2- - lTnyo
2//6/ 7/= elS=O.4(-t -2) S m.
S dt 0.6 !.fos =:o.4j'?t-Z)df ~ :0.6 0 I
S = O. 6 r 0.2 t::l_ 0.8 t 0 '~ ~S .5~C
-02 ~3 4-LlS = 5 -5 .
0-3 "3 0
= (0. (; + O.Z "3 '1.- O.8 y 3) - o. G = - O. 6 rn
]) ::: 0.6+ 0.2 + 0.2 = 1.0 /101..g/N I .X = 3t3-I.?cz.f 24t rn
v=},:,= 9t2.-36t 12.4 ,.,.,/~
cL =.x =Ifl£ -36 rn/sz.
Vrn/5I Z4~,
o ~5
o ~-12
O~------------~=+-
-3"~
cr, "',"/s- 400 mm
-'S == J (300 ~7St~)df% ~ ,
== 300 t - 25 f'Jf =- -17S mm~
"oo~ S, ., t, .•I 2 S~
-37S
]) _ 400+175 " 575' mm]) '.stancc -"., t AS " 400-/75 .=<' 225 mm.v ISp'QCe ,.,..,," s-s
.2/I~ I LIS z:Jv-dt s: Qr-ea t.o?d<:r l/- t cor»-e .
LIs:::. L t1t ~v = 6~ (e.7+ 7.0 +-6.0 «.s:»f- 4.6 + 4,2)
- 6'0 (35.7) = 0.60 rn/(nC1U.)
•. _Z_/_7_7....Jr A Oy =- ~ J ~ =: g r..>'~
£ast I-I
" hCl I
)< 1- '9;(. __ L __
s.50 d= (Zh//2 !JW C05 Q s = 2 liwh·1ji cos r
'J 3 I .3 Yg-
Cl)(.= Z vyw co s Q z: 2g tG<.J Cos ir. r:t" 2
X = J Q)f. dt = 9 t w Cos ro
t:x= [xt:lt = f9t-~~C05;r = d
B UT t';",~0/ fa/I IS 9;v~" Dy
Y ,.I x. PI ~,I ---------t .'I~---Yo X y. -6 w7-S/ncUt /; s,;"e
=;': = 2 x: _' __
-0 U}7.eos.vt b CClSC?r J
/.5 /,17 ",/tree 1/ar ftRn-t P f<> 0,
8ur
Y,I "" :: /,'/' 1I •• 1
____ 1- '"
.s o 1<.= r- eEo""P
~ r= es, t ': ~?t)"""p
y = ~£o (t f- ";::1;" ,Pt) +- ~""-f' r- '
(.)It:I~.1 t- =: Z1r/p) 0.,,1
,. .Y =: e. Eo z rrrn.p2
lA
x.=ut
co •.••..,P/e,t'"
~ =- ZJru/p
~=:o
2/n I ""'v,= 2,_2/p J VI <A, /1 '(r=: 30o(o.'<5x'1.f?I)
=:48',5YY1/5 or 174.1km/h
2/8;J a= q~= -v~~p = 981-2 =/e;>0 "",
'2.0/ '"~C(YI
V-~ 100 (0.4)(9.8'/)=- 392.4- J ZI=/9.f?/ ';
V= Iq, '?I (3. r;) z: 7/. 3 km/;'
2/~41 Horizonl:a/ v~/ccif, co"stant
at 72 k"'/h ;z 20 •••/s
At top of irq/t!ccor 'j
.,.Z (20)1.z: 40,8 InT· 9.8/
•.•..&a,., "'_"'!j,,.
8 '" '1.11/ _ 0.04'1/720 "'lo~/.36oo-
== v-t9 == 9, r"d.$
o = 0.04'71 180 ==rr
or
i----r--------· ----- ....----.--.-------_ ..•
2/8'61 a = qn = V-g = 2.0(1· 8'52).l!. _/ = 0 269 /. z-3.6 2 Go ' h7 S
2/371 Cl = v5l(-l /J:: 5/& = -,300 = 382 Hn ) r : Jrl4
1/:: 45 44 = 66 R/scc30
a = Qn = 66/382 = 11.40 /fj.5cc2
2/8'?1~ 2/&'0/ ~7'. '"'/ ,P47',t,.. ..y X 1-4n 8
Lc::-r I"Y1 = IUM8· -1- = 1~r>'1:/.) Co",'<9
Th«", Y= rY1X _ ~lC.l. (/~n'1~
Zuzor" rrr 2 _ (2t(. ')'n1 .,. (I +- 2«. 'Y) ~ 0
:3 '>'- ." 1/.'
lZoo/::5pr~ eyua/ /J d/.$C,rJ;""",rl4rl( • 0,
7JJu~ 4"4 _ 4(1 r zu.y) = o· y= 2.~' -;]2'1.' :JiL') j
2/c9'?[ C?t:=1.5rn/:,2 a'l..=a2./-q2.. 2./96 I FOr ~.: Ct:Jrt.st; Z~Z V-2 2:/ - A = q 5
»>: ~ -~if V' ,n , C t",r: = Zoo ny5 tr= ~ In/$ _ I!JOz_ 2002
_ 3 m; an=(-i ::Va2_~2 Cl - - - - 38--A 3,(' J C ,J.6 / i- .17,2-(2)(200) , .52
an v.; 2= 7/Z+ 2q 5 (200 '2
----a=2.5rn/s~ 8 A t A-B = 3.G) f2(-,J.38)(/t/}o) .;24//7. V Z 2
v.::: 2 241/~= 4'1./ ht/sV = zoo 2,52_1.52"", 200(Z) = 400 rn/s
a{ (320 ('3 ' G) = 72 m.,/h t;. = 81f' -= --:= [],04 h-t/52
V= 20 n-7/5 or n 3L?O_ / Z z
~7~ IT'Ijs2Cl,- y?, 04 -f 3,38 c-
2/'10 1 a = 5:1:1,- = 7.33 /1-/sn2 CL:: VC?",1- f- 0i 2~ .30 )
.2/971 i~ fS'a :;:V-2= ya2_t?t2 7. V 2 - , '1
V=- 600 /0 - 7,33 ,-:4080 &It=-J:$; vdv =- -ksds, v-'1" if2-f J:.s~) .,. o '"h P
43.5 I'n-i/hrD 0
tr = 63. 'I f~ec 0"-/4.., '" -'6 +k(324J, K=
, ~-z. at- s: - if 8 = - (. m/s ~5' J
a" ::-v!QI -a./ "/101-6< = , ••.•/s" = <fA/( J /" ;~ :: 4.S m
2/'11 I a = 20/3,6 == 5:5'6 Y>7/S2
t- 2 2. 2/96'1 Llv-= 2V5/n ~B 20 S/n ..:1&Cl 2. s:-_ ..z Q L
2.3(9,81) - s.~r:; ;;:- S'35.Z ==a= f-c? .::rt , n
rn/sa LJt= Ll5 z: rLl~ 2/lB LlBan = 2!? 90 = ----;0 = $if zr2-
(a) =- ~-~ z: /00 51/'7 LJB/2CL = v~ D::: (gOO/3,~ -: 1709 rrt /1 Civ Ll t L1Gn (J, I 2£'10 o.esee z
(a) LIB=-30~' LlG-=- 0.5236 rCl~ (<2nL, = /00 0. 5236 = 4'1.4 ;If/sec
~~an =r /a.r - a" -IIOO~- sz= (6) LIB::: /SO 'Llttc: 0.2618 raa' l,zn) =/00 o. 1305 z; 49. 9 JI/JC?cz
r ) J ov 0.26/9
= 80 m/sZ c) LJ&=5:',JCJc:o,0873rao; (~)Qv=IOO 0.0136:-::49.98 :fI/.xc2
an'" ""z/r, r::vI- jO.r 0,0873- = __ :: 1-25,..
L'a,. SO --- /ns/~n faneou 5 c?Y)=Zr /'r :: 100/2 = 50 '/h'J':=G 2
2/931--
Z/99 1vz,
Y=-I~(R~II/V=.!.Z -= -1- /f/kC a,. z: 'i = wl,"r6if= r&) 8 4 I'
vir 42/8' :: 24 h'/scc 2 (11.fi ICl= 4 =:t:~(~~~ZV'2" In I n / i!Z
I' ,. ~ 9. 1.2/ (10-')••-eoo 32 ooe; ~ 90'10 Je •••
.::2="£6 '2 /r74z/Z "'ODCi =re =- 12 ./1" 's.=ct ) c -1-
V -2. _2 2-
z/lool V-YR (/7260[44])2t2e = 24 -/-/2 :: 26.? ,II/.5t:c0= ~l:= y z: = 30
(200 + 3958)(5 2l?O)2/'14 1 S = 4- ,t t% tr=s~-t~ C2 = .5.::: Zt-
) t-2'1,1'1 fI/.5cC 2
Cin: V- ita:: -t'/, 34_ =i=-3.5j
= 9 m/52.( er -
C-hec-k: -J (:R ( 3958 2." 2 t- 2(3) ;:.6 ""'/5
:2 /1- °Rfh/:=3Z.ZZ( )=Z9.1'l.!!..a. "-6 200 I-39n .sec, 2-
a~ vQ~2~Qt-Z :::Vq2f-6"2=.' 2
/0, 8' 2 rH,'S._----
.2-/10/ I a ;: J!:..l # £ ",here R", /73K /(In
2/951 CL x: I, S /l/~ec 2 cons!'. 2 2- n /' Il+Ir•••/s") <-;; = ~ - 2~ S 9.. '" f.O.t-
V:.2: 402_2(/.5)(40°)= 4-00) t.-;;=2oH~tU a" ·9·~ (--'l-t = /.6Z (,n.Jf /= /.35'1 "'/5"8 " 1(";' 17~8+ 1&0
U',IZ. = v;; /;~ = 4°7200 c; .2 J-1/S~,;;2",,:I.:: (/738+/60)/.3S'1(.U;oo)~(lo-,J) ~ 33,42 (/0·; (Ic•••lh)~
CL z; va z, f- t? z, c: 0.5- 2..,.. 22- s: Z, 5' .ft/$'::'~ tT = S781 1< ••• /1>t- /1
2/IOZ I Fo r: ;; t?n == V)j- J if= VO. / (40) == 2_rn._I_s1/----=2--2. y- 2. -2. 0 :La = y a f q = 40 1" 30 z: 5i:J rrr 5
I n t
Qn= V-~/r =- L/O.05" = rfo rn/52
az=Van2-ta/=V?o2f30Z. z: J'5.4 m/s"Z.
2/1081 r= zt2/ -f .!=.tJj /11.- - J -
Y=i=.4tj+2t-~/ I !3=!:=4!, +41:-JI-Z-//-O-4-'~I'--/-t--a-n-=-1J-~-;;--1 -f-::::-r-4-0-0-0-; -2.-
0
-=-3-.3-3-(-'0-0-)-I;=or !: = 2 s) ~ =4 t 1.= 4 (2.) = f? m./scc
11./ 9.6 GOS 60 m Y = Zt = 2(27.)= J' //J./scc----- y 1y ~. s: 4- 11?-/5CC. ~ Si == 4(t) = 4 (2) = f' /n.j"f
a . '0 /;2
/~
n V=Cl./::'-'1.6Sln60 ::::-E",3/rn 5 rt.. /,'t: 2. 'Z .2. 2 e? ,z, 2.Cl 60°/", ", tY;/1/ if = :. +- y :~,.,., =. /28 m/scct fIn ". /45: (2..,=y CoS 45 - X. Sin 45'"
"......I '1
, 2 ,/ ~ !£"> z: 6'/ri - 4/vz =2Vz l:n./SCCZ
Q,6 rn/5-- ..-.-- .. ------ x. an == tr If) f == ~2. =- 12 i? e- 45. 3 I~
Cln 2rz .
l{- = 1500 (0,866) rO z: 12.99 fr/s«
~ =/500 (0.5) -32.Z(/0)
= 42.9 .;I'I/sec
if= VV' 2f z-: 2:: 1/29921' 428'2X Y
= 136.5' fr/Scc
an" q sin30"r 1.32 (o.S)& 3.6(, m/s'
or an = 3.66 (.3~00)' '" 47 400 k,.,/",a1000
11'" ~ ( ••a,., s: - • f = ..!:!:. =40 000) :.55 700 t",P ti" 4- 7 400 ~-'-'-_cc:.
c:r,..at"7.2-7.32-:!": /.28 "'/$1.
2//061 Cl. =:1 .5/',., CJ) an::: V' 'Y.tt-
O Cl = a s in A Q = Cl cos;9cc r-' 1'1
[0 .l
1/[ Q,'I '-.
A
2/1071 t-/~
t<=V'=r&l.-frr3' "- - -I 5::: re t- ,
50 ]/=-r(- !,)B :::re!1of r8 ((y;) = raj, ~_c~~
a" ." 1.557 c038 - I. $57 (O.IG~)
.= 1,$48 ,.,/s·~t ( )1a; = - 1 f = 17 if 70 :: 1$ 480 kmI /.348 (lo-Jj(3bOO)'
2.///21 if= ZC; /cos 30" x: 2/0. ?6C; = 2.31 rn/5
Cl =- Vir z: 2.312;'0.250n
Cl = Q ran 30"= /2.32 rn/s?I: rr /'
'2/1/3 J X. =YY3 -t X = U- Q. ConSTanto )• ::z. z(" . 2) .. .2/
?C = 3' y y) ')G z: 0 s: "3 yy + y ~ -Y = - y i.,YFOr y=z. YY!J 1- =4/3 '? , y=.i..Y:.::.1- tfo =..l.v:.
Y m . /t Z Y Z 2 -I a) Y -f7:V .' (3tJ.:0
2 _ 9 22 ---------k-.-~. y = - _0 Z - --l{;
/' I ~=V: 4- 32/ I' 0 "Z. •••• .2 "Z. 3 )J z5 If z.
I / ," V'=X +01 = t..; +(4-v;, = 16 0
I : \1"1.I I,
.,o '--__ --1 X.o 4/3
ir
4..,.
/J = ;:'1..f ,~'.1. = 0 + (_.i v"I(~) = - o. 169 v-2V(.t 5 -'.5 32 0 ) 5/ 0
v'a=-n ~'
m/s'
~Z/I~J If=VO.lS2t-O.22. =O.25m/s (d/rection oTPofh
a~t = 2-,5 - /-2 cf,.,,~·., (/5"0 -100/0.86")('3 = S9.1 ~••/s~
~ = 118t /150 = /.5];/ = 0·3'19 r••d/s", jj - - 0.399 rdd /s~
Va =(r&)t=- z = (41-4)2. = /6 tn/5
V -(r ) = 2.2 = 4 ;n./sr c= z!:!== 4.!"; i- / b §J, rn/s
2//201 r =- re. Id r= rke id " Jr2 Ido ) 0 ) r==t; e
&=Cc e=c 0=0) )
if- =- r ::::r:ke Irt: e: z; re x: t;;Cekcr 0 ) ~
V=-yv;.?fl1;2.= r,;elrtj/k2.+CZ
a = r_roz = r:Jr2e/l-t - relrtc2=r:ekt-(Ir~c2Jr 0 0 0 /•• , ' Ir-r
.Cl = n3 +2r& = 0 1- 2r;;kCe61
a::Va/,..~Z= r;elrtl/(4_2Jr2C2+c1r- 4h~z
= t;; e Id )1.4-4,.. Zh/c'tt- c" =;;; (jrLfCZ) e /rl
(1::r'-riJ~,a =rO+Zr&r: 8
2tr /9 = 80 ~o '" 8.38 ra d s
if :: -280 £!! =: _ 29.3 ra.d ($1eO
== -17.SSr - 12.36 (; •••/s""-I _ I
.2//23/ a = r -re.l.· 0:-;': - 0.25(+)", r" 4- m/s"r, .
-- .-.------- ------- - 2/IZ91
z//25T r: = 15 t- S:::20 ;", r = --f::: c /"/~L!r. /:cj:: --?In,fNc.1-- - , / /.J Fro,," jn:l.fh, at t ~T :s,
e = 174 rad' (/J:= ~ 1?7,%<!c B':c - 10..!!:...= - 0 171S ,,-n"':I / /S'!)' sec". For r~ 0.5 m a ::I -L (z) 'C 4 ""/S1Z/;=r '" 6>/17/.1~/ v;;=-nil = 20 (;1'/4) :15:7///7/S« • ~ 0.5
&.- =- VZ-.;;:/-", ~-z. =- VG2-f t's,7lZ = 16. S'Z i,.,';.s~~ 2/130 J f l' V; ;. -I ;;~i-~-;-,-;~ q~~:-:-s=/or;iv-eclor.. . '- IT 2.
Qr::::.r-r& :- -4-Z0(4):: -16.34 ''''ls~c2. rt z: $u,CL tr/fu;-.=r)
Q(J=riJ ,1lrt9 = 20 (-0017'1S) r: 2((,;)(04-):= 5.'13 /i--;.jsa2 r: 'I-a SMCC r: IS 0"& One ,bort 0./ n-IC>:J'?Ifucle of Qr
v' 2 2 d 2. j:----:Z . I 2 C·...;r_r, .sroce r= Z- a/so conj'ql;'s a. i9-co"-"pone"I'a=- r tlr J. Qt9 ~, (-16. 31) +t's'flJ) = /7 3S'':'_''.~_.5ec. -r-
2//.2''11 r =- ISf'S:: ZQ i"'J r =- i = - 6 117,/S<:C., r =- 0
& z: 60..3:... = /F"/3ratlike I} = 0180 r . )
v,:~r =- -6 /n/.s~c, ~;- r a » 20(,%) -= 20. q In.j.s<<
V = Y'Z..j- Z -f itj."i: : 1(-0) ~-f(z-;;.-;;ji = 21.8 I;"/~~., ,2 ( I ,2 . / zt(r::' r=re =- 0 - 20 173) z: -21.9 /11,/.$«
t/. =-rG'f 2.r&; 20(0) f2(-6Y'1'/3)= -12.57//J./xcLe;
Cl = V~/f~2 = vI-~~9)'+(--;2..5.7_'2. z: 25. 3 iJJ/fO:>C2
r = h/sin 8, r = 8:r ., 9.24 km
9.24 f-O.025') 2 "" 0.00S77 km/51r=
Ir= 8000(0.02$) = Z67 rn/s = '160 km/h(./3/2)1.
r
v:, = Ft?=30 (103)(0.020)B
= 600 .:ff/.5~C
1/= ~ /(;05600
'" 600!O.S=-IZOO/r,.{,ec.
'2a = r - rer:::70 - 30(103)(0,020)2
:::S~ :If/secz
0.= C(r/S;'n60o
= 5'5'/0.f!6G = 67.0 .fI;&<!cz
Z/13~jf- -:cif)
. r~---
;'::j. rt" .5lnce a"". COrJION1.5 pnolJ,er r-con-?/'oneJJ!I
Cind a/s» a. &-Co~'7pOrJ~#,t
r;;i r&t9 51nce r=v Cl/Se> Ccnr-QI;'~ Cl,.., r-cor>7pOTlent- - / --
Z/I3( I A== ron-I 8'000 = 30° r:« 16000 kY>'1r-: /31i'60 J
"r\Y' /J • 0' - 0778 12t/ \ cc a =-1.55'6> 51,.,30 v=q - -. rn/5--~-7>tt t J to _
t: '"IIC;\ I •
~ilf3~' if=re e= ~ = Vcos/3\)1 I ar=-/.S.56 ""'/5£ GJ ) r --;:-->cl •():):r e &= !.2.!.!.E.co.530o= 0.973 nu'l;,~L __ ~A/ 16000
8'oOO.~ .• 6 0. 973 z .'a.,.= r <": r z: -1.55{, + 16(10 I( 3600) =-0. 38!i ""/5'
.2//331 For 19=0)
•• • 2 ->_1Cl :: f" - rs ~he"~ 19= v If",..a ::.V"y~
n :<;So r: - v Yr = - 1//1'
~(( ') ,-/, ')r: = 'V -;:.-;:0 e, -{/ ';p -r
2/134!r=6-C<=DSB r:» c co s s-« ;;=Ccuzws8 2/1381r==Bo:2R$/n.§! r:-,tGcos!'!. ;:~_.&.GZSI;'6'.. . z "\. ) )) 2. z • ....J Z ) 2 ) 2 Z
a =r-r8 = cco CLJ58 - (o-cc.osB ((.) =(2CCl>5f)-/')W ;=Or &=30" r=2(I:T)s/nI5°", 7764/n.r v ,
Cl. = r;:;'.,.2;8 = 0 of 2cw~~,;"'B r: = 15i4)c05/.5Q= -.57'f$"/,>·/sec19
a. = Vt2/,..~ z = W2y'4<; 2 _ 4.6c =CJ +- /:, 1 r == -~
2/141 I r = r.; + it, ,5/;' 2lTnt) r == 2ff-irz CDS 2n-nt"
;: :: _ 417:16 n. 2 ,5/n 2TT/l tD
2.//421
4a
:J5
soZ5
20E~
/S ~~/0
':l.•."
5..•..<,'J{
0
r,1on
i I
IS
r -; 10, ,
o
r: •• 20 100 '"
;. :: - 77. 5 •••Is8 " - O.01+ rQd/~
er '" .;ry.-z=-+-(j-r-8-;-)'"'&'""./07. sr + (20 fOO)'(O.01+) l := 214- ",Isor er <: 1020 k•••/h
60
~2//441 (r.$/"J)2 + (r",58 - ~l= hZz J. t1\ r2- 2re coSt9 t: e = 6
2t90cv .
=epnslr-",,,,;'"/i jrr-j;eW.58 t,JreW5,""t9: 0
-& -'" /0: Fo r t9 = fij.z. ;. = - eu.;LeJ_--~ )
,: 2r rr' - ;'eco5& + rews",t) ;. rews/"o ,...reW~61<"Fo, 6? :;~'-;'J ;zf" r r: -I-- Zrew = 0 ",. = et., I
J ~
v;. = ,;. = - ew) ~ s: n9" Vbl.-,:2 t.V.
17=VC;"'f ~l = wI6~-e2~e3 = 6w•• -2. '1. 1.. :t.
a = r - reJ ; e.w - Vb ~ e l tu ==r: V 6 ~_~.
e, d~9 4/<9=.r8 f-Z':'& 2.. (-ew)w • - :tr=t:..J I-
= 0 f
100 va 1. ~ <? lol.cv z
a= =,- 61 V62._~180
~.21/4sl r= 100- 7S':ostJ, r= 7$'8 sinS, ~= 758 "co.8, 8=0___ -'40 r.v/_'~,
~
" 8 '" 40 (2".) _ 4- J Ir _ 61 I 1.0 - Y 11''-'' 5___ I
=t: --t-- (}".30£!!. = TT' r"ells6'. ~ '0---5: 8;:: P, + B~. {} = &, + 8•. s: Z!!... ra d /!
30 r.v/"'i~ 3•• • & •.• .•
ar:: r - rtJ, , a" " re, + 2r~
For (I • .30·, r: = lOO - 7Sfi ,: ;'5.0 "'••• r= 75 71T;.!. z 27$' "./'.2. I '3 :z
r ~ 7S(Lft7 '"~+'10 ~m/.·
a; • .3490 -"5,0(4,"')". 287S "'~/!~
IJ.,= "s.O(0)-I2{:l7S)(~It')" 2i10,3 ,"",Is"
a >I /2173'1.+ 2303 a - 3'80 ",,,,/s' ~ 3,68 m/s"
21/47\ if {2 IS nc>rl77Q/ 10 V- Ih-::n a :=cznQ!C2=zr=O
6-!:::. q =- (2;' -_y-2!!-)- (2./ -I- 2./ - j) := 4- - (, r- Z = 0
50 if /; norJ-Y7O / /0 a .
a =-It3:/=VZzI-Z'tr/<- =-Y9 =3 .fl:-jsec2
if = (VI=!2~.57.-f22.::fi7 z: -f./Z Jf/s(!c
a =. C?n z: V ~ J f' =- 'V '-::: .!.Z = 5. 6 7 ./rqn ..:1
~= Cl Sin 30';::- /O(O.S):Srn/s2
1/ ::Iy-I = V3 '2..+ 4 z: :;;;.5 n-7Is2.j -2,; 52a = V//-, I.) = V/e;:= - =:: 51YJ17 I) t ' n 5 __
~ =:: CZ cos 300 = /0 (o. 6'61.) =- g, G6 rn /s 2-
Z//SCIB = R~'J g. = «s, ~R(<;i!, 'UjC4$~ ~,)
Ii =/i~, ~,e(~PJ r6coS(1J~)
t- R.~:A f- ~~'~ .J.. ,ej; (- ««. -; SI;' ~~,)
.•..R.ec-o~rj>§;, .•.. Rd/c<>~ '" ~J - Rei> .!SI;" ~#J >Z/1591 /-or k = ;nox f?' dl',-~cr<=<?' r~"-'p~'" ~
••.Keco~';(-&UJsl>~,+Bs/r) tf;!)~ron-? t=:": /-0 Pr06, 2.~/5R > ,~/;" 47Tr: ~
/' JI I: .I-_~. ~_-' L' Co:; 47Tt' -0 !/ /G;o /0117, R= 0 .e -::-3217 /n,/'.5_,2-<-(>//I:C '''3 ,~,,..~ ~,.q Co-.-al'?"".) ..yl"'~ , I , ,:' , •. -~
•• " 'z. 'l ~",) and tA/lrh &=4rC7d/.sec. e~o ~-:;60d"J rp=<;b=o£7 - ( e - x ~ _ A: <9 co s )" K ,') f ) /
~ - - J acceleraTion c::ornpon~~1/..5 /ro"".,., .5o/uT-/;)-} f-o
+(~!!c(R.'&) -:U~t9~SI;".p)f) Prvh 2//58 becomeR. It- _J 2, L Z
Q,e= -32772-0 -/O(47J){'2} = -72fT In./sec2
d,//~r~"hd/ p'/$~",C.~ pion.!! cC.,,"~11 ",,., d;,..'c~':on d/
c-on~elern~"tr = k Z 1'0";:>' z- blh
h' 1-
df== ra'& ca" ~etr = -,/.! 'sI'7,8 z: - /0'& ~ T .5J;';B,., e
/ ~r : _Ta" 1/,.5,;"/5/d.9
" -KB 0 ,- --- r -:;b e ",h~r1(? K. z: !vn T Sfnp~ "-<, , ' -iCB 2' 2. -Kt) '.Y r=-bK&c:. r-:: bKe e 8=0
) }
, , • 2 'z. -Kt9( 2: It2,= r - r& = be e K -/)
,2 - (9 tt/I? r S/;'(3 ( 2 2 )ar: Cl,= b9 e r-aY1 0" SI;" !3 - J
UJh~r(' f-3: ivn-' ~
8
Ir P rcKi
_-'asdf
~2/1571 R '" 24 m consl., t9~w~ 2(zrr)= ~ r"d/3 (} =0.0 /$ ,
~. 30: f. ~ -j1, -1= -) a -0.10 r"d/" ,i", _ji mO• , (tr) I
~ - R - 0, ~ = I?tJ '"'' 'f = 24 is ii = 2, SI", Is~ = R.t> z: 24 (-OIO)'= - 2,4 m/3
",.:/2,51'-1 rz: = 3,+8 ",/3
,. , z ' • 2 •. /tr)'( ')"a = R -RIt -R8 ~<>s '1=O-2~(-",.lo) -241iJi :;;~r -0,503 •••/s~
a = c., If .i.(Rz&) -2R8;s;~f' = O-Z(24){,.!!:.){-O.lo)ffI} R dt 15 ~
::: 0, 8 7 I '" / S z.
«: .•.i£.. (R"~)+RB.r.r,.••<tc#s"'~ O+24(~)·.g!..-r R. clt lis ~ 2-
= O,~S", "'/SL
2/IS8' IR;: f? f- 2 -5/;' 477! R. ~ 877Co.$ 401..)i =-32772.5/,.-, 1'-7Tt
B=C4Tr ;::) e'" 0) p: 17;2 -;3 := 60°} ~ =~' = 0
,c-or /2 yy, ax/rnwYn. c o » </-17t := I $' .51",., 417 t- z: 0
So Cl = k' _R~2_ R&Zcos2<j'le 2, z
= 0 -g(O) -8(4TT){'2} = -32Tr2/n./seci!.
Cl - Cos </> d / 2.; I "')B - --;e- clt ( le &/ - ZR9 l' Sin <p
=- 2R & :::05 cP - 212Q'i SI'r; q> ;: 2(877)(4lT)!"320<~'2. sec.~
c?A.=~ cl (R2<)) -1-1<:82.51;",0 cos~v-: R oIt
= 0 f- 8'(477/9 i s: 32V51T'Z.;J?/5ecZ.
CL: 327/-/(-1)2-/- / 2.f- v}2 =3ZlT'/S= 706 in,/secC.
c:261= 0 - 0 == 0
( )2 (J ) 2. ' 2.
Cl" = 0 r 10 477 z"2 " 40 V3 7T /17. /.5ec.
a.= tY77'2.fi9)1...f-(SVj}1... = 8172viS6 = 936 ;/7.j5CC'
2/19
-6
L1x= ./"Xdt ;::area under- i-t"curve
X,-(-6) =-1(4)(10) -1(2)(5)
')(. = et m
X -(-6)==.,f(4)(lo)max. """
1/;~X)m/5
o !5 t 5o 10 J
-2
2./20
y;.ft/6
00
Y ..ff/gC)
05,
-{z,j
Lls =Ju-dt ;: area under t/- t- curve
For ~nf/re I;' fervo~ 0-/6 = 52 -5/
- E (5/v;;O f- -i- (~)(~ 10;,1)
= (-z + 3.. \ lu: 1= -!2{ v: I2 10) '" 10"
(vcl= 10 16 "' 10 ft/.sec16
l.{; = -/0 .:rt/secs
t; sec
Z/23/200
tr #/S~C '""'60" = 20 Hh~c) r--"':"-~
II
'hI, t
"2"2 •tr : 1{ r2aS;-220 =0f-2(0.4)(32.2)1-)/
h, = 15.53 FI
A2=300-2(/S'.53) = 269ft:
I
!h,i;, t,
if = vI-at· 20 = 0 I- 0.4(32.2) t-) t) = / ..55 .sec./ ) ;
S=v-t· r;> 26'1::-/3.45sec.) a: 20
t: = 21:) f- t:z =- 2 (1.55) 1-/3.45'= /6.55 sec.
2./24 400 km/h = Ill. I m/.5t.r:l.=2a$. , s = (/11.1):1. =199,.,
I e ro. 7)(9.81)
5••.=> 8000 - 8'19 [.2)
=- {. 200 mt, 5
t: = ~ ;;: "I. II '" 0.7(9. f I)
t ::;L" 6200~ IF /1 t.1
"./8 $ }
= SS.8.$ J ~ &8.2 s or 1.47 "';"
2/2/ Z./UF Z{= z...:;f-Cll1t.u 8-c ..i
(60 -IOO)/J.C;4-
z: -2.7 S' rnl:/'
X= x::: + Vt: +2/ at2. O=4S-6o(3)f';'a(3~I /) ,
a. ;::30 in·ls~cZ.so )C = 45 _ 60 t r 16-t; 2
>0) in.45 t, Sec.
o f-_~:----'i2=----::",3_vuh~n t=/5ec) ><,=0
C == Z sec, ,,:::-/51",-;.o / ------15
X) in/.5t!c
2.o t-=---;----::o>I.-<:::.--+_ t seco 3 )
-60
Z/2.Zv; "'/5
l1i r, st, ta
s s L Cl t I!. t =/2 (138· S):;l. I I O.7(¥.61}
5, + $.;l :;: 300 '"
Thus y $2 + $", = .300} 5.1. rI6/.5,..0.7
S, z: 300 -/~/. 5 ~ /38.5 ",
= (,.3.5"~5..:..,~_
t~~/.2(I"I.5) = 7.4/5O.~(9.81)
t~t,+t.2 z::1;.3S+7.41=13.7{' 5
==«, IT = 0.7(9.81)6.3.5" = 43.6,"/5 = 157.0 k",/h
LtS.. :: ic LJ t.. I- -f CUlt:Zv-C •.•• <.J-C.c. 8-C
== ;~604 + ~ (-2.76')42. = 8'8'.8'1 m
.15 =65 + L15 f- <:15A-D A-a 8-C C-D
3000 = /00 t" I- (18, 8''1 r .se. t: t;:; 65.6053.6 3.6 )
S::.c1~_8 = ~~:(t&,s;5"O) = /:il/9n? orS=/.819k»1
2/26
O~-r--r--+---~o I Z
S s
-3 ----ta) (b)
2//6/1 Uu cf' 2../19 "'hu~ <i::-j. R=L,
9 = 4'1l (0.12) cos 4rrt =0.4Irr r-dd/sl if = - 1.9Z,,"SI;' 41Cf=O
all = O-I.2(-ft-I.2{a.43n/(1/'"= -4.75 ,.../s"as z ~stp(2R6+R8')-2~8tfs;,,9'
>= -r;2(o.9)(0.-fI7t}-Z(/.2}(0.UIl}(-IH = S.07 ".,/.s Z.
Q '" 2 R. tf + F? ;; + R e 1S in If CpS 't cf = 0f ,-= 2 (o.9){-1) + 0 .;-1.2 (0.48 n:t;i "I -= -/.518 m/s"
a=/r-4-.75)'-+(S.07)'+(-I.S/8j" = 7./1 ,.,/5'"
2//64IL,1 erD = ytrlocltJ of dllsf""j'trr •••'10 (/.8S.2)
•• SS'b Ie•••/;,
II
.~
.z/I~ol c/s e Ef 2//'f wf,~,-~ 9' z: -(3; R =: L, 67 =co
a = 0 -1.2(-1../-1.2(2fi =: -5:/0 rn/S2.R Z ~
~ = s~(3 (2U:'.eo + 0) r2Lwj cos/-i.::2UJ(L·S,;""/3 t-L;3 cosf3}
= 2(Z)(O.9~ + 1.2/-:l../;)=:~8 = 7.64 rn/s2rz 2'2 rZ ----
• • 2 . ~) 3 ( 21 I IU<J' :: - 2 Lf3 + Lw C05/-i sm;5 s: - 2( O.9/"""2 .;- I. 2 2 I v2 v;?
=: - 2. 7 f- 2,4 .::- O. 3 In /s Z.
o r ~ % SS. " .•• 15.43 ",15___ .!.6
A, (VMfo). :: 7S Cos 30· -= ~S.O '"IsI #,r'~.
'r - "'.., .: V,/ocitylI"ril':. '" I .,.,£ "'iss;/~c""'p. I
I(,0· I si"
'" = 11.88 •
2/lb6-1 Yw= Ye"" rWk
V-w 6.5
$/" 350 S/n /.50
6_5/0. 573~)t/.= t :w O.25/Nl
--- ::
= /4.4 knots
I
.s"" th of Wllst51· -ISIVI e -
) ~o= I. 3S _~~ofs
1.38IS;'niO° ..5/ne= 0.6280 B=3l1.9C>/;) )
(3= 270 - 38'.9 z: 23/"
w h e r e If V-- r = -6/,04
// VA = 500 k •••ln/ v;.. ~ "" 500~ ';'500
2"," 2 (500)r.500)c~~Sur
// \. :: K 54 000 (km/fo//
<,~',
,,~.5.:I
.,.~"$00 Jr.,./"
t.r,. -= '1Z4- km/I,
L" / h = p o s s i bit! cons!<".,! d;f{l'ro,ce in
affiEu/1!
Since ...iLrkh) =_0, vtPrf/c,,/ separat;on hasdt -
no inl"lutt,.,cc on re/alive -.-«lociE:! e1L1D-!ions.
m/s a
or E,04I~
0A=I.5 a{J"""'/s""'/f~
"A/e =/US-+/S" = n.o m/i
in dir, sh~",,,
a,4/a '" 40,S ,.,/s'"in d/r, shown
x
2//75'1 Cl. s: Cl =0-~ -8
reloT/I;<!: n, 6',I
so ;: = r,92
SO & == - Zr.-;/r
2//76! Yi = Ye -f- J.:4/g
.Vc." .: VB ,t- Ycl13
~ = 2{ -I- Z~VC
" gA z: g./5 i- gAle
CC = {Z8 I- Qc/~-c - /1;;>
~ .: <?c I _CZA/cFrom /~ I-U/o
~- K = :?'Ale - rc/8Cprnb/l'1~ ?V/rh 1'h/r./ k,-:j/~~ ('/AA: = U:;/s - v.;-,6>or ~/8 = ?A/C f ]/CM S/~llayly?U /1), arce/.
Io<.p"~ ~A/i;~= ~A/c f g'c/&
vc ~ 6378 (0. 729}(IO-+) (3'00)
;; /674 km/!' (~asl)
~ x: 27<140 km/h (North)
{} 0: ran -I .az«. '".27 U'O
5at~I/,.t", af'p~ars
w~st of north
$.43°
to i r a r e I 3.4-3"
~ar5 will Ql'lNIar te> be approa,j'irtJ
Spacecra It h,aa! On when JrM/S /s alonJ I/n e
--""..t;~kmI5
~ = 24.( km/s
~ /'7.0 %~ 2-1.1 :I. - .t (1'1.0)(24-.1) t:os /S·;: 57. 2 (k",lsi
Jl"M = i'S + ~/s
trM/S = ZS6 k"'/$, .24./ '" 7.S{'sin {rr-{3J sin IS·
sin (rr-I) = sin! '" 0.8246, 1-;; 55. 6·
2/ 2'I • t t . 'r· 'tS./8 I ii ;; - g. J = U s i,. 4S - 9 ) J = u ·I1n ~ -.z '1I------.J
X ;; 0, ,;:'" u cos.(.S", x:= ut cas 45"
d'l z: ? ::(_ J..! rz:= /_ ~ ;;('"~(I- d'f)dx;< u ",2. ' 29 dx
!!:t :: ian 20" ee 0.3640, J(" (IS)Z;O') -3 (l-o.3t4)r613d»: (3{,OO (2)19){IO ) le",
t "" x • % ~rz:: 0.OS78 h or .Jr.-un 282Cu cos -+5 /~ 000
h" 'j s: IS(/O~)(0.0578)_ 9(/0--') (0.OS7K/(3600/ = 418 kmff 2
2//~3 f Vecrors ore p~rp~nd/c"/or I'~.£2 of- I'n., -n2 of- n,"2 = 0
.2. 4 1- ~ .L - .!i ~ ;;: 0 So -u: ¥ a. art! p~r-Va. ra. 1/0-pend/c4l/ar. /n r.h~ oSG~/a/"n"y ~/ane rh/~mean:> rha:t' rh.,. a.cc~/~rQT'/on /$ norrhQ/
.tf, rhe curv~ .pt he/?ce ~ht: ~o"!le"r/t:2/a· cc e/e r- a flan /s z~ro.
B to Cj t" 10h =,.5 $
.r= or; +at:; 0 s: er; - 9.81(.5)
11"'," -1-".0 m/S
trZ::: /.'j '" +"za s; 0 = (M.tJ) 2 r2 (-~ (I) h;z
htl c: 122.6 '"A ~ B. "'~"V-:/.~24S. (of.'l.0)2::0.,.,2(-I-o!lJh
I I I I
h, " 3.07 m
Z/I&'7IReal Coordinattfs )(aO.25J(, 'fz:: 1.2.5:/, T~o.o-l-tSQ x;;:4-'x, 'j=<J,8Y, t.2.5'T
.f 4X= .5"0 +0./ (Z.5' T)%, .x== /2.5 + /5.62 r:l
t----:-Gi.....j------~--------------I O.BY:: 50 + O.OO/S' (20' 7)0/3 - 0.04 (ZS r)2/2,Z//P.::> I EA :: BB + !!AIIJ ) ~A. = v;/Ij" (SO/J.~J'/60:z3.22 f. Y•. 62.5 + 29.5 T'l'.3 - 31.2.5' 7%/2
(1.,q/IJ:/rJ.22"'!+I.SXC.,.O.22{1/ZJt For actual ;71otion -:t:: dX/o'T= 31.2.5"7
"Y ~ dY/dTr 29.3 72 -31.257
(Ix = .31.25 ",/s::'
"r " .5S.6 7-3/.25
T.,2 5) ~ = 62.5" ,..'&J "Y = 5..,.73'a :: hl.252+85;9 «; '71.4 Ir7
zt:1x::: :31.2Sm/5I, ((y ••S5.9;i
3I
Q,q '" 3.22 f.,-I0,
,1
2: 4. 58 mls~
~••" -10.3752.
2/16'1 r .,.dv-=ads.J
a ;r
fV' s
m..,-d.r _ dso T-4.50cr'" -f J'
5 = ~ __ T__T- -1.5"o,,"
a" =I.~ l!l.s•.
::: .20. ~ • Wlst .1' hortJ,
9.00
s = /6 170~ 6, -=-,:Z=-=-5",,0-:-_:--~=:;-;-;-If. o o 2S0 - ~•.501/4 /' (152)%.3.~
= 1 778 t. 15.08 '" I 776 re. 714) '" 48Z0rn er '-/.'22.k"..,
4. S'2.--:-- .:-I. SS 2
2.{Oo ".,. (nQlIh'cal)
2j;5'F If = tr + If U" = If - er-A/{3 -A/e - ell! , -Ale -AIB ""C/lI
~/c =- (6.8 J - 3.6j) - (-8·:Zi - 10.6'; = /5.0 i + 7.0./~/c"/is.02 .•.r.o~-16,5!) kt/Db GZ]------ ("Ale
7.0 !3 ,A "" tan -I ,S.O = 65.00
r 7.0 -- 15.0
ex ,. ta" -1.5-" 41. 2•v: ---...
et, A b Y?4j:J f .:r c~e/~rH PQ7fl .=,," Id
J7.4 t ;- Cause aj;ruj>1 forcesj//"cZr : wh/~J, u.>oufd Ig un-
'--- .•..,--+-; --+1 5 Ccn--H70r'-T-Ob/C for ~"'55~3t!'rl
A B C A frt::ZM ~/h;)H .f""ch~ •.•. /0
chan:;c c.t4r'vap,rc ..In?,d'u4/1y oVer Clnl'n!e,.-val Cl.! ;/ra,k ~4ul4' .le rc:9u/red.
2//90Q]18 - .: 1,,;_~__l~~
1~ 4S".,
C ~o ..,
:/-dir.J
v: ~ sx: ,,;2 +245
.s~V;t+i4t7.
o ~ u; - 29 os) A -to B
tlJ
co ,/.307 •• /7.16 m/s
-43 c /7.lht - j 9.8/ 'Ct = £2.5' s (or t=-!.749,,)
x-d'-r., '0'" tI", fS,25}, tI",;: /1.44 m/S
t.I" ,.111.4-1-% + f"'T.7Z'l. ;:: 2.0.6 mls
2/19/ I tf5.i.::km/h I I
, I h
h,=~11/60~
I/r'"~6357
km
So 6357.;-1-7
_ ,2.a..--;;=-JO(rrS)2
o fh r2.(V-dV-= -..9, oSJ' o (r-l-5)2V; h,
Z;: -.9 r2_i_]h:= .9or2[/. n - r~h ]o r-f-5 h '
1
t t 1/ Z- __ - __ 1-
rf-h rl-h, 230r2
s: 1 _(/6000)2.(1000)6357.;-160 2('7.233)(60)+(6357)2
= 1.534('0-4) - O. 2<;'86 (10-4)
=0 1.285(10-4-) I/lrm.
= 104 - 7777 h1.285 - ) ;:- 1420 km
-------
.<;1/941 ~r = 3 rn./s 1 -51.7- rn-/51./:= 4- n-r-/s
&-/ur} j3 z: 3/4-
-/0 rn/.5 Z
_.:;-rn/.52ttr =c:z =eczn::-Iczrl CcJs(3 -I 4alSinf3
= /0 (4/5) - 5(3/5) = 0"-3:= S rYI/S2
Ur}::- 1/2./-~) f = V"/C;n= 5:;1':)-;:- Srn
015 x: 2 YVt
CP z: Sn?
\(-y C:oordt:'ofes of Care
----------------- -----~r=25 000/1
I
& = 0.03 reo/sece:= 60°a..:= 64 :ff/5~Cz
Cl z: - /5 fOr} 30° = - Sv3 .fl/scc2y !
~=-KJ ,p=-0J R.=b R=R.=O
Fn>m Ey. z.j/9C2 =- 0 - bK'- b&2p251-"~tC05Z",A' 0
Qf) =- (nosi> (-~.t/C05;Dt) - 26(-5j,PS"?;DI)KS,;'.j>/' . eC?", =- 0 f- 6{ ~ P Sin pt) .5117 rj Cos <f
A~ A, cos pt z: -I, SI;' pt =- 0
aR =- -bK', aB =- 6p2c:, COS" -' a", =- 0
.50 a. = bY K4 t- p4q/u>sZcj
At f3) Co.sj:Jt=- 0) SO? pt =~ t/>=Jr/z
u,e =- -bK.~ a", =- 2hpKi{,) Cl", = 0
.50 a.= 6KVK.lr-4p'6{l
when t = 2 n?1'n
cl t = 7 h?;r7
2/ C 8'J For / er f? =: z.;;::c a /: ~ = / Ill) = 8 -Is6=..{at" 5=.!..(1)(8~)=32h'18 2 ) g 2
Eh rrr« I;" lerya 1/ .:1V:-J 0 c/ C-)
o =(1)(8') +2Cit-/OIZ) =0Llt-=- 6 5
V:= uf-aLJt V-:: g+ 2(6)=20;'/4 it J 1"'1
S = 5 f V cJ [ I- ..!.. u 4t 2/4 8 8 2
2z: 32 i- [;(6) i j(2)6 = 116 rh
S .::;Sf-V- ,[J t f- .: C2 LIt 2/4 14 2
= /I~ t: 20(10) of i (_2)(102)
=- 21(, h7
r= 4:1=iVi92 ) t-= I. 69 5
( or t =- 6,3/ s )
Z/= d5 -50de t
/:5 =Jr-4f-3t - : t~dt2 0
3 2 I tJS = 2 - 4t f-"2 t - ~ In
2/3110-2
V= 5.9tZ,2. )
, a-AC..<_dv-, r;v jtsec.J - -<5' L - c/t )J dv = "::>";9t" dt-
5/2) 0 0V;~ -;z:J(2 = /03 .::322 ,:If/sec
2/36 I Z/= U e-kco
lA/hen t ~0 if=- v: = 10 knol-srv' I 0 _ 6k
t-Z--:--:/.3.:-:::'3~I--------------------~ Wher: t = 6 m;'nI V- =- 4- = loe .-1
Vdv = act's 0,- 21 d(v1):. ads 6k==/I1.!..E- =0.9/63 le::: 0,/527 rn r n
d(V~/d5 15 cO}f~f if= ci5 ~ Id~ z: iie-k.t-clt- s= 1{ (1_ e-let)de ./ '" .J 0 I Lo 0 11:-
Thus .s = 10 _/ (/_ e-O.9/63):: Q:..£.~~!.'(0. /S2 7) 60 (na v I-/c:aI)
Compare':) k<//Ih 0.60 /rorn PI"Ob. 2/leancl 0.61 fro!?'? Pro b. 2/35
I1 " t- Ll 5 -= 2000 - 93.3 - 90 z: 18/7 rnz: )II 1/ i'j I u= a..L1t-=-5(6) ~=30 rn/s
I
II 1/ i:'z. J Lls::: V'".d t- t-2 = :;;7= bO.6 5.J
a=.!..z
Thus
_ COh.51: If900-3~00
= -2250 Irm/I/2(0,6)
2250x 1000 = -0,174- YYl/sl.(3600 tZ
a.. = -
2/351 v=- ~11-7.{,/et
when t:. 0 7/= -V- = 10 Jrnols/ 0
when t-= 6 rn/n = 110 hour v=- 4- knOTSI
50 4 = 10 /) k z: /. 5 rn/ -/ f- lok I~
at 5 5 t'if:' dt- SO PS = /; v,,; clt
o 0 If/{kt
S:: ~ /17 (I f V-le t) J t IL 0 =-_/I7(I-f-/O!</'S~.!..)r<. 0 /.::> /0
:. _/- In Z. S" z- 0.9163 - o. GI In/ /IJour/cQ/)1•.5 1.5" - --'-1_' _
Compare:; k//rh 0.60 ./rorn Pro62.//8
if= 6(27r27r'h= 2213" /,5"'1 I;'.
sec
Z/3r I vdv = adxI
V-Z_ t<] x,z-T-X. ;
oFOr x,:: L - % )
/;:;;v = />'- K dxo !7L-X.)2.
V2: 2K;<-L(~-}(.)
V= 2. ,IK(L - .0/2)V' LD
2/4<J I a:= vdzr _ I d(t/Z)75 -'2 --;;;;s =A =': 400 -1600
Z .15 - 2(500 - 2.00)
z: •..2 ff/s~czLl if =..la d t )'?/:: 40 - 2 tAt- B 20 = 40 - 2 t- t = 10 sec
) I10 ] 10
/JS=/'Vdt=j(40-Zt:)dt =40t_t"2 z:? El ---
44/1
2/1/ I a « 1r:.I;c If d er::: .5.. d)(-~ , )t
fIT" er d v: :: k.f" Px i IF S =- le /" .1!...
• .," 2 =,Thus ('OO)&.=k[" 750 k~ o.~, :::o.O.39/(Ir,"'./1-
~ 7.S '~(4.t;oS) I
af x - .J 75 "'''', a c 0.03'11 ::. 104.2 km/51
375 (IO"'~)
2./42/(.1""1.;;: kls, er:::i ./k/s wh<",.~ k"'(5"O/.t.25
z:O..5b.2 (10·) m"., •S to. sz.1 rs"tls "'IO.S6:1(IO'J 1tit, fs~:z.J =7So(t-a)
. Z2S" tJ ~:zs-
$s/;l.~ 3375"+ 1125t, but .,..:/k.s-~ s« I
V"z: 7So (.3375 +-1125 e)" '/3 .f at t .•..35, crw 750 ('750P
= 39.7 "'".;:5
.30 ...35 .5.8 104-.0 10.2.. 2OL-~--~ __L--i __~
1&1.Z•.••/s 10 IS 20 2S .10 35s, •••
.I" <IX:: _ K xor, JK:: .L, er, I., //0 -I
X ~ - ;: - <,., -"'- -418(IO ....1jIn~ 400 .10 ---' _
dv- j"i- = -Kt.r2. dtr - -Kt t _ I ( I __ I)di ) 2 - - - -<,;V" 'K~~
t::: ~ (...!.. - .L )3.& :: .2~.3 s+,18 30 160 ---
2/45/ if= 40- K. 5) 0 = 40 -I< (~o») 1<= '12.50 1/ =- 40- S/2
Vdv:::odS' dV-;dS=-~v=-'/Z) a..=--~v-j
T/;""e t 10 reach 5: 'dO1't.J' a= dV/dt 50 -.!...v" dt/o t: 2 dt
J:!!:! = - ...t..fdt . In 0 - In 40 e- -.!.. ~1/ ,2) 2
40 0
t = Z In 40 - Z I¥) 0 6u f In 0 ::::- 00 So t-+- 00
V" = 40- 60 = 10 :ff/s<:c a.=...~ =-.5" ffz. j Z. s<:GZ
y = ~ In ~5h..;;e;t
tr~k~ .le;: =
ifo
if= 'if e. -kt-o
/17 1./ = -ktZ{; )zr= dx.-
dt-kt= ve..
o )
x=
Vdv = a dxJ vdv = _ le dx.
1/
v=v-lexo
2/481 a = dv /I/dtr f- 1tdt- ) ~-k if .!
o 0
/ ] if-k'ln (S-/ev) = z:o
kt = In .:«: ).9 kt 1/=.9 (1_ e-kt-),J-kv .J-kV x: e) le
--------ay iY tV= dt ) Jdy = ~j(/_ e~kt)dt
o ko
y=.!..(t~// e-kt)t y= .9[t_..{(/_e-k~]k ~ 0) k k
~2/47'I/V-dt.r e./ct cix-a .#/~c-z
Jo~o~r-- I.:~_I I r-- =
A, I f1000 r- - -,- ~1- -
: A I Io I A, I 2. 1 I ;( If
o '12. X I /
x-!...__ L looo = 4cOO(K-.!..)
.!... . 22
~2/s/1 12= d2,X =Kt-;tax d2X".~~=k't"
dtZ J df-2X- = X ". X. = A s~;;-A t + 8 CC5 .If. t + ':z. r-
e P ~
X =- A~. cos ~t-- tJi s/;.,~"t -f :'2 =0 tor t=o
o = A li - 0 r ~2. I A = - K/ ~ 3
;t:= 0 V(Jh~n t:o) so 0 = 0 f-B rO/ 8=0
Thus X' = 1<= (it - s/n ~t)~3
I I ) "f2/5_'A, = 2" '2 (/000. = 250 J; (."~1----'
Az= (~-i )(000
A '::..!...(X-!..)40oo(-r---'-)3 2 2 2
AI J- Az. +-A3:= 2000 X. 2_ /000 X- -r- 250
So -':600 = -2(2000 X.L-loooX_ f2.S0)
or x: 2_ O. 6-X - 0. 2 75" z: 0
(x>o.5.H)
-et A. -6t v=o ~;'I!" t·"= AI! r-e 9c-b C J .1'.. A = ..!. _ ~
G C-6
2/5:;1 £:i (.lt~-3t}i +.:t! j Fro,", SDffTpf~ Prob. z/& ranJ~ ;0$c
- J- u.£"" ~ (I)2s = sin 2O, .50 10(103)~af.z2 (4f -3) ~. + t~ 9 9.81
t :z 3 (18-<i)J + 9,/ 9./ + 9/ ".,/~ ~ ::: (9.81 (04)=- 313 ,.,/ss, f!:= :=
a =- (I2 -3).!· + 9' s: '1.i + 9j ••.•/s%:!
2/53 r x;: .tt2+ s r , x := +t +3, X = 4- m/s'"
!I = t~ -8, :J ~ =", f= e e ".,/57.
Wh~n t- 3 s, x = 4(3) +3 - /5 m/s:J := (3)2.:= q m/$
x=,4- ,..,/s"-J •• 2(3).e {. m/5%
ir =: /(15)2. + 9:2 := 17.41 ""/5>vi fh ():= ton -/ 3- = 3/.0·
c /5--
a :::/4:Z + (..2. "'" 7.2./ '"/s~
with Bv = ran-I £:: 56.3°~ 4- ---
'la "'/s}!!I E have same direct-ion.
5" at i: •• 3 S f'article IS at-ii7I"!t~c-t/on l'o;lTt on c o r ve ,
2/561
2/5"8' I ~= -3 So x= 0 - i:J"c~')1
'u.r'=.<:y.LOm .•
ve ».,1
f = V2y =-(ZOO:3 9.~/
:::1.2715
t..L'= 30//.277
= 23.S m./5
2/5'11 x,=/ooorn cc=600 n1./S &= ZOo
//0 Fron-? ..5an->p/e Pro6/ern 2/G/- - - gx:-Z 2
" ~ - y y = x. fan (9 - --2 sec "GI~- 2~
() XI 0= 9.8;(iooo)Z _1__Butyfd=)(.·Am~.)(1 2(600)2 (O.'13?7)
J = 1.:7.43 rn
2/60 I A n9fe. B lor ,"<lxi,.,.,."..,hence Ini,.,l mum a is
2/61 I For y-mt:Jn~n ')r"" J = 'i. f?/ rn/5z
y=fgc'
.,,y
& = /?ly) -/ J!... = I-an-' l!!.5: = Z/. 70
x: 25/ _
2/6"21 y= .x:-%) X:-= t 7"2. I i = t 0: ~
zy z; y =- X- X- =" f i =: t;/2 ) S,. ~ = i:tZ
• . Z4'x ::.~ = / In,/.5~C
. I 3·SO Y= t-t- + 2t .:!
2/66 I Titn" 10 fravel
.1: t, ---lr---: ~------ dh7--~---b - ~- X-I 0 I
U-pon
2/65' I ~:: y = Pt) y:: 41: Z-f C,./· Y = 2 ./1' ~hen r::.O
50 S =.2 d Y =- 4c 2. r 2 ./t•. '2Cl =x..=4t ;K.=2C -feZ' 1<:=0 t-vhen r=o SoC=o.x j j Z
;t::= Zt;j+c . x.=o "'-'heM t=o ..50 C=o3 3.1 __ :33/qJL Z1/ .5'::1' y= 4 V 4 -r z3 2.
or (y-2) ~ l44x
Wh~ n X' = 18' ft, 2 t J= / g t- 3= 2 7 If ~ 3sec3 ) j
d i= 2(3)z~ /8.H .; .: f?(3)::. 24 ;NScc ) sec
,/ - z. 2-V=VI8 -1-2.4 = 30./t/scc
I at t:::c2.Yes Y= 2; -14..; in.jJe£_____ -___ El /yn;'na le zf
AI"' d., _ . _ m.'Y:: .>':: Zt;- ay-v- -2-) dt. r .Y SZ
Y = cl.
'V=-Y92.f62
j =- ID. &'2.. m./s
J a...=Y'Iz.r z. 2
= 9.22 rn/5z
/L-~_____ X-3
Yl~VX
a,= x:
Ihro"'"yh ,bmlb 15
t= ZIV' SI17ce 1/'=//=., )L 0
Y=at=e£1-:Y.Y rrr er
D
A/50
2/681 ;:::r~;¥7 results 0/ .5QI'Yl,b/e
Prob/en-> 2/6if
.....-·---1------).--
/ t/ Z . 2A Z .I h - U. Sln-el' 5 _ U. Sin 2&/ h=l5'oo - ~- J - 23
/ IrYlo'YI.
~
I 50 h _ s, n 2& _ I.L-- - --rwn<9e I S ,51",2<9 2
__ - . - __I
S=400""'Wt Cl=- l<7n-121,= Ia,,-Jz(~n!)o)= 6f?Zo.s 4<;>0 __ . _
2 Z!) h 2 (tU?I)(O.5) ~I'2.U = --- z: ( • = 11.3 i ..,.,'5 U. =: 3,37 nt/s
sin:l.& o, 92fS)~ I
1/= cA.;Cose:= con5f. v= 3.37(0.3711-)= 1.25"I'>1./s
2/6'11 A/rcra/I: x-= vt = 600 ~ t: =- ?frOt30
517ell: x: = a:t c-os B = / goo C cos B.,L' I .•.Zy= ccc s;n& -z.3~
20000 = /s ao t; $117& - 32.2t2., 2.Frofrl /:!1' J"t,-vo e9L.n:Jh'ons;
?gossc t = 11100t CosB COSCl z: -- = 0.4f1l?9/ /floo
5/n8= 0.87Z3 (3=60.7°
50 20000::=: IJ'OO(O.8723)t-IG./t;2J
tZ- 97.szt- +/Z4Z =0) -t-=IS.I..5~c (it=O'z,Ssa)
= ~3.33i%/5S = utCOj e
= ?3.33 (0.7071) t = 58.93 ty=UtslnB r-jOyt2
-soo::=: 83.33 t (0. 7071)
f--j (-9. e.) t2.or t 2 - 12.01 t - IN.'?'; = 0
1.-= IZ.oII vi 12.o/~r4(IDI.'f4)= /77Ss
(or -5.74 s)
-5 = 58. "13 (17.75) = /046.2 m. 0 r s;:: 1.046 km
2
S :: tJ t S/n 45· x: 278 (7.69)(0. 707)
:: /.5"/0 ".,.
t4n(!l-8)~ /5/0:: 0.8369-s- 1800
3'1. '1'1., .r,. -I-S· - 39. '19·:: 5.0/·
2/721 //rn~ or drop h=-;igcZ
SOOffWl/h t= '12 (s-o 00)/'1. 81 :: 31.'1.5- - --. ~ s= v-r =(100/3.6)31.9 z: ~!J7 TY1.
b I- 5 = ut:: (SOD /3.6) 31. 9= 4434 rr:
b = 4434 - 5'87 z- 3547 rn
~I~:~I
IIL.
/
b e---~----
y
R=2u..2.
:: - -hnt9sec&9
)C
V3~(0.oOG4L/){. -2/(3) =0)
Hence O.CJOG441s.::. ~
A/50 Y= X faY! 3o~
¥ s:: X./:05 36" :: :J... Y-.V3
) s= 207 cif
z: Ic:i'h-' SOD 0
.?h'oV2 (sooo) T 500032.2 Z.
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