การเคลื่อนที่แบบโพรเจกไทล์ by วรท ศรีรัตนโช

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Fcosle uz cos' c[

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t o nttnJdautulat amx ticntinqdt nd auffi,uuuo2 tnauitlrttfi a nltu tit

qngr"l ro r. qcrfiu.irgrJnrln,tdflilPQRr{un'u*radflilAoB rflus{rurudeuafi'ltlar'.rriuqvld'jr

f-rlr =4s1 PQ! IP R

tflowrn6rrq.^6",,ddffiffii.'rilt5,]1.ruv1nrrruvr,rdlfufuurnrtvi"rri,udmrrrfud..oo=u, o'rqfiolfidrnmrt = v firi'nqtnrdoufinrnlar A trJri.4nn 1{nnr At du'i A rns n e;oq1nfrfi'ulrn eg d.tflrilrYirriuvAt (nrn S = v X t 6.rtiuu1fr'ir) urqrfiunx AB firirr,vi.rriuFlclils.lr.iro.:c{ru1fi{

/at

r v

Av AV

At = r"z Av Av A y .=- y 2 Ar r .,A t = a f' uns a fifirtvi.:rd,raryr4uJnar.r = a^ fi?'rurirrinudnnN . r! c

lrto

uuno rirr,luoti

v = dnntirto.rinqiirndauduuu'l{nnu r = infitor,:rnnr tl.:UU

2 n1€$1rrffdd11f{norn6aud$uui0na$ s' dwd- a- r J

clnn0n.lttnnouvl'ilovrfio{1'|€.rulnu$nln,ltlnnouYtitt 1l0{nail ut{nT{fytNln?vyt,.}mo.lnn

fiune'lutdrra.:inqnvfifinnr{16urfu fiofin{.:r{ru,yr4udnnt d{riflurflunrnrrlfr.jr

lfr"ir

IF=ma

1XF 6o F.)

F" =.ma"

t""=ft

uta

dotarEerfiql (Angular speed) .. d'nnrirtordnqiirnfiouiiu'uur':nnrfineh'llnufrrfufionrlxijmlo$duld.:diorqm6oudtff lutcnr r iurfi dlrn o'rqrS (v) r ndneijr{udr.jr d'nmriarfiordu un'luiidftfidnnrirdnrlrvrnnudr firrilunrrusndsrnnrlrJdduu:laworryrfiyr4utinn"n tfio.:crnnmnmnlrltarinfi iu'rfi lu r lSun'jrdotrrEzr8rqil(C0)rirudr Iar$rir f,srrrdomrEerfisilil(ol) fio qrdinfinmnhflAlu r 4urfrilulirsrfl u rrrfiiluziurfi a.rr {nnrndaufiuuuxnouiafi nrngrJ, r dqild'nerr5') n{ficrn A fir B lur.rar t 6urfi lsufiinfi r n.rrprhl rflu4u 0 rlfiilu ru vru.irrln A 1 -nnqu{nn}rro.r?{nn:.r qcfi yruuinfi oA nrlndsudirkft oB'lurontdncfiu fiamyrril rilrffu (v) lil rrirfi'u (rvactm{Di'r,rri'u1urc nr rvirri'u)usifid'nnritiqpr'(frl) rvirfi'ut4uiirfin{u rvirriulur"inl vi-rriu) 4

.qY

71 nlTLnAOunUuUNnaNnltUXUnlnluHnAl,rLUnilUUUn,l

rirorqldurjrflrflurr16su (radian}Ie uiifi,sredr utl n1ruon{}llronqrnc;fiud':stiluosnrud,r J

1at

-. sorJf,?ulacraffiililrtsmrrirfiufnfi I rtronrl iln1rfi1nuilruenn$sna1otror?irnail .ttgt

.itnrlundrs rtrdsu 6a dnzrdruz:u'i'udrutdulisdsaoiur.nrfiuieirar?ona$

fir

a 6o Rr1]ru'l'r1ro.:riru1fr.:firoliu4x r 6o infirotriculfrr 0 fio r1:.rdlnr4utinnttilut:rfiuu

q sl d.j r

l' 0=r ul !

unvqr nfisrildnnr5.:rt.rr.r ucrlfi;. l-rri 9f

raiondrrlfi

dauri"rantr?au nmrndaudruutnmn ntl ld$dude:ndroullunriz

ilrJr fi}

gann,rfiur*dsu

.-{_-r--

n${crau r..rtil rdsnuh

e d s r sy da tilaflc']til10{nnN vrul'rtJiltouqnnuunal{?ro{'}{nnrJtyl'tnu 360 o-!FtltnddculA{frralfuxlJfr6o 9J

tnutou?.:uuto.r a.rn a = '

rro

vet

2TErcylfr,jr

= Ir

ry T

zn ltrdslr

a 'l l *

dtriu nsill#j,uu s6o ao61rfiauwirriuxil 2it nrGisu dA-.:i4v rilafrqrrrur{nnrnfioufr [uuc.lnnldrudnnr5,lnlfi nru r rouna6 s1n

C I-

l{U.j'}

2.tt tflAetu

T Qu r fi( nr u)

F =+l,*EC

rylff.ir $ d ;c1n !

f----i

or'rriu

u n rv = 2 ? f , d

,Drl

l-_

dufi u nmudl.rr{u6xu'j'wd'nrTr5.:ri*&u (v ) uacd'olr'lril ti.r4r (tg;

f,?rlcnrin1ritm ornrt rndaudsuu?rnru F"

= 'moc

F ^c =lnv2 r

--)'

tr =

111r= + rl

= zn rf

*=?=+

= zllf

0 =?

rllo

4 .' ruurnetndrr{gurinar$sauinndradaud*uuzunal{nfi4 ulq? drsi'n:tie zo rumsiaiurfi Sratr:rileurf rdaf r = d m, v = 20 mus frotnr:rara. ..) q'rn rc = f

= Egl' 4

-td, OtIUX[n?1Sll€$dnxlgnAT qu

5 -tU

)-]r.!)J-,v)

2gtfinailuuuflntuang,l?

0AA$?Agl$61 t

t Oo ltlO€ nA?ll.lll-

| urnl| ua?Ltn?g lu[aaaililluu?nnaila"gfr"1ila

' e$$1n?1$r.indaurinnrraauicn UU i =s Hz f ro ln r: llra " tfiaf r=rm,f iu n fl :r ilenri C1n

a^ cr

v 2,

a nzrf

.,22,2x1xs2 4\7)

'lc =

,,- tv = ZIErt)

987JG

mls2

d'uriunetnisdnurfn aroIirir eaz.zo urnt aiaiurfi 2 u!

5 ?ailoa?u1?l

urir___lg_

nrr rrrirlsgurteurinrt rndauduan{nnt rruu?$na$ '4

,- d..-

,4! , av -'--g -'r. lnqvliln1tInnauv]ttuur.:natJild?snuunludnlrruv uoifr{umoun,rrrrffrTegrarrr,rfiouri'u 6a _draia-.

t,uuuilln"lrtnnouytlro.:qhq.

2.'[ri ur{finrc?irrio6'nqtlnrruilnrr.r 3. fierrrur.nn4udnnrltornT :rndaud udlunnur.:finrvvrhoiainqhiagluuurai,runn4utinnrr (uurinfi) un;luuurri.rornriuuucairuqnrnurlnnrr A. rurudvrf (xF) luuurrirular4utinar.l.uo.:n'rln6aufi6auruqi4utinn-r.: (Fc)rF = F" = ma" 5. rrx'lorr!.:rirurlnrnueinartfirirrfluu'rn (- ) 1+; t{ruur.:1ffr!.roonn,rnyr4ufnn-r.:firirrilunu 6. fonriauudsernariuu[d: dsnruldrauunurruads drsfinrrnu$rtrs o.s rrqr sirdrt rltttndu os neruriisqrnrut:udrofir{nqfiuTfr:fia'r o.s oarnrntrur'rduld:firiru ploirh dqodrauzurfi fqn6nfnhinr:rduernld: irnn:rdteuri riuugrjtd;raruufrr6rrqaqjuuldrlfirinon.ir{nqrndaufiuuun,:nnx fiq1ru.r, trxnr;rirrioinnr^ru'jr ur{16flnuru(fr) rJrrvrqfifr'rriluur"rqiqutinn,x (F")

fiouarnm*d rrl riou q1n

F" mv2 -T-

pmg

an2 rf

ttg aa.,

(v = 2firt)

= o.r 52 _g{ 9_ a(zzrt)2o.s = 0.3e rsuriurfi = 0.39X oorsuzurfi l,tto trfr:ulurEeden= 2s.4 aauzurii ag.crausiaurfi f

rlmngdrufiandun

oooriu uuar:du eaanL?a1rirrirufnrao surdnar,rruandlf,fi ir r fir6'u esln$rsdnuri na,toue:R2ur.iuuan{onduarsriian g{

I' I

tu l* * y-,ira4o

irnnn{trnd

? nnfi iln1 EJtX OnIndo ul1u,llu?nnail 10tnil.llln31ronflilnn J.

'[rinxnr;rirripinq ttnyrarfisrri'uuoFl

tsm 30

ng{nar-- r.':rvr *{iinuh dauri'rant#su nrandauduuufinaqntnduc;ur-ru':.rnnr

ufir-:E-

'F T

rf,-

Tcosed pc

EFv

o.aee N

Tsin 3oo

.mg m

nkg

1r'tac ; nrn F"

ffiuc

(fi1 o"

o .8 6 6 a-

1 7 3 2 m /s2

r z.sz rlntdaturfiz d'sriu uro{gurinarufiir o.aeofir6'u ra:n?1illd6dnurinarur*aCIi'n$fid'r {t

B . erngildnr€sununiiurirnrtuoaasr4arursdnurfnatt {trngdrtnr:{noaridruaan$1e1n .t

unl'l$illl t J t

o.a ula€

$a:agsnao$1e1n[tu?€36|u

o.1 ttJ6t aglnvr€1il?'lg6u,3uu2qa$ao€1

uEtlrirlo

o . rml

o.l m

.l.

iratrtrilwrd

.li

q vJ

-; v t'tr?luDlouLtnurull1l:o.1|{nnN ?FlQUilAlEJttOnrnnoUvluuu?{nnlJF.|o{n1tu'l

(r. nrragrJ d.l{arint s. $nnu:{uuutinunutin^"on^"urrrfr.a'rn z.'l,ciur.: EF: Fc) 4,A (frnrrrurfri'nq) ;

IFx =

Tsin0

^c mv2 -r

=0 -

fcos0 (r)/(2)

ttl

= .

g€? a Hln€oa?u1$ oulltl?0lo5tani1 I -

--(2)

v' rg

tan0

-e

v2-_

o . 8(r o ) 8 m/ s

/r (rl\

dauri?snrflisu-. nnndauid$rutil?n .l

ne{nau

u.qeyrr6inuk

uri'r**:o-_

n1€rs6aufiuurnlslir r.

rgiuuuilrcldamu J

lilCtntnuctn.: uunuutn{r,tu n{ri$u'lyt1{n1t i

tndoufrrilurirul6wo.:.Nnarud1rll 12 { ( ta- - ?.

no.:r.[t?.:{nuunnl{ntcyllF}o?nn

") d'lriu

tilafic,'tttu1

u:r nr vvr:rsiorn'lu rru,tr ; d'un u.ir srurt,dur ro tt

€

v{fl1u''ltJlnno o nq1n [n{ s.:u ut.:ttluttyt"lu li?iu

nr: rirsiofrornlufinu'rlli.u{rluuurairunnnud notnrnulfrr ta g

3rl

uttn:xitrb.,ntruzt{acuunuuliltnu tl.luu $?.:r6rJnytlu nr.rqi4utinnr.r t

]

f- s=

d,tsorEszdasdntr uieaoanloirtaanfin ut fr(max) F"^ mv' pN r mv2 pmg r v2

tio p 6o fiirJr;Evrf,ro{nmilrd flnflrurvu.ir.:darnfi'unuu v 6a d'nnrSoqoqrn*rdnr:nldrJoaari'rr r fia infiaryulfrirolnuu nrzur$udsoaaceoin: rnusuefuot;rdsz ' n*rrdrmnnrunryrirEiornfi mg,N un; f drux N un; f yru.j.r riluur,rdn r.rr.rri'uld fR ur.:r.i'r,{ #triairuTr4urinnl{ro{ F.=f*N t qcriah;rfinlrr&uo{ rrn C.M. rirhirnnr'trru;rdu.rd'rg:J r s(n)frrlrifro{rnr'hirnnehfre*6 u.rrnhil nqurinntr ow ra Rcos 0 airuuurnx R r,ru"rdur rn{.:crrdflq'tfrlnudnoor.rTa hilr6nn":"rd'rgrJ r s(r ) frrrdarrnfrnudnnrSros4or nu.lr ('luuu"r6r) Rcos0 - mg (t ) ".................. fluuutrcn'u)Rsin0 = F" Rsin 0

= mv2 r

til (t)

{€rl

13 tvtanuJan, rit eio ndnt at u auEi'tt wyt{a tu un uulit n u .i-

n€

r,l.rou no illytrncntglutofl{q1n(uct{.t o-taJ,

v nO OFttlttlltuCtnUrj <

r nO tnilfi.l''l:J [n{{nr3[ail.J

(2) ..........,........

vfil' tt

z. r$asuuurnliusiigtd* I .1 vc J ! J se 4 v r ' otir{ilnonJi'ufiuotjrTrrirdurl:vdvr6 rnqvrha.flntiquSotr c'rnnlldur:nuunuuldr:ruwu.ir urifirfiriaarnrdflrfirsdnrlr5rei'r frrfiurnrnrdflr1fr6,rad'nmrFcos rsrnrilrfisnflruasu.irrdufi'ufra d'rduir rnlriarur:ntdsclfi.:tfirnu unvfird'rrJxfrvrGrorncrx16rrnynurvlr.jtfrurdulfi'udorflu4uti !_

ee, - ddr :

J =.- - - .- - .,

4 r d ), torulrifiolornu frnrrrfr'lnflnrl6ellfrunuurfioorfi'EJursr.lflfi?urfrfrunrcdrriornriluuxq{4utinat 4v

Itt{!6flttvl'ruFt{t1J {

ru.jtn.:Nsin0fifinvnwlrairu fiqrrnnuxnrvdroio{nqr ncrulfrq6'{liu yr 4utinnr.rraq (uurr:fi'u) Nsin0 = Fc : mv2 ................( r ) r (rru':r;fr'u)Ncos0 =mB ...............( z ) = +ut x'l (r ) -r (z) N!4 r mg Ncos0

Ncos 0

Nsin e

3rI 14 uronrrirri*rrzizt{acl&uunuudwfit4x 0 iuf,urzilu g

lilo u no ]]ilvrflunuilro0{qlnlluQtsnu

v fio dnrtS.noart,*r,dotuulfi.:rdar r fie infinr.r,lfi.oronorl i

d.|iln1€Lnaauiluoszora I

g2c)latl

r. uutdrs'ruo.r*rdr"ro ' r 2 Jq

l; trL

,c w2 e !

."; z.

uulfsldss

( p

rnrfiurlfrrinon"riil 4e

= tl

3nilonr"ltt14{nnl?Ntna

> p

.rnr6flc'lil't61nnoonuonfl1.!

rre:dnautnEsrtdo

I ) ^ = u2 l rniiauriuurinrrritnrrdrarTulasnuu1dur6sutiiaufiu$r$rdu (r ) ernleie itan0 iI u I; r rg I !Y : a e n,!tn? g'|uga 92 tilaunu$l{?nr

(a) rainaarurdselis tanOS p rrirrtu '9. tnduufissdsetiauuo$ur1ild?${niineT$Iio

roo r$ns if$rll:infraoae.rrrrilgsrr.r$fo

rTuouurfiu o.aenfiude:rdsedesdntrrEatrriXuqnwirlr ? r n m*i l e n ri rru ;rn rd s" fft{ r =r oom, M= o.a fiolnr r utvC{ qn q'ln

rr=b

uz =

prg

Y -i i j osuu?ole

0.4 X ' 100X 10

20m/s

g?nx gafr €1t€?6lg6l6r 2 0 Lllslf oo?141?l

dourirsnrr6u,r ntrn6oufiuuulnnqnlndun;uuu'1{nnil ng{aau

x'Qifl difrtuh

ro{rlnzoiaturfiloafifnfiaar*rtio ro.rn{nsgusu0i6uudori,rftrdtedzu6'ermsi" * nudr:datr6so:ndr{1fi'uuuzt:duwiTls irnsr:ritrnri ro{nrrl.lrurritditolftiu = ior6 m/s,r* to{* uld'rfi'uuuqrvdu fiolnrrurrl.tdro16

.uft 18

ro€

nrquriuuurdn(0) riou -v2 -rg

tun0

c'ln

(ro16)2

rofi (ro) =

tun0

./;

lfr.jr0 = od (riiuuri'uuutfit) uiiariul ed fiu$ueE:oiu I

d'oriuaudiaordtlffi qdrql so aanrfiu*urtdu

11.rogudfiuudauriurdgrtiauuouuldrinfi 1oo rgort $a: 4oo rloarirsdntrrEr zz fita -' *ruroduiirllo rir6lrjz:frnduaoamurf,somuarvrd'rtduorilrriusrsfii'to-e asrnuttud't rne:rdszldil aaofisu€ald v 4i4et = prg nruvrnrdudfulfrilnoorfitj tra:rnitenri v urnn'irv 4{qt}rnq;lnnoonuonvll{ frrrdu'ifi.:frqfl t n*r,du, = tz kmthr u6oeomrs ,1 2 p = 'c r n v d l q o ;c1 n ,2 =

prg

v=G tfi or =toom,

('roo) t o = 14.14 ml s

LI J O r = t r O 0 m ,

0.2(roo)

= 28.28 m/s

r oo ,.tn! dn*riaiiiurnndr

driiurerdsruunrslirinfi

?r1! dzuvnotiainfi aoo j3rnr dnmriefi itiasndr

6oerrEegnqololoaaan*an

dn*rriexoqelord?tzldrlnonfitl

'12. r osud durndaududzs6or,!rir r oe fitar$osdai'tT$a uuouutft diiinfi amlliu 3ootrqr' rdo.|,rif, n uru riln ouruduouuatre

lfiu uual:du rriTll *6 uarir 11

r o€sr:rf;furIdill.

rl aoofiulrilo aaanuan$rtl tiiaf u = toe km/hr n?o 30m/s, r = 300m irnn:rilqud fiorn'irur 0

c1 n

tano = # = tan0

= $

(so)2 3oo('r o) 0.3 tan-l o'a n€a t o'y'

t o.?oasnl to6n*rdnzlioldrtaaofia d'uriud*or.g""oc:rdsnrirrilfiuuuee:d'urilu4l ----' -a|

ntrndoudguuliltentfl6uffi$.ru?.:nnr.r ng{rtau---amudfnull--ur1'r

iiauri.Ianr#s*

.:r

n,,l?tnaa$$ Hluu??t| neil lutrtl,luo€ 'iunrrfinurnrlndaudtotinq'luriurxnnr'iut;utu6lartfic1rrurlfrc'lnnrni'r{nqqntiirn'a vJdq-'-dt4

*dctuie*fronlur:,urufi{r'rqfirJnrartenqrrn6oufrluuuerlnnN'lurvurufilfrinJ ]s

gr.l rs urtnxitoiainqiimdauffluuucctnnx'lurtutufrt ru shuvitgitt urln:;tirinqfi z ur.rfio unfi.rriontr) uanh i'nqrr,dauoglurirsuriqrir.:1 urinto,:inn(mg)frr{nqilrnm rnfioufifrefld'mflr5':nrfi u infi r trur.i'nqoqjdol"rnraril A, B,C,D tlig, q,tnn1nrrtndaud E r6onfiurl6l T4, Tg, TC, TD uns T" ntrtiror'u {a z mlfirfi'u unsrntl rndoudluut .re.rnnil fi crrrurur,rdln{ fluuur airulor4urinnrslfr.J,r EF

r nA

TA -mg

JTe

TB - mg corO

mv2 r

4Tg

fi c

$v-2 *mg r

* mg"otO

,uv-2

TD + mgcosy

ry:

"r -a,^D

fi e

TE *mg

mv2

4TE

#-*t

mscos T

) Jr e A ! J tsla?ootaaaufl o"sao€1r€?att?r usrdqrfian or druurioriro q r:lrirrirfiu I

n6tuanileoo1coiln1il1nndo litautt

ttu.tgl$st It

gba;$eododoiln,lua gvtf, o

nraradaudriluaonauloinauaasrrafiritufntil:aoa{nntdas.r rniarrdrsr

?$flar ua:ail,!6seoosooioo*fi6'nmlieriogdan lstar

rilutnaili

rirdet,rdasn'irdu

c:lrif,'lrnrsmCeud

na:uuitfianfinafio:Eiu utcnsriian ar qo€n*eiinirfiugprf (T = o)

ru1n{{En XF '

T+ {v4

f:J€ v u€sflqrl -t tl.lUU

*ooodr rda{nqrndaudriluronnulurnnufislsinad

tosnrr

dourjqsn.pr*n, nrffdolrfiuuulnrnntndunvuu:tmnnrnX{cou -u.rnr dlfautt v =0 o . fr u r fr

s'lnfrl rir s| roeron sagl

urir zo

ioulnvnnnttr'lurturhl nuuu,l(r ) inqnnntntur(z) yrxrfiuuuulnlqnlyrd (3) {orqrndouilrflue.:nnil0nuuu1

d r e .l i J ue A Av dqt i ' yrileltvmnRfllxttil 2 $uq nlonntndourincgant.ttt?$Jn{rit{unltmnaun$uuQ.:nnNtut;ulun.:

a - o r. azrliol$$urdnur[nert (a") dudru&fu (a11 dm; firuTrr{uoqifrunhurariw o.:{orq 2. aer$r.iolu$uzrdudfl cTc n1?u1nmuitdr rfrdo.rqrn dc+-ar rflila

,r-q

adud= \q.4

nTntrrJ srfn A srroB I

aT =8

a, = gsin0

inqrne r fitlnil gnfinfiurte,sf,rundouatrdurfionu,rzz urf,?ude*ndrhilre r filenil raiarrdriluernls luuuernnudrsiradoarienudrio$dqo rcurrrsiisrfianurrrlinqalid A, B, C,I) *a: E

iramsritruri .,. t ^) iierronii

E ;

uln6utinnr.: r.'lciur.rnrsrho'ia{nqr1n1n'rificunnur.ilfraiT u riaoqndlnq'rqor'tE) z. inqrndorjdtuuuq.nndulurvurud.r - {F ur v rioud{nd unvTg r'.0 '

u r t d A ,B ' ,Uo tfr sm = i kg r, - 2 m TE = o Er

v riouqn -) q1n

mg+T" 'l 0 + 0 u2

f,mronCA;

TA T" - mgcos ad

r"-rotll TB

f,mrond c i

I (v' ) 2 20 mv2 -T-

TA-*g TO-to'

f,errslrCn ;

F"o TL\L.

r (29) 2

zoN !0f r r ( zot ,2

r cN mv2 -T1(20)

--Tro N

To + mgcosod :

f,mlanC u ;

= ,{ P

r o * r o 1]1

. 'n .., - o -j'N_ -D , . _& - dbriuumfiortanaosiono{fi e,, B, c, D ue: E iii't zo, i6, ro, o nr: o Sedunud, du

j3; qln{nq$?r o.u fi[ani$ itsriNanrura,r?2 $lnt udbrcndtrilumnt$m$?suludr uqr:tic pdrqofidnnrir ro unrrdaiurfi qwn$rrdtttan ar r.n ri, qariluuhk 1{ud'nnrlrfiilcgurr?o{e{nnil irnm:ritenri mnXrJufirldur.:nr;tiroia{nq Ir

c1n

F"o

T - m s = rnv' t

s:!-gf

T-5 -o -

T = - ! ' - - tnrti.ulr.tcntttEan

soN

30 u2nu ot eon,ltaun,r !t

.rl----ajlerl

ao rum daiurfr +r+. rflla€urrtcnrnr I zoo fiteni! fiuriluzonnflutnn$ardbsdnr'rrizeirqn rwrriniisocn,trihr siornia$u ru ln{{{rl rieugrJunnrnl?rnfioufito.rtnialiuu6.:1eiur.:nr;ri, irannileuri = F " . IF

rn!'+N

rfio v fiaflntt -)

LF riouon --) N = o

r =';

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