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leqcia 3
Tavisufali vardna
veqtorebi
2-ganzomilebiani moZraoba
Physics 211: Lecture 1, Pg 2
mudmivi aCqarebis SemTxvevaSi Cven vnaxeT:
atvv 0 +=
200 at
21tvxx ++=
consta =
x
a
v t
t
t
v)(v21v
)x2a(xvv
0av
020
2
+=
−=−
sasargeblo formulebi:
TanabaraCqarebuli moZraobis modelireba
Physics 211: Lecture 1, Pg 4
TavisufaliTavisufali vardnavardna
y
ay = − g
y 0yv = v - g t2
y00 tg21tvyy −+=
y
a
v
t
t
t
gay −=
Physics 211: Lecture 1, Pg 5
Galileo Galilei (1564-1642)
Physics 211: Lecture 1, Pg 6
g ar aris damokidebuli sxeulis masaze !
Galileo Galilei (1564-1642)
g = 9.81 m/s2
ekvatorze g = 9.78 m/s2
CrdiloeT polusze g = 9.83 m/s2
Tavisufali vardnis aCqarebis gazomvebi
gamoiyeneba geologiasa da geografiaSi
g -s ganawilebis ruqa aSS-s teritoriaze
videofilmi Tavisufal vardnaze
Physics 211: Lecture 1, Pg 9
vardnis dro da saboloo
siCqare ?
1000 m
Physics 211: Lecture 1, Pg 10
avirCioT koordinatTa
sistema.aTvlis wertili damimarTuleba.
v0y = 0.
20y0 gt
21tvyy +=
20 gt
21yy −=
1000 m
y = 0
y
Physics 211: Lecture 1, Pg 11
y = 0 da y0 = 1000 m.
gavixsenoT:y0 = 1000 m
y
2.14gy2t 0 ==
20 gt
21yy -=
y = 0
)( 02
y02y yya2vv -- =
140gy2v 0y −=±= m/wm
wm
Physics 211: Lecture 1, Pg 12
ormoSi vagdebT qvas da vzomavT drosromelic dasWirdeba qvas ormos fskerismisaRwevad.
Tu y RerZs mivmarTavT qveviT:
y0=0 ; v0y = 0.
20y0 gt
21tvyy ++=
2gt21y =
amocana: rogor ganvsazRvroT ormos siRrme
vTqvaT t=3 wm.maSin y≈45 m.
Physics 211: Lecture 1, Pg 13
oriori adamianiadamiani dgasdgas simaRlezesimaRleze HH. . oriveorive isvrisisvris
burTsburTs sawyisisawyisi siCqariTsiCqariT vv00, , pirvelipirveli isvrisisvris qveviTqveviT
dada meoremeore isvrisisvris zeviTzeviT. . burTebisburTebis siCqaresiCqare miwazemiwaze
davardnisasdavardnisas arisaris vvAA dada vvBB .. romeliromeli pasuxiapasuxia
sworiswori: :
(a)(a) vvAA < < vvBB (b) (b) vvAA = = vvBB (c) (c) vvAA > > vvBB
vv00
vv00
2211
HHvvAA vvBB
Physics 211: Lecture 1, Pg 14
radgan moZraoba zeviT da qveviT simetriulia,
intuicia gckarnaxobs rom v = v0
marTlac, SegviZlia vnaxoT:
vv002 2
HH
vv = v= v00
( ) 0HHg2vv 20
2 =−−=− )(
y = 0y = 0
Physics 211: Lecture 1, Pg 15
SegviZliaSegviZlia pirdapirpirdapir gamoviyenoTgamoviyenoT formulaformula::
( )H0g2vv 20
2 −−=− )(1:1:
vv00
vv00
11 22
y = 0y = 0
( )H0g2vv 20
2 −−=− )(2:2:erTmaneTiserTmaneTis toliatolia !!!!
moZraobis aRweramoZraobis aRwera
erTganzomilebiani
moZraoba
organzomilebiani
moZraoba
saWiroa
SemovitanoT
Physics 211: Lecture 1, Pg 17
veqtorebiveqtorebi ((mimoxilvamimoxilva))::1 ganzomilebaSi mimarTulebas avRniSnavdiT + an –niSniT.
2 an 3 ganzomilebaSi ar aris sakmarisi niSnismiTiTeba:
magaliTisaTvis ganvixiloT mdebareobis veqtori rr2 ganzomilebaSi.
kiTxvakiTxva: sad aris rusTavi?avirCioT aTvlis wertili
TbilisSi
unda mivuTiToT aramarto manZili aramed
mimarTulebac.
Tbilisi
rusTavi
r r
Physics 211: Lecture 1, Pg 18
skalarebs aqvT mxolod sidide.
skalarebia: masa, moculoba, temperatura.
veqtorebs gaaCniaT sidide da mimarTuleba.
veqtorebia: siCqare, aCqareba, Zala.
veqtorebi da skalarebi
Physics 211: Lecture 1, Pg 19
veqtorebiveqtorebi......
veqtorebis ori aRniSvna:
A A anan
A A =rA
rA
SeiZleba gadavaadgiloT veqtorebi rogorcgvinda, magram unda SevinarCunoT sigrZe da
mimarTuleba !!
veqtorebis paraleluri gadatana
Sekrebis Tanmimdevrobas mniSvneloba ara aqvs
veqtorebis grafikuli Sekreba
ganvixiloT ori veqtori AA and BB. movnaxoT maTi jami AA + BB.
veqtorebis Sekrebis ilustracia
Physics 2211: Lecture 5, Pg 23
1 2 3A A A+ +r r r
movZebnoT veqtorebis
jami
1 2 3 4 5
1Ar
2Ar
3Ar
Physics 2211: Lecture 5, Pg 24
1 2 3A A A+ +r r r
1 2 3
t2
Slide 24
t2 11128teacher; 25.08.2002
Physics 2211: Lecture 5, Pg 25
veqtorebisveqtorebis gamravleba/gayofagamravleba/gayofa skalarzeskalarze
Physics 2211: Lecture 5, Pg 26
kerZokerZo SemTxvevaSemTxveva, , gamravlebagamravleba --11--zeze
Physics 2211: Lecture 5, Pg 27
magaliTebimagaliTebi: : veqtorebisveqtorebis skalarebzeskalarebze
gamravlebagamravleba
Physics 2211: Lecture 5, Pg 28
2 A B−r r
1 2 3 4 5
airCieT swori pasuxi
Aur
Bur
Physics 2211: Lecture 5, Pg 29
swori pasuxia (1)
1 2 3
Physics 2211: Lecture 5, Pg 30
veqtorisveqtoris komponentebikomponentebi
veqtorisveqtoris komponentebikomponentebi arisaris veqtorisveqtoris
proeqciaproeqcia koordinatTakoordinatTa RerZebzeRerZebze
{xa
aya
y
xxauur
yauur
y
x
a
x ya a= +uur uur
a
Physics 2211: Lecture 5, Pg 31
veqtorisveqtoris komponentebikomponentebi SeiZlebaSeiZleba iyosiyos
dadebiTidadebiTi an an uaryofiTiuaryofiTi..
b
y
x
by > 0
bx < 0
{}
y
xxbuur
ybuurb
b x yb b= +uur uur
Physics 2211: Lecture 5, Pg 32
veqtoris sigrZe ar aris damokidebulimimarTulebaze
2 2x ya a a a= = +
rar
ya
xa x
y
absoluturi mniSvneloba (sigrZe) rr veqtorisamovnaxoT piTagoras Teoremis gamoyenebiT:
Physics 2211: Lecture 5, Pg 33
trigonometriistrigonometriis meSveobiTmeSveobiT veqtorisveqtoris
komponentebiskomponentebis monaxvamonaxva
θ
ar
ya
xa x
y
cossin
x
y
a aa a
θθ
==
ay = ax tan θ
tan θ = ay/ax
Physics 211: Lecture 1, Pg 34
33--ganzomilebiani ganzomilebiani kinematikakinematika
sxeulis mdebareoba, siCqare da aCqareba
3 ganzomilebaSi:
rr = x + y + zvv = vx + vy + vz
aa = ax + ay + az
mudmivi aCqarebis SemTxvevaSi:
aa = constvv = vv0 + a a trr = rr0 + vv0 t + 1/2 aa t2
(sadac aa, vv, vv0, rr, rr0, veqtorebia)
Physics 211: Lecture 1, Pg 35
romeli burTi davardeba miwaze ufro adre ?
a) pirveli
b) meorec) erTdroulad
1 2
h h
y=y0+v0yt-gt2/2v0y=0 da y=0t=(2y0/g)1/2
dro ar aris damokidebuli v0x-ze
moZraoba x da y RerZebis gaswvriv
SegviZlia ganvixiloT
erTmaneTisagan damoukideblad !
22--ganzomilebiani ganzomilebiani moZraobamoZraoba
Physics 211: Lecture 1, Pg 37
sad unda daumiznos zooparkis
TanamSromelma rom moartyas maimuns ?
( man icis rom maimuni Camoxteba gasrolisTanave ! )
Physics 211: Lecture 1, Pg 38
gravitacia rom ar arsebobdes cxadia, rom
pirdapir unda daumiznos.
gravitaciis SemTxvevaSi mainc pirdapir
unda daumiznos. r = r0
r =v0t
Physics 211: Lecture 1, Pg 39
rr = vv0 t - 1/2 gg t2
rr = r0 - 1/2 gg t2
bravo !
Physics 211: Lecture 1, Pg 40
x x = = xx00
yy = -1/2 gg t2
igive situacia sxva perspeqtividan.
xx = = vv0 0 ttyy = -1/2 gg t2