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GERAK DALAM DUA DIMENSI
TIU
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A
Dimanakah A berada ?
OKerangka acuan
Pusat acuan
Vektor posisi
rjarak
θ arah
Y
X
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PENGURAIAN VEKTOR ATAS KOMPONEN-KOMPONENNYA
X
Y
O
θ
ay a
ax
a
a
ay = a sin θax = a cos θa2 = ax2 + ay2
a a ax y= +2 2
tanθ=a
ay
x
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X
Y
O
ay a
ax
a
a
VEKTOR SATUAN
i
j
jia ˆˆyx aa +=
- Menunjukkan satu arah tertentu- Panjangnya satu satuan- Tak berdimensi- Saling tegak lurus (ortogonal)
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PENJUMLAHAN VEKTOR
a
a
b
+ b
R
= R
b
= b
a
+ aPenjumlahan vektor adalah komutatif
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PENJUMLAHAN VEKTOR MENGGUNAKAN KOMPONEN-KOMPONENNYA
a
bR
X
Y
oθ
ax
ay
bx
by
Rx
Ry
R a bx x x= +
R a by y y= +
R R Rx y= +2 2
tanθ=R
Ry
x
jiR ˆˆyx RR +=
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PENGURANGAN VEKTOR
a
b
-ba - b
a b a ( b)− = + −
Apakah pengurangan vektor komutatif ?
-a
b - aa b b a− ≠ −
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PENJUMLAHAN BEBERAPA VEKTOR
a
b
c
d
R
R = a + b + c + d
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P,ti
O
ri
Posisi awal
Q,t2∆rPergeseran
rfPosisi akhir
ri + ∆r = rf
VEKTOR PERGESERANY
X
∆r = rf − ri
C
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O
ri
∆r
rf
Y
X∆xxi
yi
∆y
yf
xf
xxx if ∆+=yyy if ∆+=
jir ˆˆfff yx +=
jir ˆ)(ˆ)( yyxx iif ∆++∆+=
jir ˆˆ yx ∆+∆=∆
jir ˆˆiii yx +=
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KECEPATAN rATA-rATA
O
ri
∆r
rf
Y
X
t∆∆= r
if
ifav tt −
−=
rrv
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KECEPATAN SESAAT
O
r1
Y
X
tav ∆∆= r
vif
if
tt −−
=rr
∆r
r2r2r2
v ∆r∆r tt ∆
∆=→∆
rv
0lim
dtdr=
dtyxd )ˆˆ( ji +=
ji ˆˆdtdy
dtdx +=
ji ˆˆyx vv +=
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PErCEPATAN
O
r1 r2
Y
X
t∆∆= vv1
v2
v1
dtdv=
ji ˆˆyx aa +=
12
12
ttav −−= vv
a
tt ∆∆=
→∆
va
0lim
∆vaav
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Gerak dalam Dua Dimensi dengan Percepatan Tetap
jiv ˆˆyx vv +=vxo + axt vxo + axt
ji ˆ)(ˆ)( tavtav yyoxxo +++=
)ˆˆ()ˆˆ( jiji tatavv yxyoxo +++=
to avv +=
A. Kecepat an
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jir ˆˆ yx +=221 tatvx xxoo ++ 2
21 tatvx xxoo ++
jir ˆ)(ˆ)( 2212
21 tatvytatvx yyooxxoo +++++=
)ˆˆ()ˆˆ()ˆˆ( 2212
21 jijiji tatatvtvyx yxyoxooo +++++=
221 ttoo avrr ++=
B. Posisi
Cont oh Soal :
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GERAK PELURU Asumsi-asumsi :
Selama bergerak percepatan gravitasi, g, adalah konstan dan arahnya ke bawah
Pengaruh gesekan udara dapat diabaikan
Benda tidak mengalami rotasi
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0 X
Y
vxo
vyovo
vxo
vy vvxo
vy = 0
vxo
vy v
vyo
vxo
vo
g
θo
konstan== xox vv
gtvv yoy −=
tvx xo=oov θcos= tv oo )cos( θ=
gtv oo −= θsin
221 gttvy yo −=
221sin gttv oo −= θ Pr oblem :
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TUGAS
Dalam gerak parabola tunjukkan bahwa lintasan part ikel
dapat dinyatakan seperti ber ikut ini :
222
)cos2
()(tan xv
gxy
ooo θ
θ −=