Page 1 3 - D KINEMATICS [ HOW THINGS MOVE : ( ) r r t = ] 3

3 - D KINEMATICS
r = r (t ) ]
[ HOW THINGS MOVE :
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3 - D KINEMATICS
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1 - D KINEMATICS
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ao = ( 0 , − g , 0
PROJECTILE MOTION
(v
vo =
o
cos φ , v o sin φ , 0 )
g t2
y = −
2
x = v o cos φ t
;
y = 0
or
→ t = 0
t =
vy = 0
→
{ x (t ) ,
y (t )
0, 0, 0
)
+ v o sin φ t
2 v o sin φ
g
= τ
: Time of Flight
ymax
}
→
v o2 sin 2 φ
τ 
≡ H = y   =
2g
2
y = y ( x) :
sin φ
g
x −
x2
2
2
cos φ
2 vo cos φ
Parabola
(
2 v o2 sin φ cos φ
R ≡ x ( τ ) = v o cos φ τ =
g
Range :
y =
ro =
;
)
≠
→
Orbit ( Trajectory )
x

y = tan φ x 1 −

R

Ellipse ! ?
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Plane Polar Coordinates :
x = r cos φ
r =
y = r sin φ
φ = tan −1 y x
r = R :
CIRCULAR MOTION :
Uniform Circular Motion :
o
(
a = − R ωo2
= R rˆ
o
− sin ωot , cos ωot , 0
)
( cos ω t , sin ω t , 0 )
o
a = − ω02 r
r = R
φ = ωo t
( cos ω t , sin ω t , 0 )
v = R ωo
Constant
φ
φ
r = R
;
,
x2 + y2
a ⊥ v
v = ω0 R
o
≡ v vˆ = R ωo vˆ
≡ a aˆ = − R ωo2 rˆ
,
r ⊥ v
,
v2
a = ω R =
R
,
2
0
a ↑↓ r
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General Circular Motion :
φ = φ (t ) , ω (t )
r = R
( cos φ
v = Rω
(
d φ (t )
d ω (t )
d 2 φ (t )
≡
, α (t ) ≡
=
dt
dt
dt 2
, sin φ , 0
)
= R rˆ
)
≡ v vˆ = R ω vˆ
, sin φ , 0
) + R α ( − sin φ
− sin φ , cos φ , 0
a = − R ω2
( cos φ
= a aˆ = − R ω 2 rˆ + R α vˆ
a = R
ω + α
4
2
;
aˆ =
−ω 2 rˆ + α vˆ
ω4 + α 2
, cos φ , 0
)