Aula_04-Movimento_em_Duas_e_Tres_Dimensoes

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Prof. Anderson Coser Gaudio 
TecnoLab – Depto. Física – CCE - UFES 
http://profanderson.net 
anderson.gaudio@ufes.br 
   1 ˆ ˆ3 m 4 mr i j     2 ˆ ˆ5 m 3 mr i j 
x y r r r x yr r r i j
x y z  r r r r
x y z  r r r r
x y zr r r  r i j k
ˆˆ ˆr xi yj zk  
 ©2004 by Pearson Education 
A AB Br r r  
AB B Ar r r  
,
AB B A
m AB
AB B A
r r r
v
t t t
 
 
 m
r
v
t



m
r
v
t



 ©2004 by Pearson Education 
 
0
ˆˆ ˆlim t t t
t
r r dr d
v xi yj zk
t dt dt

 

    

dr
dt ( )r t
0
lim t t t
t
r r dr
v
t dt

 

 

 ©2002 by John Wiley & Sons 
tan
y
x
v
v
 
2 2 2
x y zv v v v v   
 ©2004 by Pearson Education 
,
AB B A
m AB
AB B A
v v v
a
t t t
 
 
 
m
v
a
t



2
20
lim t t t
t
v v dv d r
a
t dt dt

 

  

dv
dt
2
2
d r
dt ( )v t( )r t
0
lim t t t
t
v v dv
a
t dt

 

 

 ©2002 by John Wiley & Sons 
( ) ( )v t f t
0
lim
t
v dv
a
t dt 

 

m
v
a
t


 2 1v v v  
 ©2004 by Pearson Education 
aa
 ©2004 by Pearson Education 
 ©2004 by Pearson Education 
 ©2004 by Pearson Education 
 ©2004 by Pearson Education 
ˆa g j 
 ©2004 by Pearson Education 
 ©2004 by Pearson Education 
 0 0cosx v t
  20 2 2
0 0
tan
2 cos
g
y x x
v
  
  20 0
1
sen
2
y v t gt 
2
0 0
1
2
yy y v t gt  
0 xx x v t 
 ©2004 by Pearson Education 
 ©2004 by Pearson Education 
 0 0cosR v t
2
0
0sen 2
v
R
g

  20 0
1
0 sen
2
v t gt 
0 0xx x v t 
0
0
0 0 0
0 0 0
0
0
0
cos
sen
x
y
x R
x
y
y
v v
v v








2
0 0
1
2
yy y v t gt  
   1 2R R 
2 2
0 0
1 2sen 2 sen 2
v v
g g
 
1 2sen 2 sen 2 
1 22 2   
o1 2 45
2 4
  
 
o1 2 45
2
 

 ©2002 by John Wiley & Sons 
 ©2004 by Pearson Education 
0 0A Pr v t
2
0 0
1
2
P P Pr r v t gt   20 0
1
2
A A Ar r v t gt  
2
0
1
0
2
P Pr v t gt   20
1
0
2
A Ar r gt  
P Ar r
P Ar r
 ©2004 by Pearson Education 
 ©2004 by Pearson Education 
r
t
v

  sen
2 2
v
v


22 sen sen
2 2
2
m
v
v v
a
rt r
v
 
 

  

2
0 0 0
sen
2lim lim lim
2
m
t t
v v
a a
t r 

    

  

2v
a
r

ˆ ˆˆ cos senru i j  
ˆv vu
ˆ ˆˆ sen cosu i j    ˆ ˆdvu dudv
a v
dt dt dt
 
  
ˆ sen cosˆ ˆ
ˆ ˆ cos sen
ˆ r
du d d
i j
dt dt dt
d d
i j
dt dt
d
u
dt
  
 
 

  
  
 
ˆ
r
d
a v u
dt

 
2
2
d v
rdt t r
v
  


  

2
ˆ
r
v
a u
r
 
ˆ ˆcos senru i j  
ˆv vu
ˆ ˆsen cosu i j     ˆ ˆ
ˆ
dvu dudv dv
a v u
dt dt dt dt
 
   
ˆ ˆ
r r ta a u a u  
sen cosˆ ˆ
ˆ ˆ cos sen
 r
du d d
i j
dt dt dt
d d
i j
dt dt
d
u
dt
  
 
 

  
  
 
uˆ
PA PB BAx x x 
 ©2002 by John Wiley & Sons 
PA BA PBr r r 
 ©2004 by Pearson Education