4.1 Position and Displacement

filercaliforniaMechanics

Nov 14, 2013 (3 years and 6 months ago)

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Ch. 4

MOTION IN TWO AND THREE DIMENSIONS

4.1 Position and Displacement


4.1.1

Position vector

k
t
z
j
t
y
i
t
x
t
r
r
ˆ
)
(
ˆ
)
(
ˆ
)
(
)
(






o
P
r

z
y
x









)
(
)
(
)
(
t
z
z
t
y
y
t
x
x





0
)
,
,
(
0
)
,
,
(
2
1
z
y
x
f
z
y
x
f
k
t
z
j
t
y
i
t
x
t
r
r
ˆ
)
(
ˆ
)
(
ˆ
)
(
)
(






2
2
2
z
y
x
r



r
z
r
y
r
x






cos
,
cos
,
cos



o
P
r

z
y
x

4.1.2

Displacement vector

)
(
t
r

)
(
t
t
r




r
O
y
z
x
)
(
)
(
)
(
)
(
1
2
t
r
t
r
t
r
t
t
r
r












Note that:

s
r




but

ds
r
d


4.2 Velocity and Acceleration


4.2.1

Average and instantaneous velocities

t
r
V





dt
r
d
t
r
v
t









0
lim
v
r
O
y
z
x

ˆ
v
v


t
r
v
v
t








0
lim
dt
ds
t
s
t






0
lim
In Cartesian coordinates, a velocity is in a form of

k
z
j
y
i
x
k
dt
dz
j
dt
dy
i
dt
dx
dt
r
d
v
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ












2
2
2
z
y
x
v






v
v
v
v
v
v
z
y
x






cos
,
cos
,
cos
4.2.2

Acceleration

Average acceleration:

t
v
v
t
v
a






1
2




r
v
dt
v
d
t
v
a
t
















0
lim
Instantaneous acceleration:

In rectangular coordinates:

k
z
j
y
i
x
a
ˆ
ˆ
ˆ










4.3 Motion in a Plane


4.3.1

Motion in a plane and the principle of superposition


In a two
-
dimension case, we have the component form of
kinematics variables as:

)
(
)
(
)
(
x
f
y
t
y
y
t
x
x















dt
dy
v
dt
dx
v
y
x









dt
dv
a
dt
dv
a
y
y
x
x











t
t
y
yo
y
t
t
x
xo
x
o
o
dt
t
a
v
v
dt
t
a
v
v
)
(
)
(











t
t
y
o
t
t
x
o
o
o
dt
t
v
y
y
dt
t
v
x
x
)
(
)
(

If and are independent of each other, the
x
-
direction
motion and the
y
-
direction motion are independent (see figure 4
-
11
on page 56).

x
a
y
a
4.3.2

Projectile motion analyzed


A particle is thrown in a vertical plane from a position, ,
with an initial velocity, .

o
r

o
v


o
v

y
x

j
g
i
j
g
a
ˆ
ˆ
0
ˆ











g
y
x




0







1
1
D
gt
y
C
x


)
ˆ
sin
ˆ
(cos
j
i
v
v
o
o














sin
cos
o
o
v
gt
y
v
x


j
y
i
x
r
o
o
o
ˆ
ˆ















2
2
2
2
1
sin
cos
D
gt
t
v
y
C
t
v
x
o
o














o
o
o
o
y
gt
t
v
y
x
t
v
x
2
2
1
sin
cos


Take
x
o

= 0
and
y
o

= 0

2
2
2
cos
2
tan
x
v
g
x
y
o












This is a parabola equation.

4.4 Acceleration in Circular Motion


4.4.1

Natural coordinates


Take an “arc length” as a coordinate
s


)
(
s
r
r



n
ˆ

ˆ
s
s
= 0


ˆ
: tangent unit vector

n
ˆ
: normal unit vector

n
A
A
A
n
ˆ
ˆ





4.4.2

Acceleration in uniform circular motion

i
f
v
v
v








f
v

i
v

B
A
B


v




f
v

i
v

A
t
v
a
t






0
lim
t
R
v
t





AB
lim
0
t
R
v
t




AB
lim
0
R
v
t
s
R
v
t
2
0
lim






n
R
v
a
ˆ
2


4.4.3

Acceleration of circular motion



f
v

i
v

B
A
2
1
v
v
v








2
v


'
f
v

f
v

i
v

1
v


'
f
v

2
v


1
v


v


f
v

i
v

i
f
v
v
v






'
1
'
2
f
f
v
v
v






2
1
2
1
0
lim
a
a
t
v
v
a
t














n
R
v
a
ˆ
2
1



ˆ
2
dt
dv
a


n
R
v
dt
dv
a
ˆ
ˆ
2




Tangent acceleration

Normal acceleration

4.4.4

General curvature motion

By introducing a curvature radius, ,


n
v
dt
dv
a
ˆ
ˆ
2





'
'
)
'
1
(
2
3
2
y
y



4.5 Polar Coordinates


4.5.1

Polar coordinates


ˆ
r
ˆ

o
P(
r
,

)
r







)
(
)
(
t
t
r
r


)
(

r
r


Trajectory equation:

r
r
r
ˆ


Radial unit vector:

r
ˆ
Transverse unit vector:


ˆ


ˆ
ˆ
A
r
A
A
r



4.5.2

Velocity

)
(
ˆ
)
(
t
r
t
r
r


dt
r
d
r
r
dt
dr
v
ˆ
ˆ



r


2
ˆ
r
2
ˆ

1
ˆ

1
ˆ
r
o

d

r
ˆ

2
ˆ
r
1
ˆ
r

1
2
ˆ
ˆ
ˆ
r
r
r
d




ˆ
ˆ
dt
d
dt
r
d



ˆ
ˆ
dt
d
r
r
dt
dr
v



2
2
)
(



r
r
v


4.5.3

Acceleration









ˆ
)
2
(
ˆ
)
(
ˆ
ˆ
2











r
r
r
r
r
r
r
r
dt
d
dt
v
d
a










ˆ

r
ˆ

2
ˆ
r
2
ˆ

1
ˆ

1
ˆ
r

4.6 Relative Motion


BA
v

PB
r

PA
r

BA
r

z'
y'
x'
z
y
x

BA
PB
PA
r
r
r





BA
PB
PA
v
v
v





Because

C
v
BA



we get

PB
PA
a
a



But if

C
v
BA



then

BA
PB
PA
a
a
a





Problems:

1.
4
-
9 (on page 67) ,

2.
4
-
11,

3.
4
-
26,

4.
The gun of a battery is emplaced on the foot of a hill with a
slope of . It fires a shot with a muzzle velocity of . What
is the appropriate elevation angle, , with respect to the
horizontal, of the shot to hit a target at a distance of
S
along
the slope? Using a rectangular coordinates with
-
axis along
the slope.

5.
4
-
52


o
v

6.
A train is running along a circular
-
railway from east to north
(figure 1). In the considered time region its kinematics
equation is (length: m, time: s). At , the train
is in the point of
O
, where m. Find out the speed and
the acceleration after the train moving a distance of 1200 m.

7.
A particle moves in a plane. Its speed is a constant,
C
, and its
angular speed of radius is a constant, , as well. The initial
condition is at , . What are its
kinematics equation and the trajectory?

8.
4
-
63

2
80
t
t
s


0

t
1500

r

0

t
0

and

0


o
o
r

9.
A ferryboat on a river has a speed relative to the water. The
water of the river flows with a speed relative to the ground.
The width of the river is
d
. (i) Show that the ferryboat takes a
time






to travel across the river and back. (ii) Show that the ferryboat
takes a time






to travel a distance
d
up the river and back. Which trip takes a
shorter time?


2
2
2
V
v
d

2
2
2
V
v
dv