Biomedical Engineering Reference
In-Depth Information
Solution
The velocity of an element is calculated as the time derivative of the position vector.
2
s
2
1
t
2
i
-
d
-
ð
j
-
t
Þ
d
dt
5
d
dt
-
ð
3ms
2
1
2ms
2
1
t
Þ 5
5
ð
Þ
t
1
1
ð
Þ
t
dt
i
-
j
-
3ms
2
1
5s
2
1
2ms
2
1
5 ð
Þð
1
1 ð
Þ
t
Þ
1 ð
Þ
Acceleration is defined as the time rate of change of the velocity vector.
d
-
ð
i
-
j
-
i
-
t
Þ
d
dt
d
dt
-
ð
3ms
2
1
5s
2
1
2ms
2
1
3ms
2
1
5s
2
1
t
Þ 5
5
ð
Þð
1
1 ð
Þ
t
Þ
1
ð
Þ
5 ð
Þðð
ÞÞ
dt
i
-
15ms
2
2
5
TABLE 2.2
Time
(sec)
X-position
(m)
Y-position
(m)
X-velocity
(m/s)
Y-velocity
(m/s)
Velocity Magnitude
and Direction
X-acceleration
(m/s
2
)
3.61 m/s @ 33.7
0
0
0
3
2
15
18.11 m/s @ 6.34
1
10.5
2
18
2
15
33.06 m/s @ 3.47
2
36
4
33
2
15
63.03 m/s @ 1.82
4
132
8
63
2
15
FIGURE 2.16
10
The fluid
particles position as a function
of time.
The arrows represent
the magnitude of velocity at
each time. Please note that the
x and y axes are not on the
same scale.
8
Position at t = 4 s
6
4
Position at t = 2 s
2
Position at t = 1 s
0
Position at t = 0 s
0
20
40
60
80
100
120
140
X-Position (m)
The formulations for velocity and acceleration are based on the Lagrangian assumption
that you can define discrete particles within the fluid. It is a requirement that each particle
has a known position as a function of time; therefore, velocity and acceleration can be
defined from this position vector (see previous example). However, in fluids mechanics, it
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