Problem 4.156
[Difficulty: 3]
Given:
Data on system
Find:
Jet speed to stop cart after 1 s; plot speed & position; maximum x; time to return to origin
Solution:
Apply x momentum
Assumptions:
1) All changes wrt CV
2) Incompressible flow 3) Atmospheric pressure in jet 4) Uniform flow 5) Constant jet area
The given data is
ρ
999
kg
m
3
⋅
=
M
100 kg
⋅
=
A
0.01 m
2
⋅
=
U
0
5
m
s
⋅
=
Then
a
rf
−
M
⋅
u
1
ρ
−
VU
+
()
⋅
A
⋅
[]
⋅
u
2
m
2
⋅
+
u
3
m
3
⋅
+
=
where
a
rf
dU
dt
=
u
1
+
−
=
and
u
2
u
3
=
0
=
Hence
dU
dt
−
M
⋅
ρ
+
2
⋅
A
⋅
=
or
dU
dt
ρ
+
2
⋅
A
⋅
M
−
=
which leads to
dV U
+
+
2
ρ
A
⋅
M
dt
⋅
⎛
⎝
⎞
⎠
−
=
Integrating and using the IC
U
=
U
0
at
t
= 0
UV
−
0
+
1
ρ
A
⋅
0
+
⋅
M
t
⋅
+
+
=
To find the jet speed
V
to stop the cart after 1 s, solve the above equation for
V
, with
U
= 0 and
t
= 1 s.
(The equation becomes a
quadratic in
V
).
Instead we use
Excel
's
Goal Seek
in the associated workbook
From
Excel
V5
m
s
⋅
=
For the position
x
we need to integrate
dx
dt
U
=
V
−
0
+
1
ρ
A
⋅
0
+
⋅
M
t
⋅
+
+
=
The result is
xV
−
t
⋅
M
ρ
A
⋅
ln 1
ρ
A
⋅
0
+
⋅
M
t
⋅
+
⎡
⎢
⎣
⎤
⎥
⎦
⋅
+
=
This equation (or the one for
U
with
U
= 0) can be easily used to find the maximum value of
x
by differentiating, as well as the time for
x
to be zero again.
Instead we use
Excel
's
Goal Seek
and
Solver
in the associated workbook
From
Excel
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 Fall '07
 Lear
 Fluid Mechanics, Goal seek, Yorkshire Television

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