Problem 4.176
[Difficulty: 4]
y
x
CS
at speed
V
V
e
Y
X
Given:
Data on rocket
Find:
Maximum speed and height; Plot of speed and distance versus time
Solution:
Basic equation: Momentum flux in y direction
Assumptions:
1) No resistance
2) p
e
= p
atm
3) Uniform flow 4) Use relative velocities 5) Constant mass flow rate
From continuity
dM
dt
m
rate
=
constant
=
so
MM
0
m
rate
t
⋅
−
=
(Note: Software cannot render a dot!)
Hence from momentum
M
−
g
⋅
a
rfy
M
⋅
−
u
e
ρ
e
V
e
⋅
A
e
⋅
()
⋅
=
V
e
−
m
rate
⋅
=
Hence
a
rfy
dV
dt
=
V
e
m
rate
⋅
M
g
−
=
V
e
m
rate
⋅
M
0
m
rate
t
⋅
−
g
−
=
Separating variables
dV
V
e
m
rate
⋅
M
0
m
rate
t
⋅
−
g
−
⎛
⎜
⎝
⎞
⎠
dt
⋅
=
Integrating from V = at t = 0 to V = V at t = t
VV
e
−
ln M
0
m
rate
t
⋅
−
ln M
0
−
⋅
gt
⋅
−
=
V
e
−
ln 1
m
rate
t
⋅
M
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⋅
−
=
e
−
ln 1
m
rate
t
⋅
M
0
−
⎛
⎜
⎝
⎞
⎠
⋅
⋅
−
=
for
tt
b
≤
(burn time)
(1)
To evaluate at t
b
= 1.7 s, we need V
e
and m
rate
m
rate
m
f
t
b
=
m
rate
12.5 gm
⋅
1.7 s
⋅
=
m
rate
7.35
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 Fall '07
 Lear
 Fluid Mechanics, Flux, Trigraph, Mass flow rate, momentum flux, Ve⋅ mrate, mrate

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