WRITERWRITERWRITERWRITER
§
AERE 261 Homework 6
Sep 25, 2025
C
D
,
0
=
0.017
C_{D,0}=0.017
C
D
,
0
=
0.017
K
=
0.056
K=0.056
K
=
0.056
(
C
L
/
C
D
)
m
a
x
=
16.2
(C_L/C_D)_{max}=16.2
(
C
L
/
C
D
)
ma
x
=
16.2
S
w
i
n
g
=
22.9
m
2
S_{wing}=22.9m^2
S
w
in
g
=
22.9
m
2
η
p
r
=
90
%
=
0.90
\eta_{pr}=90\%=0.90
η
p
r
=
90%
=
0.90
ρ
∞
=
1.2
k
g
/
m
3
\rho_\infty=1.2kg/m^3
ρ
∞
=
1.2
k
g
/
m
3
C
m
a
s
s
=
315
g
k
W
∗
h
∗
k
g
1000
g
∗
h
3600
s
∗
k
W
1000
W
=
8.75
∗
10
−
8
k
g
/
s
W
C_{mass}=315\frac{g}{kW*h}*\frac{kg}{1000g}*\frac{h}{3600s}*\frac{kW}{1000W}=8.75*10^{-8}\frac{kg/s}{W}
C
ma
ss
=
315
kW
∗
h
g
∗
1000
g
k
g
∗
3600
s
h
∗
1000
W
kW
=
8.75
∗
1
0
−
8
W
k
g
/
s
g
=
9.81
m
/
s
2
g=9.81m/s^2
g
=
9.81
m
/
s
2
C
=
C
m
a
s
s
∗
g
=
8.75
∗
10
−
8
k
g
/
s
W
∗
9.81
m
/
s
2
=
8.584
∗
10
−
7
N
/
s
W
=
8.584
∗
10
−
7
m
−
1
C=C_{mass}*g=8.75*10^{-8}\frac{kg/s}{W}*9.81m/s^2=8.584*10^{-7}\frac{N/s}{W}=8.584*10^{-7}m^{-1}
C
=
C
ma
ss
∗
g
=
8.75
∗
1
0
−
8
W
k
g
/
s
∗
9.81
m
/
s
2
=
8.584
∗
1
0
−
7
W
N
/
s
=
8.584
∗
1
0
−
7
m
−
1
W
0
=
16481
N
W_0=16481N
W
0
=
16481
N
W
1
=
15039
N
W_1=15039N
W
1
=
15039
N
§
1.
C
L
C
D
=
(
C
L
/
C
D
)
m
a
x
=
16.2
\frac{C_L}{C_D}=(C_L/C_D)_{max}=16.2
C
D
C
L
=
(
C
L
/
C
D
)
ma
x
=
16.2
R
=
η
p
r
C
C
L
C
D
ln
W
0
W
1
R=\frac{\eta_{pr}}{C}\frac{C_L}{C_D}\ln\frac{W_0}{W_1}
R
=
C
η
p
r
C
D
C
L
ln
W
1
W
0
=
0.90
8.584
∗
10
−
7
m
−
1
∗
16.2
∗
ln
16481
N
15039
N
=\frac{0.90}{8.584*10^{-7}m^{-1}}*16.2*\ln\frac{16481N}{15039N}
=
8.584
∗
1
0
−
7
m
−
1
0.90
∗
16.2
∗
ln
15039
N
16481
N
=
1.555
×
10
6
m
=1.555×10^6m
=
1.555
×
1
0
6
m
=
1555
k
m
=\boxed{1555km}
=
1555
km
§
2.
V
∞
,
0
=
2
W
0
ρ
∞
S
K
C
D
,
0
V_{\infty,0}=\sqrt{\frac{2W_0}{\rho_\infty S}\sqrt{\frac{K}{C_{D,0}}}}
V
∞
,
0
=
ρ
∞
S
2
W
0
C
D
,
0
K
=
2
∗
16481
N
1.2
k
g
/
m
3
∗
22.9
m
2
0.056
0.017
=\sqrt{\frac{2*16481N}{1.2kg/m^3*22.9m^2}\sqrt{\frac{0.056}{0.017}}}
=
1.2
k
g
/
m
3
∗
22.9
m
2
2
∗
16481
N
0.017
0.056
=
46.7
m
/
s
=\boxed{46.7m/s}
=
46.7
m
/
s
V
∞
,
1
=
2
W
1
ρ
∞
S
K
C
D
,
0
V_{\infty,1}=\sqrt{\frac{2W_1}{\rho_\infty S}\sqrt{\frac{K}{C_{D,0}}}}
V
∞
,
1
=
ρ
∞
S
2
W
1
C
D
,
0
K
=
2
∗
15039
N
1.2
k
g
/
m
3
∗
22.9
m
2
0.056
0.017
=\sqrt{\frac{2*15039N}{1.2kg/m^3*22.9m^2}\sqrt{\frac{0.056}{0.017}}}
=
1.2
k
g
/
m
3
∗
22.9
m
2
2
∗
15039
N
0.017
0.056
=
44.6
m
/
s
=\boxed{44.6m/s}
=
44.6
m
/
s
§
3.
P
R
=
1
2
ρ
∞
V
∞
3
S
C
D
,
0
+
2
K
W
2
ρ
∞
V
∞
S
P_R=\frac{1}{2}\rho_\infty V_\infty^3SC_{D,0}+\frac{2KW^2}{\rho_\infty V_\infty S}
P
R
=
2
1
ρ
∞
V
∞
3
S
C
D
,
0
+
ρ
∞
V
∞
S
2
K
W
2
P
R
=
1
2
∗
1.2
k
g
/
m
3
∗
(
46.7
m
/
s
)
3
∗
22.9
m
2
∗
0.017
+
2
∗
0.056
∗
(
16481
N
)
2
1.2
k
g
/
m
3
∗
46.7
m
/
s
∗
22.9
m
2
P_R=\frac{1}{2}*1.2kg/m^3 * (46.7m/s)^3 * 22.9m^2 * 0.017 + \frac{2 * 0.056 * (16481N)^2}{1.2kg/m^3 * 46.7m/s * 22.9m^2}
P
R
=
2
1
∗
1.2
k
g
/
m
3
∗
(
46.7
m
/
s
)
3
∗
22.9
m
2
∗
0.017
+
1.2
k
g
/
m
3
∗
46.7
m
/
s
∗
22.9
m
2
2
∗
0.056
∗
(
16481
N
)
2
P
R
=
47500
W
=
47.5
k
W
P_R=47500W=\boxed{47.5kW}
P
R
=
47500
W
=
47.5
kW
§
4.
(
C
L
3
/
2
C
D
)
m
a
x
=
1
4
(
3
K
C
D
,
0
1
/
3
)
3
/
4
\left(\frac{C_L^{3/2}}{C_D}\right)_{max}=\frac{1}{4}\left(\frac{3}{KC_{D,0}^{1/3}}\right)^{3/4}
(
C
D
C
L
3/2
)
ma
x
=
4
1
(
K
C
D
,
0
1/3
3
)
3/4
=
1
4
(
3
0.056
∗
(
0.017
)
1
/
3
)
3
/
4
=\frac{1}{4}\left(\frac{3}{0.056 * (0.017)^{1/3}}\right)^{3/4}
=
4
1
(
0.056
∗
(
0.017
)
1/3
3
)
3/4
=
13.71
=\boxed{13.71}
=
13.71
§
5.
E
=
η
p
r
2
ρ
∞
S
C
C
L
3
/
2
C
D
(
1
W
1
−
1
W
0
)
E=\frac{\eta_{pr}\sqrt{2\rho_\infty S}}{C}\frac{C_L^{3/2}}{C_D}\left(\frac{1}{\sqrt{W_1}}-\frac{1}{\sqrt{W_0}}\right)
E
=
C
η
p
r
2
ρ
∞
S
C
D
C
L
3/2
(
W
1
1
−
W
0
1
)
E
=
0.90
∗
2
∗
1.2
k
g
/
m
3
∗
22.9
m
2
8.584
∗
10
−
7
m
−
1
∗
13.71
∗
(
1
15039
N
−
1
16481
N
)
E=\frac{0.90 * \sqrt{2 * 1.2kg/m^3 * 22.9m^2}}{8.584*10^{-7}m^{-1}} * 13.71 * \left(\frac{1}{\sqrt{15039N}}-\frac{1}{\sqrt{16481N}}\right)
E
=
8.584
∗
1
0
−
7
m
−
1
0.90
∗
2
∗
1.2
k
g
/
m
3
∗
22.9
m
2
∗
13.71
∗
(
15039
N
1
−
16481
N
1
)
=
38890
s
=38890s
=
38890
s
=
10.8
h
=\boxed{10.8h}
=
10.8
h
§
6.
V
∞
,
0
=
2
W
0
ρ
∞
S
K
3
C
D
,
0
V_{\infty,0}=\sqrt{\frac{2W_0}{\rho_\infty S}\sqrt{\frac{K}{3C_{D,0}}}}
V
∞
,
0
=
ρ
∞
S
2
W
0
3
C
D
,
0
K
=
2
∗
16481
N
1.2
k
g
/
m
3
∗
22.9
m
2
0.056
3
∗
0.017
=\sqrt{\frac{2 * 16481N}{1.2kg/m^3 * 22.9m^2}\sqrt{\frac{0.056}{3 * 0.017}}}
=
1.2
k
g
/
m
3
∗
22.9
m
2
2
∗
16481
N
3
∗
0.017
0.056
=
35.5
m
/
s
=\boxed{35.5m/s}
=
35.5
m
/
s
V
∞
,
1
=
2
W
1
ρ
∞
S
K
3
C
D
,
0
V_{\infty,1}=\sqrt{\frac{2W_1}{\rho_\infty S}\sqrt{\frac{K}{3C_{D,0}}}}
V
∞
,
1
=
ρ
∞
S
2
W
1
3
C
D
,
0
K
=
2
∗
15039
N
1.2
k
g
/
m
3
∗
22.9
m
2
0.056
3
∗
0.017
=\sqrt{\frac{2 * 15039N}{1.2kg/m^3 * 22.9m^2}\sqrt{\frac{0.056}{3 * 0.017}}}
=
1.2
k
g
/
m
3
∗
22.9
m
2
2
∗
15039
N
3
∗
0.017
0.056
=
33.9
m
/
s
=\boxed{33.9m/s}
=
33.9
m
/
s
§
7.
P
R
=
1
2
ρ
∞
V
∞
3
S
C
D
,
0
+
2
K
W
2
ρ
∞
V
∞
S
P_R=\frac{1}{2}\rho_\infty V_\infty^3SC_{D,0}+\frac{2KW^2}{\rho_\infty V_\infty S}
P
R
=
2
1
ρ
∞
V
∞
3
S
C
D
,
0
+
ρ
∞
V
∞
S
2
K
W
2
P
R
=
1
2
∗
1.2
k
g
/
m
3
∗
(
35.5
m
/
s
)
3
∗
22.9
m
2
∗
0.017
+
2
∗
0.056
∗
(
16481
N
)
2
1.2
k
g
/
m
3
∗
35.5
m
/
s
∗
22.9
m
2
P_R=\frac{1}{2}*1.2kg/m^3 * (35.5m/s)^3 * 22.9m^2 * 0.017 + \frac{2 * 0.056 * (16481N)^2}{1.2kg/m^3 * 35.5m/s * 22.9m^2}
P
R
=
2
1
∗
1.2
k
g
/
m
3
∗
(
35.5
m
/
s
)
3
∗
22.9
m
2
∗
0.017
+
1.2
k
g
/
m
3
∗
35.5
m
/
s
∗
22.9
m
2
2
∗
0.056
∗
(
16481
N
)
2
P
R
=
41600
W
=
41.6
k
W
P_R=41600W=\boxed{41.6kW}
P
R
=
41600
W
=
41.6
kW