
4.10. Given the fluid drag equation F
D
= C
D
V
2
, where C
D
is a constant (with dimensions) and
f
V
(v) = 0.1, for the range 10 ≤ v ≤ 20, derive f
F
D
and sketch both density functions.
Solution: For this case there are two roots
v = ±
F
D
/C
D
, and
dv
dF
D
= ±
1
2
√
F
D
C
D
.
The general transformation is given by
f
F
D
(F
D
) =
1
2
√
C
D
F
D
1
f
V
3
F
D
C
D
+ f
V
−
3
F
D
C
D
2
u (F
D
) ,
where u(·) is the unit step function. However, since v has a positive range we must drop the negative root.
Therefore,
f
F
D
(F
D
) =
1
2
√
C
D
F
D
1
10
u (F
D
)
=
1
20 ...