ME3834 ‐ Fluid Mechanics Problem 9.111 Given: Data on a tennis ball: Mass of the ball M= 57 g Diameter of the ball D= 64 mm Find: Terminal speed time and distance to reach 95% of terminal speed
Solution: The given data or available data is
M=57gm
1.45
Then
A
D=64 mm
10
1.23
A
3.22
10
m
Assuming high Reynolds number CD = 0.5
(from Fig. 9.11)
At terminal speed drag equals weight FD = M g
ρAV C
F
The drag at speed V is given by
.
Hence the terminal speed is V
Check the Reynolds number Re
.
V
Re
.
23.8
1.05
1
ρAV C
2
V
1 10
ME3834 ‐ Fluid Mechanics Problem 9.111 k= ρ AC
Where
k = 0.0174
dV
Separating variables
t
1
dV
g kV
Hence
V t
atanh
gk
atanh
t = 4.44s
For distance x versus time, integrate
tanh
t
0
g
tanh
k
tanh a. t dt
Hence
ln cosh
So,
g. k. t
g. k. t dt
Note that
Evaluating at V = 0.95Vt
atanh
g. k. t
0.95V
Evaluating at V = 0.95Vt
1
ln cos a. t
g. k. t
t = 4.44 s
x(t) = 67.1m
2 g. k. t
k
V
g