Download 4 - velocity-dependent forces

§4 - velocity-dependent forces
[HRW §6.4, Y&F §5.3, F §2.5]
• velocity-dependent forces: fluid resistance and terminal velocity
normal fluid resistance:
F(v) = - c1v - c2v | v |
where coefficients c1 and c2 depend on size, shape, surface roughness and fluid properties
(Reynolds number).
c1 ≈ 1.55  10-4 D, c2 ≈ 0.22 D2
For sphere in air:
with D diameter in m [SI]
• vertical fall through fluid:
dv
- mg - c1v = m dt
a) linear case c2 ≈ 0
v
m dv
t = 
 - mg - c v
1

integrate
m
 mg + c1v 
= c ln mg + c v 
1
1 o

vo
or
mg
mg

v = - c +  c + vo e - t/
1
 1

m
where time constant  = c
1
mg
The first term is the limiting value known as the terminal velocity vt = c = g  .
1
v = - vt (1 - e - t/)
So v increases from zero at t = 0, but never exceeds the terminal velocity. The time constant is a
measure of how quickly this velocity is approached. After 3 time constants (t = 3) the projectile
will be at 95% of its terminal speed.
b) quadratic case (c2 only)
 =
rising:
falling:
m
c2g
vt =
mg
c2
vo 
 to - t
v = vt tan 
+ tan-1 v 
t 
 
t
t
'
vo 

o
v = - vt tanh 
+ tanh-1 v 
t 
 
t
falling from rest at to' = 0: v = - vt tanh  

NOTE: in both linear and quadratic cases we have
Spherical drop falling vertically under gravity:
Given:
Derived
drop diameter
D=2r
blood density

falling height
h
gravity
g

4
 r3
3
volume
Vo
mass
m   Vo
4
  r3
3


Determine drag coefficients:
c1 
≈ 1.55  10-4 D
c2 ≈ 0.22 D2
Determine terminal speed:
v1

mg
c1
Drag-free case:

Loss of momentum:


Mean force during impact:
or

mg
c2

speed at impact
Time interval:
v2



v

2gh
 p  mv  m 2 g h
2r
2r
t 

v
2gh
Fmean

Fmean

p
mv 2

t
2r
4
  g h r2
3

mgh
r
Solid object falling vertically under gravity:
Given:
mass
m
falling height
h
gravity
g
Determine drag coefficients:
(depends on shape but probably neglibile)
Drag-free case:
speed at impact
Loss of momentum:
Time interval:


Mean force during impact:

v

2gh
 p  mv  m 2 g h
difficult to derive from theory (?)
easier to measure ???
e.g. using video capture (Ralph has a HD camera at 60 fps)
[Ian at CFS has one with 1000 fps]
p
Fmean 
t
RNJ 2-Dec-11