Why does a constant
force
on an object in a viscous medium produce a
constant (terminal)
velocity
?
You know that the viscous drag force
F
drag
is proportional to the velocity
v
(and in the
opposite direction); we’ll use the symbol
f
to represent the coefficient of proportionality:
F
drag
= –
fv
When an object reaches terminal velocity, the drag force is equal and opposite to the
constant external force, so the velocity is constant:
F
ext
= –
F
drag
so
v
=
F
ext
f
We seek a
microscopic model for viscosity
that can explain this observation. As the
object is being pulled through the fluid by a constant force, the object is also subjected to
numerous microscopic collisions with the molecules of the fluid. Let’s assume that these
collisions are periodic: there is one collision per time
!
t
. Let’s also assume that as a
result of each collision, our object’s velocity is “reset” to some random value
v
o
. Then,
before the next collision, the object moves freely, influenced only by the constant
external force. The constant force produces a constant acceleration
a
=
F
ext
/
m
, so the
object’s velocity is given by the usual kinematic equation for constant acceleration:
v
=
v
o
+
F
ext
m
"
#
$
%
&
’
(
t
For a constant “downward” force (for instance, a gravitational force) the velocity of such
a particle might look like this:
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View Full DocumentTake a close look at this graph. Each black circle shows an initial (random) velocity that
follows a collision. On average, those velocities are zero (i.e. the random velocity is
equally likely to be positive or negative). However, during the period between collisions,
the object is subjected to a constant downward acceleration, so its velocity decreases
linearly between each collision in a constant, predictable way. Thus, the average of
all
the velocities on the graph is a
negative
number: on average, the object is falling “down”
with a
constant
velocity.
If we integrate the velocity we get the
position
as a function of time. The kinematic
expression is:
x
=
x
o
+
v
o
"
t
+
1
2
F
ext
m
#
$
%
&
’
(
"
t
( )
2
and a graph of position versus time (using the velocities shown on the previous graph) is:
Each dot on the above graph shows one collision. Can you see that the object is in
parabolic freefall between each collision? The dashed line represents the average
downward trajectory of the object. We see that, although the
instantaneous
motion of the
object is a constantacceleration trajectory (as expected for a constant force), the
average
motion is a constantvelocity trajectory (as observed in a viscous medium).
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 Fall '10
 LOGANMCCARTY

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