terminal velocity
We have examined
quantitative differences
between the fall of an object in the atmosphere and its fall in a
vacuum. In particular, a skydiver falls more slowly in air because
drag forces always oppose the motion of an object passing through a
fluid.
Drag forces represent the interaction of the medium with the surface
of the moving object. Since a vacuum is devoid of matter, it has no
such interactions and consequently exerts no drag forces on the moving
object.
There is also, however, a qualitative difference between
falling in the atmosphere and falling in a vacuum. In a vacuum an
object, having fallen a distance y, has velocity
The velocity increases indefinitely as y
increases:
By contrast, for a skydiver falling in the
atmosphere, the velocity approaches a constant or terminal value as
y gets large.
The terminal velocity is easily calculated from
by letting y go to infinity. Denoting the terminal velocity by Vt, we see that
We can normalize both the velocity and the
distance to make both dimensionless and recast eqn. (4.1) in
dimensionless form as
vf/Vt is the
dimensionless velocity function and
ky/m the dimensionless
distance.
The acceleration of the skydiver was given by equation (3.2) as
The acceleration here is not constant, as opposed
to the case of falling in a vacuum where a = g. Combining eqns
(3.2) and (7.1) we see that the acceleration is zero in terminal
fall. Thus
It is not necessary, therefore, to solve the
differential equation (3.2) to determine terminal velocity, since
af equals zero in terminal fall. Thus setting
maf = 0, we have
Solving for vf, we obtain
Copyright
1998-1999
Rensselaer Polytechnic Institute. All Rights Reserved.