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

    (2.5)
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

    (4.1)
by letting y go to infinity. Denoting the terminal velocity by Vt, we see that
    (7.1)
We can normalize both the velocity and the distance to make both dimensionless and recast eqn. (4.1) in dimensionless form as
    (7.2)
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

    (3.2)
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
    (7.3)
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
    (7.4)
Solving for vf, we obtain
    (7.1)

Copyright 1998-1999 Rensselaer Polytechnic Institute. All Rights Reserved.