Comparison
Consider again how the velocity of the object
falling in the atmosphere compares with that of an object falling in a
vacuum. The qualitative difference will be seen to depend on the
values of k and m.
Case 1: k/m << 1
For a fixed
y, if
k/m is small, it
can be shown that eqn. (
4.1) reduces to
(
2.5) by using the
Taylor series for the
exponential function and taking the limit as
k/m ---> 0.
Thus
Similarly eqn. (
3.4)
reduces to (
1.4) since
approaches
Another way of seeing that when k/m is
small the velocity function for fall in fluid is well approximated by
that for fall in a vacuum, is to compare the differential equations
for the two cases:
Eqn. (3.2) reduces to (1.2) if the term
(k/m)vf2 (representing the drag force)
can be neglected. Certainly this is so when
(k/m)vf2 <<
g, or for fixed
vf when (
k/m) is small.
A physical example of this occurs in a common laboratory experiment to
determine the value of the gravitational acceleration, g. Click
here to view the experiment.
As you have seen, a heavy weight is dropped vertically. During the
fall, the location of the weight is measured in equal intervals of
1/60 second. The velocity vs. time data is excellently represented by
the equation for free fall in a vacuum allowing g to be
evaluated accurately. Although it may seem surprising at first that
the body falling in air may be represented by the equation for free
fall in a vacuum, the result is valid because ky/m is much less
than 1 during the fall.
Case 2: k/m >> 1
In this case, the function
v(y) is no
longer a good approximation to
vf(y) even for rather
small values of
y since the
Taylor
series argument used above breaks down when
(k/m)y is not
small. Fall in a vacuum supplies a poor model of the real motion of a
skydiver when
k/m >>1
(Comparison,
Question 2).
Copyright
1998-1999
Rensselaer Polytechnic Institute. All Rights Reserved.