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

  ;    y fixed   (6.1)
Similarly eqn. (3.4) reduces to (1.4) since
approaches
as the argument of the tanh function goes to zero, (Comparison, Question 1). Thus

           y fixed   (6.2)

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:
  (vacuum)   (1.2)
  (fluid)   (3.2)

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).

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