Page 78 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 61

which has as its solution

At t= 0, we are given that v = 0. Substituting these values into (_/), we obtain

from which we conclude that c = -16 and (_/) becomes

2
(a) From (_/) or (2), we see that as t —> °°, v —> 16 so the limiting velocity is 16 ft/sec .
(b) To find the time it takes for the ball to hit the ground (x = 3000), we need an expression for the position of the
ball at any time t. Since v = dx/dt, (2) can be rewritten as

Integrating both sides of this last equation directly with respect to t, we have

where c 1 denotes a constant of integration. At t = 0, x = 0. Substituting these values into (3), we obtain

from which we conclude that Cj = -8 and (3) becomes

The ball hits the ground when x(t) = 3000. Substituting this value into (4), we have

or
Although (5) cannot be solved explicitly for t, we can approximate the solution by trial and error, substituting
different values of t into (5) until we locate a solution to the degree of accuracy we need. Alternatively, we
note that for any large value of t, the negative exponential term will be negligible. A good approximation is
obtained by setting 2t = 376 or t = 188 sec. For this value of t, the exponential is essentially zero.

7.13. A body weighing 64 Ib is dropped from a height of 100 ft with an initial velocity of 10 ft/sec. Assume
that the air resistance is proportional to the velocity of the body. If the limiting velocity is known to be
128 ft/sec, find (a) an expression for the velocity of the body at any time t and (b) an expression for the
position of the body at any time t.
(a) Locate the coordinate system as in Fig. 7-5. Here w = 64 Ib. Since w = mg, it follows that mg = 64, or m = 2
slugs. Given that v : = 128 ft/sec, it follows from (7.6) that 128 = 641k, or k = j. Substituting these values into
(6.4), we obtain the linear differential equation

which has the solution

At t = 0, we are given that v = 10. Substituting these values into (_/), we have 10 = ce° + 128, or c = -118. The
velocity at any time t is given by