Page 79 - Schaum's Outline of Differential Equations

62 APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 7

(b) Since v = dxldt, where x is displacement, (2) can be rewritten as

This last equation, in differential form, is separable; its solution is

At t = 0, we have x = 0 (see Fig. 7-5). Thus, (3) gives

The displacement at any time t is then given by

7.14. A body of mass m is thrown vertically into the air with an initial velocity v 0. If the body encounters an
air resistance proportional to its velocity, find (a) the equation of motion in the coordinate system of
Fig. 7-6, (b) an expression for the velocity of the body at any time t, and (c) the time at which the body
reaches its maximum height.

Fig. 7-6

(a) In this coordinate system, Eq. (7.4) may not be the equation of motion. To derive the appropriate equation,
we note that there are two forces on the body: (1) the force due to the gravity given by mg and (2) the force
due to air resistance given by kv, which will impede the velocity of the body. Since both of these forces act in
the downward or negative direction, the net force on the body is —mg —kv. Using (7.3) and rearranging,
we obtain

as the equation of motion.
(b) Equation (T) is a linear differential equation, and its solution is v = ce ( ''-mg/k. At t = 0, v = v 0; hence
( /m)0
v 0 = ce * - (mg/k), or c = v 0 + (mg/k). The velocity of the body at any time t is