Page 80 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 63

(c) The body reaches its maximum height when v = 0. Thus, we require t when v = 0. Substituting v = 0 into (2)
and solving for t, we find

7.15. A body of mass 2 slugs is dropped with no initial velocity and encounters an air resistance that is
proportional to the square of its velocity. Find an expression for the velocity of the body at any time t.
2
The force due to air resistance is -kv ; so that Newton's second law of motion becomes

Rewriting this equation in differential form, we have

which is separable. By partial fractions,

Hence (_/) can be rewritten as

This last equation has as its solution

or

which can be rewritten as

or

f
At = 0, we are given that v = 0. This implies c l = 1, and the velocity is given by

Note that without additional information, we cannot obtain a numerical value for the constant k.