Page 68 - Schaum's Outline of Differential Equations

CHAP. 7] APPLICATIONS OF FIRST-ORDER DIFFERENTIAL EQUATIONS 51

FALLING BODY PROBLEMS
Consider a vertically falling body of mass m that is being influenced only by gravity g and an air resistance
that is proportional to the velocity of the body. Assume that both gravity and mass remain constant and, for
convenience, choose the downward direction as the positive direction.

Newton's second law of motion: The net force acting on a body is equal to the time rate of change of the
momentum of the body; or, for constant mass,

where F is the net force on the body and v is the velocity of the body, both at time t.

For the problem at hand, there are two forces acting on the body: (1) the force due to gravity given by the
weight w of the body, which equals mg, and (2) the force due to air resistance given by —kv, where k > 0 is a
constant of proportionality. The minus sign is required because this force opposes the velocity; that is, it acts in
the upward, or negative, direction (see Fig. 7-1). The net force F on the body is, therefore, F = mg-kv.
Substituting this result into (7.3), we obtain

or

as the equation of motion for the body.
If air resistance is negligible or nonexistent, then k = 0 and (7.4) simplifies to

Fig. 7-1 Fig. 7-2