1550078481-Ordinary_Differential_Equations__Roberts_

Applications of the Initial Value Problem y' = f(x, y); y(c) = d 143

If we let m denote the mass of a body and v denote its velocity, then the

momentum of the body is mv. Hence, Newton's second law of motion may

be expressed mathematically as

(1) d(mv) = kF

dt

where k is a constant of proportionality and F is the magnitude of the force

acting on the body. If we make the assumption that the mass of the body is

a constant and that the units of mass, velocity, and force are chosen so that

k = 1, then Newton's second law becomes

(2) m dv = F.

dt

Consider an origin located somewhere above the earth's surface. Let the

positive axis of a one-dimensional coordinate system extend from the origin

through the gravitational center of the earth. Suppose that at time t = 0 a

body with mass mis initially located at Yo and is traveling with a velocity Vo.

See Figure 3.12. Assuming that the body is falling freely in a vacuum and that
it is close enough to the earth's surface so that the only significant force acting
on the body is the earth's gravitational attraction, then F = mg, where g is
the gravitational attraction- approximately 32 ft/sec^2 in the English system

of measurement and 9.8 m/sec^2 in the metric system. Substituting F =mg

in equation (2) and then dividing by m "I-0, we find the velo city of the body

satisfies the initial value problem dv/dt = g; v(O) = v 0. Integrating and

satisfying the initial condition, we easily find the solution of this initial value

problem to be v(t) = gt+vo. Notice that the velocity increases with time and

as t --) oo, v(t) --) oo.

0

At t= 0

y=yo

v=vo t

y

m

Gravitational center

of the earth

Figure 3.12 A Falling Body