calculus book - Mathematics and Computer Science

1.4. CALCULUS AND THE “REAL WORLD” 25
is fundamental, and is discussed at length in Chapter 3. Because the
quantity of interest—the height of the stone at time t—is determined
by a single number, this model is said to be a “one-variable” problem.
To say anything further, it is necessary to borrow some results from
physics; mathematics says absolutely nothing about the way stones
fall, nor in general about anything other than mathematics. In our
idealized situation, the motion of the stone is governed by Newton’s
laws of motion: there are “forces” acting (gravitation and air resistance
are the most important ones), and these determine the “acceleration” of
the stone. Acceleration is a concept of differential calculus: the velocity
of the stone is its rate of change of position (in units of meters per
second), while the acceleration is the rate of change of the velocity (in
“ ‘meters per second’ per second”). In the Newtonian model, the forces
acting on the stone determine its behavior, and it is this predictive
power that answers the original questions.
It is convenient to make a couple of idealizations:
• The acceleration due to gravity is constant during the stone’s fall.
• There is no air resistance.
The first assumption is justified as follows. According to Newton’s law
of gravitation, the force acting on the stone is a certain constant G times
the mass m of the stone times the mass M of the earth, divided by the
square of the distance R(t) from the stone to the center of the earth
at time t. According to Newton’s third law of motion, the net force on
the stone is equal to the mass of the stone times its acceleration. In
symbols,
(1.2) F = −G mM
= ma;
R(t) 2
the minus sign indicates that the force is directed toward the center of
the earth. Because the distance the stone falls is very small compared
to the radius R of the earth, the ratio R(t)/R is very nearly equal to 1
throughout the stone’s fall, so the denominator in equation (1.2) may
be replaced by R 2 without much loss of accuracy. The assumption that
there is no air resistance is not realistic, but modeling the air resistance
on a solid body is horrendously complicated even if the body is perfectly
spherical (another unrealistic assumption). However, the point to be
made concerns modeling, and neglecting air resistance illustrates this