y = 16t 2
The Calculus
A Simple Gift from Galileo
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The way these rules work, and the reason for their enormous usefulness, can
best be illustrated by applying them to the classically simple equation y=16t2,
devised by the renowned Italian astronomer and physicist, Galileo Galilei.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
This brief, unpretentious expression is one of the most versatile in all physics
because it shows how gravity acts on a freely falling object—an elevator run
amuck, a hailstone or a jumper descending to ground.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Since almost all movements and changes on earth are heavily influenced by
gravity, the equation of free fall indirectly plays a part in innumerable human
actions—from taking a step or lobbing a tennis ball to lifting a steel girder or
launching an astronaut into orbit.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Timing an object as it falls from a given height is the most straightforward
method of gauging the effects of gravity.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
It was this technique which Galileo used, about 1585, to arrive at his free-fall
equation. According to legend, Galileo dropped small cannon balls from the
colonnades of the leaning tower of Pisa. According to his own account, he used
the less fanciful means of timing cannon balls as they rolled down a ramp.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
In any event, Galileo ascertained that the equation for free fall was y = 16t2,
with y representing the distance fallen in feet and t the elapsed time in seconds
after the start of the fall.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
y = 16t 2
By differentiating this equation twice—so as to shave away successive layers of
change and inconsistency—Newton uncovered the essential nature of
gravitation.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
y = 16t 2
d
2
16
t
(
)
dt
y ' = 32t
Differentiating the equation once, he found that the speed with which a jumper is
falling at any moment equals 32 times the number of seconds which he is falling.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
y = 16t 2
d
2
16
t
(
)
dt
y ' = 32t
d
( 32t )
dt
y " = 32
Differentiating the equation a second time, he found that the jumper’s
acceleration—the rate of increase in his speed—is always 32 feet second,
every second.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
g = 32
ft
s2
The fact that in the free-fall equation acceleration equals a constant number,
32, indicates that end of the trail. This 32 need not be differentiated further; it
does not change, and its rate of change is zero. It represents a law of nature:
that every free-falling object falls to earth with a constant acceleration of 32 feet
per second, every second.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Having ascertained this fact by calculus, Newton was able to set his
mathematical sights far beyond the earth and to deduce the law of universal
gravitation—one of the most important results ever to be achieved by
mathematics.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
It is the law which governs the movements of all celestial bodies—from human
beings in orbit to entire systems of stars.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Looking back with awe on what a little deduction could accomplish in the mind
of Isaac Newton, later thinkers have ranked him as the greatest physicists and
one of the greatest mathematicians the world has ever known.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Albert Einstein wrote: “Nature to him was an open book, whose letters he could
read without effort.”
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Newton himself said: “I do not know what I may appear to the work; but to
myself I seem to have been only like a boy playing on the seashore, and
diverting myself in now and then finding a smoother pebble or a prettier shell
than ordinary, whilst the great ocean of truth lay all undiscovered before me.”
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Newton began to use his astounding inventiveness while still a child, to build
toys for himself, including a wooden water clock that actually kept time and a
flour mill worked by a mouse.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
His brilliance did not really catch fire, however, until he read Euclid at the
age of 19. The story goes that he rushed impatiently on to Descartes’
relatively abstruse La Géométrie. Thereafter his progress was meteoric.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Five years later, while still a graduate student at Cambridge, he had already
worked out the basic operations of calculus—the rules of integration and
differentiation, which he called the laws of “fluxions and fluents.”
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
Newton put together his great invention and applied it in a preliminary way to the
problems of motion and gravitation in a two-year burst of creativity, while
rusticating during the epidemic of plague which swept England in 1665 and 1666.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
In retrospect it seems as if the whole framework of modern science arose from
his mind as miraculously as a jinni in a bottle.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
But as Newton himself said, he “stood on the shoulders of giants.” Many men had
wrestled with the same problems; it was his genius to fuse their separate
inspirations.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The twin processes of differentiation and integration in calculus, for instance,
were rooted in two classic questions of Greek antiquity: how to construct a
tangent line (a line that just touches a curve at a given point), and how to
calculate an area which is bounded on one side by a curve.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The problem of the tangent, or “touching” line, was equivalent to the problem of
finding the slope of a curve at any point and therefore of finding the derivative of
an equation.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)
The area problem was equivalent to the problem of integrating the equation that
gives the rate of growth of an area.
The Calculus
Benjamin David (and the Staff Editors of Life Science Library, Mathematics)