76
Velocity increases more gradually on
the gentle slope, but the motion is
otherwise the same as the motion of
a falling object.
The v-t graph of a falling object is a
line.
v
t
Chapter 3Acceleration and Free Fall
speculated that in addition to the force that always pulls objects down, there
was an upward force exerted by the air. Anyone can speculate, but Galileo
went beyond speculation and came up with two clever experiments to probe
the issue. First, he experimented with objects falling in water, which probed
the same issues but made the motion slow enough that he could take time
measurements with a primitive pendulum clock. With this technique, he
established the following facts:
•
All heavy, streamlined objects (for example a steel rod dropped
point-down) reach the bottom of the tank in about the same
amount of time, only slightly longer than the time they would take
to fall the same distance in air.
•
Objects that are lighter or less streamlined take a longer time to
reach the bottom.
This supported his hypothesis about two contrary forces. He imagined
an idealized situation in which the falling object did not have to push its
way through any substance at all. Falling in air would be more like this ideal
case than falling in water, but even a thin, sparse medium like air would be
sufficient to cause obvious effects on feathers and other light objects that
were not streamlined. Today, we have vacuum pumps that allow us to suck
nearly all the air out of a chamber, and if we drop a feather and a rock side
by side in a vacuum, the feather does not lag behind the rock at all.
How the speed of a falling object increases with time
Galileo’s second stroke of genius was to find a way to make quantitative
measurements of how the speed of a falling object increased as it went
along. Again it was problematic to make sufficiently accurate time measure-
ments with primitive clocks, and again he found a tricky way to slow things
down while preserving the essential physical phenomena: he let a ball roll
down a slope instead of dropping it vertically. The steeper the incline, the
more rapidly the ball would gain speed. Without a modern video camera,
Galileo had invented a way to make a slow-motion version of falling.
Although Galileo’s clocks were only good enough to do accurate
experiments at the smaller angles, he was confident after making a system-
atic study at a variety of small angles that his basic conclusions were gener-
ally valid. Stated in modern language, what he found was that the velocity-
versus-time graph was a line. In the language of algebra, we know that a line
has an equation of the form y=ax+b, but our variables are v and t, so it
would be v=at+b. (The constant b can be interpreted simply as the initial
velocity of the object, i.e. its velocity at the time when we started our clock,
which we conventionally write as
v
o
.)