Acceleration
Due to Gravity
Introduction
Fig. 1: Aristotle (
NIH
)
One of the more famous explanations as to why and how things fall was given
to the world by the Greek scholar Aristotle (384-322 B.C.).
He believed in the
concept of teleology, which is the philosophy that the explanation of an
occurrence was found in its final causes.
Applied to the motion of objects,
Aristotle believed that object always wanted to return to their beginnings where
they could be with like objects, i.e. a rock that was from the ground wanted to
return to the ground to be amongst other rocks.
Aristotle did not stop at just giving a reason for why things fall down.
He also
went on to describe the motion.
In particular, he stated that heavier object fall
faster than lighter objects.
For instance, he posited that an object that was
twice as heavy as another object would fall twice as fast.
This, too, was
accepted by the public, based somewhat on the authority of Aristotle and
somewhat on observations.
For instance, if you drop a light piece of paper and
a heavy rock at the same time, the rock hits the ground well before the paper.
Today, we know that the cause of this phenomenon is air resistance.
However,
it was not until the early 1600’s that Galileo Galilei showed that this theory of
heavier objects falling faster was wrong.
He did this by running experiments
wherein the weight of the object did not affect the rate at which objects fall.
The
experiments that Galileo performed to study gravity were done with a ball rolling
down an inclined plane, rather than a ball falling through the air.
The reason for
this was quite simple: Galileo wanted to time how long it took the ball to move a
given distance, and the clocks of his time did not allow for accurate
measurements on such short time scales as a second or so.
By having the balls
roll down an inclined plane, Galileo lengthened the time over which the balls
moved.
Using the most accurate pendulum and water clocks of his time, he was
able to make fairly accurate measurements of the elapsed time.
These measurements allowed Galileo to do more than to show that Aristotle was
wrong.
He was able to test and discern mathematical relationships between various factors, which led to the
first accurate equations about the motions of objects.
What Galileo found was that, under the influence of
gravity, the distance that a ball travels from rest is proportional to the square of the amount of time that the ball
is allowed to travel.
Mathematically, we write this as
Fig. 2: Galileo (Sustermans)
2
xk
t
Δ=
where
Δ
x is the distance traveled, t is the elapsed time from rest, and k is a proportionality constant.
Knowing
that the average velocity of an object is related to the displacement by
v
avg
=
Δ
x/t
and that the final velocity of
an object is related to the acceleration (for an object starting at rest) by
v
final
= at
, Galileo deduced that
2
1
xa
t
2
This leads to the following equations for describing the behavior of a freely falling object:
2
1
2
ii
x(t)
x
v t
at
=+ +
and
fi
v(
t
) a
t v
=
+