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Universal Physics Journal
Article IX: Galileo's Law of Constant Acceleration
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Author: Ethan Skyler
Publication Date: September 9, 2002
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Purpose
Galileo Galilei had a fascination with falling objects.
His extensive experimentation was undoubtedly spurred on by Aristotle's
contention that large heavy objects fall faster than small light
objects. Galileo reasoned that if one stone fell at a given rate of
acceleration, and a second similar stone fell at the same rate of
acceleration, for Aristotle to be right then after binding these two
stones together, the resulting larger, heavier object should fall with a
much greater rate of acceleration. This made no sense to Galileo for
he could not think of a reason why the simple binding of these two stones
should provide any cause for an increased rate of acceleration during a fall. Galileo had
to wonder if Aristotle ever actually performed an experiment by dropping
light and heavy stones together from a height to test his
prediction. Galileo went on to perform his own experiments which
revealed a truth contrary to Aristotle's thought experiment. Galileo
was so encouraged by the power and truth of experiment that he spent years
performing and writing about experiments involving the acceleration of
falling objects. Over the course of these experiments, Galileo
discovered some basic truths regarding the acceleration of objects.
Having found these truths to be seldom published, it is
my goal herein to once again bring to light the brilliance of
Galileo's Law of Constant Acceleration.
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Article IX
When Galileo set out to study the acceleration of
a falling object, he was immediately beset with two problems. The
first problem was the simple fact that the vertical fall of an object happened too
quickly. In just a few seconds, the event was over for the falling
object had impacted with the ground. The second problem was the
timing of the increments of the object's fall. Galileo wondered if
the falling object fell equal distances in equal units of elapsed
time. But the precision timers we take for granted today did not exist back
then. So Galileo had to resort to means of his own invention to
first slow the rate of the object's fall and then to measure the passage
of equal units of elapsed time during the event.
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(2) To slow the pace of the event, Galileo selected
a spherical metal ball which he allowed to freely roll down a long,
straight groove cut into the surface of an inclined wooden plank.
While introducing some curious effects due to the changing rates of
rotation of the rolling object, this technique of Galileo's effectively
reduced the object's rate of acceleration which made the accurate measurement of the
event more achievable.
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(3) To time the event, Galileo built a water timer
where the water that dripped from a vessel at a constant frequency was
first suddenly allowed to drip into a test pan at the start of the event
and then suddenly prevented from dripping into the the pan at the event's
end. After the event was over, the water caught by the pan was very
accurately weighed on a balance beam scale. In this manner, the
amount of water in the test pan was a direct indication to Galileo as to
the amount of time that had passed during the event. Although crude
by today's standards, Galileo's water timer proved sufficiently accurate
to enable his discovery of the Law of Constant Acceleration.
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(4) Galileo soon discovered that a falling object
did not traverse equal distances with the passage of equal units of
time. Instead, his experiments revealed that the object's speed of
fall increased with the passage of each measured unit of time. It
was while analyzing the data from the falling object's rate of increase of
speed (acceleration), that Galileo discovered the Law of Constant
Acceleration. This law applies equally well to all examples of
constant acceleration whether the object's acceleration is a vertical
event caused by internal forces of gravitation toward Earth, the Moon, and
the
Sun, or instead is a horizontal event caused by
external (contact) forces such as the momentarily constant acceleration
experienced by an object inside an automobile or a rocket sled.
As long as the object's rate of acceleration remains constant, Galileo's
Law of Constant Acceleration is applicable.
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(5) (Note: Galileo's law is sometimes referred to as
the "Law of Fall". I see "fall" as a vague term
in general use that gives not a hint as to whether acceleration is present
or absent. For example, when a skydiver exits the basket of a
high-altitude balloon, he or she suddenly experiences a weightless
acceleration at the initial rate of 32 ft/s/s. This rate of
acceleration, caused by the internal force of Earth gravitation being
generated within each component of the skydiver's matter, immediately begins to diminish until, after
a number of seconds of
"fall", the skydiver's acceleration comes to a complete
end. The skydiver has reached the uniform speed through the air
known as "terminal velocity". For the remainder of this
event, the skydiver will "fall" to Earth's surface in a uniform manner
with acceleration generally absent.
Since Galileo's discoveries have to do with
acceleration (a change in velocity including a change in speed or a change
in direction) and even more specifically with constant acceleration in any
direction, not just vertically down, and further since the vague non-scientific term
"fall" makes no distinction between accelerated motion and
uniform rest-motion, I think the "Law of Constant Acceleration"
is an appropriate title for Galileo's discoveries.)
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 (6) While analyzing the data he collected from
accelerating objects, Galileo discovered that the square of the time or
duration of an object's constant rate of acceleration, multiplied by the
unit of distance the object traveled during the first unit of time, yielded
an accurate prediction of the total distance the object traveled
from the point where its acceleration began.
Understanding this
discovery is not as complex as it may at first appear. Click on the icon to the
right for a graphic representation of Galileo's "Times Square
Law". First note how the constant acceleration of the red ball
yields the same results in both a vertical event and a horizontal
event. The base unit for distance is set during the first time-unit of
acceleration. In this case, the time unit is 1 second. (The
Times-Square predictions are equally accurate with the use of any other unit
for time.) During the first second of constant acceleration, which in
this event is at the rate of 32 ft/sec2, the red ball travels a
distance unit of 16 feet. Understand that when an object is
experiencing constant linear acceleration at the rate of 32 ft/sec2, at
the end of each second of acceleration, the object's instantaneous velocity will be 32 ft/sec faster (or slower) than at the beginning of the
second. In the red ball event, its relative velocity is 0 ft/sec at
the start and 32 ft/sec at the end of the 1st second. The relative
distance traveled by the red ball is found by averaging these two
velocities. 0 ft/sec + 32 ft/sec = 32 ft/sec. 32 ft/sec / 2 = 16
ft/sec average velocity during the 1st second of the red ball's
acceleration. With an average velocity of 16 ft/sec, the 1st second
of travel will yield a Distance Unit of 16 feet. Now we are ready to
apply Galileo's Times Square Law in predicting the relative distances of
travel during the remaining seconds of the red ball's constant
acceleration.
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(7) As we proceed, keep in mind that in this event each Time Unit
(tu) is 1
second while each Distance Unit (du) is 16 ft. As we perform the
square of each Time Unit of acceleration, note that all squares share the same
point of origin which is at the start or 0 sec. Although we already
know the conditions of the first second of acceleration, realize that if a
Time Unit of 1 is squared and the answer of 1 is multiplied times the
Distance Unit of 1, the distance of travel for the red ball
during its 1st second of acceleration is correctly predicted as 1 du or 16
ft.. Next, if the Time Unit of 2 is squared, the answer of
4 multiplied times the Distance Unit of 1 predicts a total distance of travel
from the start of 4 du or 64 ft. Since each Times
Square begins at the origin, as long as the Distance Unit is known, one is free to
skip ahead to predict that after 5 Time Units of constant acceleration, the red
ball will have traveled 52 x 1 du = 25 du or 400 ft from the
origin. After 10 seconds of constant acceleration, the red ball will have traveled
102 x 1 du = 100du or 1600 ft from the origin. Discovery of this
Times Square Law must have been quite exciting for Galileo. Here he
was perhaps the first to witness an undeniable association between a Universal event
involving constant acceleration and the precise predictions of a man-made mathematical
process.
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(8) Next Galileo turned his attention to the
relationship between the predictions of the Times Square Law for each
subsequent second. Second 1 predicts 1 distance unit (du), second 2
predicts 4 du, second 3 predicts 9 du , second 4 predicts 16 du and so
on. From this information, Galileo discovered the "Odd-Number
Law" when he noticed that the number 3 separates the predictions of
seconds 1 & 2, the number 5 separates the predictions of seconds 2
& 3, the number 7 separates the predictions of seconds 3 & 4, and
so on.... Thus he discovered that the advancing gap between his
Times Square predictions was always filled by an advancing sequence of odd
numbers. Overall, these two discoveries form Galileo's Law of
Constant Acceleration.
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(9) Galileo is best known for his
light-hearted ability to question, dismantle and reassemble all of man's
creations that, for him, lacked sensibility. Against his intellect and
wit, no nonsensical concept could stand firm. If we learn anything
at all from Galileo let it be this unusual ability of his.
Question Everything - Accept Nothing. I say "unusual" for
ironically, the polar opposite, "question nothing, accept everything" is the
standard guide by which a student is most likely to succeed in ascending
through the ranks of the modern version of the ancient science of Physics.
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(10) Following Galileo's discoveries, in Galileo's
tradition, have you formed any questions regarding the red ball
event? I have a few. First of all, although I agree that the
red ball is experiencing acceleration at a constant rate when friction is
absent, which is due to the fact that in the vertical event the
acceleration/Action force of gravitation of the red ball toward Earth remains generally
constant during the ball's acceleration, I have a hard time
resolving this recognition with the actual distances traveled. To
me, if the red ball's acceleration is truly constant, then if it results
in the ball traveling 16 ft in the 1st second, I can think of no reason why
this same rate of acceleration should not continue to add 16 ft to the
ball's travel during each subsequent second. Yet this prediction of mine
appears to be unsupported by the chart above. On a separate issue, I am
left in a complete state of wonderment as to exactly why it is that Galileo's Times
Square Law so accurately predicts the outcome of constant linear
acceleration. Overall I think that our understanding of constant
acceleration will benefit if we spend some time applying the rest-motion based concepts of Universal Physics
to this accelerational event.
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 (11) I will choose the horizontal event where the
red ball is experiencing constant acceleration in a vacuum caused by a
constant impressed external force of an unspecified nature. In the
chart to the right, I have broken the ball's 4 seconds of acceleration
into 4 separate one second events. In each event I have inserted the
rest-motion reference frame of what may be thought of as the nose cone of
a non-accelerating rocket. My purpose here is to separate the
distance units traveled by the ball's rest-motion from the distance units traveled by the ball's constant acceleration. This diminishment of an
event into its base components is a technique often performed with great
success by Galileo in his various analytical efforts.
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(12) During the 1st second of the red
ball's constant acceleration, the nose cone maintains its position at
the start or point-of-origin with a speed of 0 ft/sec. I will refer
to the relative distance from the point-of-origin the nose cone travels
while in rest-motion as rest-motion distance units (rdu) which at this 1st
one second interval is 0 rdu. Meanwhile the ball's acceleration causes it
to advance 16 feet ahead of the "stationary" nose cone. I
will refer to this 16 feet as the ball's acceleration distance unit (adu)
which at this first interval is 1 adu. These are the base conditions
for this event. From here I intend to show that during the entire
event the red ball's adu never exceeds 1 during any second of its constant
acceleration. If I am successful in this effort then I hope to
finally understand how it is that the red ball's acceleration can remain
constant while the distance it travels during each 1 second interval
increases at a rate that is anything but constant.
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(13) At the end of the 1st second, the ball's velocity
is 32 ft/sec. If, for the moment, the a/A force accelerating the
ball is suddenly brought to an end, in the next second, the ball will travel in the
state of rest-motion for 2 distance units or 32 feet, relative to the
start, with acceleration entirely absent. This role of traveling the
rest-motion distance units (rdu) I will assign to the nose cone.
Thus, with the a/A force reapplied only to the red ball, at the beginning of the 2nd second,
the ball and nose cone are together at the 16 foot mark with the nose cone
traveling in rest-motion at a steady 32 ft/sec relative to the start while
the red ball is beginning its 2nd acceleration run ahead of the nose
cone. During this 2nd second of time, the nose cone, with its
velocity of 32 ft/sec, will travel 2 rdu or 32 feet while the red ball
will begin the 2nd second with a velocity of 0 ft/sec relative to the nose cone and
accelerate ahead to reach a velocity of 32 ft/sec relative to the nose
cone. This means the ball's average velocity relative to the nose
cone is 16 ft/sec which will again result in the red ball ending the 2nd
second 16 ft or 1 adu ahead of the nose cone. When the nose cone's
2 rdu, relative to the start, are added to the red ball's 1 adu, relative
to the nose cone, the combined total of 3 distance units is the
result. Notice that just as in the 1st second event, here in the 2nd
second event the red ball's acceleration accounts for just one of the
distance units traveled. The remaining 2 du traveled by the nose
cone would have been traveled had acceleration been entirely absent for
the red ball at the beginning of the 2nd second. In other words, in both the 1st and 2nd seconds, the
distance units advanced by the red ball remains constant at 16 ft or 1
distance unit..
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(14) To set the scene for the 3rd second event, I
will again increase the velocity of the nose cone to match the red ball's
velocity at the end of the previous second which is now 64 fps.
During the 3rd second event, the rest-motion velocity of the nose cone will
cause it to travel 4 rdu or 64 ft relative to the start. Again the
red ball will accelerate ahead of the nose cone causing it to travel 1
extra adu or 16 ft relative to the nose cone. A quick check of the 4th second
event reveals that the red ball's acceleration again accounts for just 1 adu of
travel. Therefore the earlier prediction that the red ball's acceleration, if constant, should account for the
addition of a constant distance unit of travel for each second of
acceleration turns out to be
correct.
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(15) Why then is this truth regarding constant
acceleration not initially obvious? The acceleration of the red ball is clear to see
during the 1st second event. It begins by accelerating away from a
position with an observed rest-motion of 0 fps.. One second later the red
ball is accelerating away from a position with an observed rest-motion of
32 fps. Then later still from a position with an observed rest-motion of 64
fps. Then 96 fps. Then 128 fps, and so on. All such
speeds of rest-motion are determined by an observer who remains at rest
relative to the 0 fps position at the start of the event. Yet the
whole thing makes sense only when the observer is forced to adjust the
speed of his or her rest-motion to match the rest-motion of the nose cone
for each subsequent second of acceleration. This way the observer is
always watching a 1st time unit event where the red ball is observed
traveling 1 distance unit (16 feet in this event) while accelerating away from a position of rest
(0 fps).
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(16) This continual reassignment of a new velocity of
rest-motion from which to observe the event of constant linear
acceleration makes more sense when one considers the observation of
constant centripetal acceleration. It is common practice to observe and
calculate the constant centripetal acceleration of an object by the continual
reassignment of a rest-motion path of observation set tangent to the
object's circular path of motion. The observer's rest-motion path of
observation is reassigned once for each time unit of the event with the
observer's velocity set equal to the orbiting object's velocity at the
instant of assignment. Then for the next time unit interval, any
change in direction of the orbiting object represents the object's
acceleration away from the tangential rest-motion path of
observation. When the time unit interval is over, the observer's
path is reassigned once again as tangent to the accelerating object's
circular path of motion which means that for an instant the non-accelerating
observer's velocity is a match with the accelerating object's
velocity. This is not unlike my reassignment of the velocity of the
non-accelerating nose cone to match for an instant with the velocity of
the accelerating red ball at the beginning of the next time unit interval
of the ball's constant acceleration. In plainer words, my solution
of resetting a constant linear acceleration event at the beginning of each
subsequent time unit interval is already accepted as common practice
during events involving constant centripetal acceleration. This common
practice extends at least as far back as Isaac Newton's time for he used
just such a model as an aid in developing his form of the mathematical
process of incremental change known as calculus.
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 (17)
My next goal is to determine exactly why it is that Galileo's Times Square
Law is correct regarding the distance units traveled by a constantly
accelerating object. I will labor over this question for I find this
direct relationship between the square of the time units of an object's
constant acceleration and the number of distance units traveled by the
accelerating object to be quite remarkable. Please consult the
Distance Units Chart below. Notice Galileo's Times Square Law at
work as the square of each time unit interval in the left-hand column is
equal to the current distance units traveled from the start in the
right-hand column.
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(18) The information in the cdu
column enabled Galileo to discover his "Odd-Number Law"
regarding the advancing series of odd numbers separating the number of
distance units traveled in subsequent intervals of time. In the
Distance Units Chart above, the rest-motion distance units (rdu) are shown
separately from the acceleration distance units (adu). In each time
interval, the distance traveled due to the red ball's acceleration is
constant at 1 du or 16 feet. This is to be expected for the red
ball's acceleration is constant in every interval. Thus the 1 du of
travel caused exclusively by the ball's acceleration in the 1st interval
can only be repeated without change in each subsequent interval.
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(19) What is changing is the rest-motion distance units
traveled by the nose cone. These rdu advance by 2 in each subsequent
time unit interval. Why by 2? The key is the ball's acceleration
rate which is 32 fps2. This rate dictates that when the
nose cone's velocity of rest-motion is reset at the end of each time unit
interval, it will change by 32 fps. Since our time unit is one second
and our distance unit is 16 feet, this means that the nose cone's velocity
has no other option but to change by 2 rdu per second from the rest-motion
distance units traveled in the previous second. This 2 rdu change
accounts for the odd-number increment that Galileo first
observed.
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(20) But what role does this 2 rdu change play in helping
to shape Galileo's Times Square Law? This is the real question for
which I am seeking an answer. Since I am working toward a solution as
this is being written, at this point I do not know if I will eventually
arrive at a satisfactory one. If not, I am prepared to leave this
question unanswered for I think this is the best state to leave a question
if no solution is clear and obvious to the author. This way when
someone does develop the true solution, he or she will not have to wrestle
needlessly with the author's supporters to displace a false solution before
being granted the position of truth.
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(21) I suspect the answer to my question will be
revealed if we study the red ball event while keeping in mind all possible
distance unit options available to the ball during its constant acceleration
run. Beginning with the 1st second event, the square of 1 is 1
which predicts 1 du of travel. Since the value of the distance unit is
set at the observed 16 feet of travel and verified by our knowledge that the
red ball's average velocity for a 32 fps2 rate of acceleration is
16 fps causing it to travel the observed 16 feet during the 1st second of
constant acceleration, I see here that there is no other option but to accept
that 16 feet is one distance unit and further that 1 distance unit is the
exact distance the red ball travels in the first time unit of one
second. Thus the square of 1 correctly predicts 1 du of travel.
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(22) During the 2nd second event, the square of 2 predicts
4 du of travel. Keep in mind that each square begins at the same point
of origin or start. This means that the results of the previous square
are included in the current square. Hence the 1 du of the 1st second
event becomes the previous distance unit (pdu). For the predicted 4 du
to be reached, 3 more du of travel must occur. Referring back to the
rest-motion role of the nose cone in the 2nd second event, we know its
velocity is a steady 32 fps as set by the red ball's acceleration rate of 32
fps2. Thus the nose cone has no option but to travel 32
feet or 2 distance units during the 2nd second event. Also set by the
red ball's acceleration rate is the 16 ft or 1 adu of travel ahead of the
nose cone. Here during the 2nd second event, the red ball has no other
option. When these three components of travel are added, 1 pdu + 2 rdu
+ 1 adu, the current distance unit total will always equal 4 du just as
predicted. Clearly, as long as the red ball's rate of acceleration
remains constant, this can be the only outcome.
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(23) During the 3rd second event, 5 more du of travel must
occur. This is in keeping with Galileo's Odd-Number Law. Notice
that the increase from 3 du to 5 du to 7 du is always 2 du. So each
subsequent second event must add an additional 2 du over the previous
event. Realize that 2 du each second in this event is 32 fps which is
the exact increase in velocity offered by the red ball's rate of
acceleration. This means the nose cone's velocity of rest-motion has no option
but to increase 32 fps for each subsequent second. So the red ball's
acceleration rate of 32 fps2 guarantees that the required 2 du of travel
will be added to each subsequent second in order for the squaring of
subsequent time units to predict the correct number of distance units of travel.
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(24) While it can be argued that I have not presented the
exact reason why Galileo's Times Square Law holds true, I think I have
presented as fact the reason why no other option is possible. From the
establishment of the length of 1 du, and the rate of increase of 2 du each
second, the red ball's constant rate of acceleration governs all in a manner
that makes the predictions of Galileo's Times Square Law come true.
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Author's Commentary
While presenting the brilliance of Galileo's Times
Square Law discovery, the red ball event is also a good demonstration of
the strength of the Universal Physics concept of rest-motion as portrayed by
the reassignable role of the non-accelerating nose cone. Observations
of each time unit made from the nose cone in the default state of
rest-motion have provided us with a deeper understanding of the constancy of
the red ball's rate of acceleration.
Ethan Skyler
Author's Photo Gallery
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Copyright Notice
Article IX: "Galileo's Law of Fall" (C) Copyright
2002-2008 by Ethan Skyler. All Rights Reserved. No portion of Article IX,
minus the exceptions noted below, may be copied by any means without the
author's written permission and even then only if the author's copyright
notice is permanently affixed to each approved copy. Requests for written
permission may be directed to Ryan Skyler, Editor, Universal Physics
Journal, 9734 Manitou Place NE, Bainbridge Island, WA, USA.
The author grants each visitor to The Universal Physics
Journal the right to make one (1) copy of Article
IX for his or her own personal archive as long as the author's copyright notice is permanently affixed to the archive copy.
The Author's Commentary to Article IX is hereby granted by the author, Ethan Skyler, into the realm of the Public Domain. As such it may be freely copied, in full, or in part, by any means so long as a reference to the author and this site address http://www.UniversalPhysics.org are included with each copy.
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