An acoustic method is presented for analyzing the time of falling motion. A ball is dropped from a measured height. The dropping device makes a distinct sound a well-determined time (roughly 14 milliseconds) after release. The ball subsequently makes a second distinct sound when it hits the surface below. These sounds are captured with a microphone resting on the surface and are readily apparent in the acoustic waveform. At each height (0.25m, 0.50m, 0.75m, and 1.00m), the measured drop time agrees with the drop time predicted by the law of falling bodies with a typical accuracy of 4.3 ms.
1
An Acoustic Demonstration of Galileo’s Law of Falling Bodies
Michael W. Courtneyi and Elya R. Courtneyii
iU.S. Air Force Academy, 2354 Fairchild Drive, USAF Academy, CO, 80840-6210
Michael.Courtney@usafa.edu
iiBTG Research, PO Box 62541, Colorado Springs, CO 80962-2541
Introduction
An acoustic method is presented for analyzing the
time of falling motion. A ball is dropped from a
measured height. The dropping device makes a
distinct sound a well-determined time (roughly 14
milliseconds) after release. The ball subsequently
makes a second distinct sound when it hits the
surface below. These sounds are captured with a
microphone resting on the surface and are readily
apparent in the acoustic waveform. At each height
(0.25m, 0.50m, 0.75m, and 1.00m), the measured
drop time agrees with the drop time predicted by
the law of falling bodies with a typical accuracy of
4.3 ms.
Since Galileo, analysis of falling motion has been
challenged to measure time with suitable accuracy
to demonstrate the Law of Falling Bodies.[1] Galileo
employed ramps to slow descent of bodies; Atwood
invented the Atwood machine.[2] Various electronic
techniques have been implemented to measure
accelerations,
including
the
60-Hz
spark
technique,[3] photogates,[4] and high-speed video.
The accuracy of these methods can be impressive,
but requires specialized equipment. Furthermore,
these techniques are often unable to determine the
instant of drop with sufficient accuracy, thus limiting
the analysis to expectations of linear increase of
velocity vs. time rather than the quadratic increase
of the distance vs. time. The acoustic method
presented
here
uses
commonly
available
equipment and offers a more direct test of the law
of falling bodies; namely, that the distance fallen is
proportional to the square of the time and the
acceleration of gravity.
Method
Figure 1: Ball drop apparatus showing ball held in place by rod
(top), and lever in position striking stop which makes distinct
sound right after ball is released (bottom).
Figure 2: Audacity screen showing distinct sounds of lever
striking stop (first peak) and ball striking floor (second peak).
A steel ball of diameter 1.8 cm is supported by a
horizontal rod that can be retracted by raising a
lever on a spring loaded device as shown in Figure
- The height of the ball above the surface (table
or floor) is measured with a meter stick. When the
lever is lifted, the spring loaded device withdraws
the rod supporting the ball, allowing it to fall. A
2
short time after the ball begins to fall, the lever
strikes a metal stop on the device, making a
distinct sound.
A microphone is set on the surface near where the
ball is expected to land. The microphone picks up
the distinct sound of the lever striking the metal
stop right after the ball is released and the sound of
the ball striking the landing surface. The digitized
sound is recorded with the Audacity[5] software
program. A typical waveform is shown in Figure 2.
The fall time can be estimated from the time
difference between these two distinct sounds. Five
waveforms were recorded and five times were
determined for each height. The average times
and standard deviations from the mean were then
computed for each height. (This method ignores
the delays of each sound reaching the microphone.
These delays are less than 4 ms and can be
mitigated by placing the microphone equidistant
from the drop height and striking surface.)[6]
Data and Analysis
Figure 3: The grey line is the theoretical fall time in the
absence of air resistance. The blue line is a best fit line which
subtracts the delay between ball drop and the noise of the
lever hitting the metal stop on the apparatus.
The measured times and theoretical models are
shown in Figure 3. The vertical error bars
represent the standard deviation from the mean
and the horizontal error bars represent a generous
estimate of the height uncertainty of 1 cm. A red
line (top) shows the theoretical fall time vs. height.
The blue line (bottom) subtracts a constant time
from the red line to account for the delay between
ball drop and the sound of the lever hitting the
metal stop. This delay is determined by fitting the
curve to the delay (t0) which gives a delay of 14.2
ms and a root mean square of residual errors of
4.3 ms.
Conclusion and Discussion
An acoustic method for demonstrating Galileo’s
Law of Falling Bodies fits the expected predictive
model with a typical drop time accuracy of 4.3 ms,
which corresponds to 1.4% of the fall time from
0.5m. This technique directly determines fall time
for a given distance and uses much less
specialized equipment than other techniques
commonly
employed
in
introductory
physics
laboratories.
It is also possible to adapt this acoustic technique
by simply rolling a steel ball or glass marble off
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