Air Resistance on Falling
Balls and Balloons
Paul Gluck, Israel Academy of Science and Arts, Jerusalem
S
tudying the effect of air resistance on falling
objects in an introductory mechanics course
has the merit of relevance to a considerable
part of our everyday experience: Leaves, parachutes,
raindrops, or soot particles do not keep accelerating as
they fall. This topic has been discussed in this and
other journals many times,1–7 ranging from theoretical treatments to various experimental investigations,
mostly for coffee filters, depending on the apparatus
at the disposal of the writers (video clips, graphic calculators, and so on). We report here a sequence of activities for our 11th-grade high school students dealing with this topic. The analysis of data emphasizes
points that do not seem to have been done in previous
articles.
After studying free-fall problems with constant acceleration g, our students measure g using a picket
fence for which air resistance is negligible. Next they
study numerically the equations of motion of an object of mass m released from a certain height from rest,
subject to a velocity-dependent resistive force R(v) =
bv n, where n = 1 or 2. This is done with Euler’s
method in Microsoft Excel® and requires no knowledge of calculus.
The aim is to get a feel for the asymptotic behavior
of both the velocity v and the height y as a function of
time before performing air-resistance experiments. In
the simulation the relative magnitudes of R and m are
varied for a fixed n, and then n is varied for a fixed
m/R. We shall not describe this part in detail as it is
available in the literature cited. Having realized that
reaching a terminal speed vt depends on the height of
178
release and the object’s mass, the students are ready for
the experimental part of the investigation.
The Experiment
The sonic ranger is fixed on the ceiling. We then
monitor the fall of two objects when released some 40
cm from below the sonic ranger. One is a hollow inflated plastic ball, the other an inflated spherical balloon. For the typical experiment reported here, the
data are as follows: diameter of ball is 20 cm, its mass
is 140 g; diameter of balloon is 24 cm, its mass is
10.16 g. The total drop is some 230 cm.
At the start of the experiment, students measure
the room temperature and incorporate its value into
the sonic ranger software module, so that the correct
value of the sound velocity is used. The sampling rate
of pulses can be varied and it is for the students to adjust it to the optimum needed: between 35 to 40/s is
usual. The software records the height y as a function
of time t. These are the raw data that they have to manipulate. We now show how to analyze the data.
a) The ball drop. From the y-versus-t data (see
Fig. 1), students construct a v-versus-t column in the
spreadsheet by the simplest algorithm, as used for
example in ticker-timer experiments. Thus, for the
data point in the nth row, the velocity is given by
vn = ⌬y/⌬t = (yn + 1 – yn – 1)/(tn + 1 – tn – 1). (This
amounts to differentiating the y-versus-t data numerically.) Students now examine the v-versus-t data
thus obtained more closely. The aim is to show that
although the height is not sufficient for the ball to
DOI: 10.1119/1.1557510
THE PHYSICS TEACHER ◆ Vol. 41, March 2003
6
velocity (m/s)
distance (m)
2
1.5
1
y = 8.5992x - 1.7254
4
y = 9.268x - 2.124
2
y = 9.6707x - 2.2515
0.5
0
0.25
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
time (s)
Fig. 1. Distance of the ball from the sonic ranger as a
function of time.
reach its terminal velocity (when R would equal mg),
one can nevertheless see that it falls with decreasing
acceleration. The velocity-time data is divided into
an initial, middle, and final section. The velocitytime graph is plotted for each, all three on the same
graph. A least-squares fit is performed for each section (a standard routine in Excel). The slopes of
each give the respective accelerations. Figure 2
shows that during the fall, the acceleration decreases.
This way of analyzing the data does not seem to have
been reported in the papers cited.
b) The balloon drop. A series of five experiments, each with a differently loaded balloon, are
carried out. At first the balloon alone is dropped.
Then at each subsequent drop, a body of mass 2 g
(precut wires) is attached to the bottom of the balloon. By this means, one only varies the balloon’s
mass and keeps its surface area (and so R) fixed. A
set of five measurements of y versus t is at our disposal in order to see how the terminal velocity varies
with mass. For each mass, students carry out at least
five drops, enabling them to calculate average values
in the subsequent analysis.
In order to extract the terminal velocity from the
raw data, the analysis proceeds somewhat differently
from that for the ball. To see why, let us look at a typical velocity-time curve, presented in Fig. 3. The velocity increases and then flattens out to a fairly constant terminal velocity. There are fluctuations. We
THE PHYSICS TEACHER ◆ Vol. 41, March 2003
0.45
0.65
0.85
time (s)
Fig. 2. Three sections of the time variation of velocity, showing decreasing accelerations.
must not forget that the graph is a result of a numerical differentiation of the data, as previously mentioned. At this stage of the balloon’s path, two quantities are nearly equal, namely R (combined with the
buoyancy) and mg, so that the differentiation magnifies the fluctuations. We use this portion of the graph
as a guide for the next step: for the restricted range only, in which the velocity is fairly constant, we return to
the y-versus-t data, plot a graph of y versus t, and fit it
with a least-squares best line. We emphasize that this
procedure is more reliable than that of Ref. 7, in
which the last 10 points of the y-versus-t graph were
chosen as the data for calculating the terminal velocity
without any obvious justification. The slope of this
straight line will be the terminal velocity vt. For the
horizontal portion of Fig. 3 this line is shown in Fig.
4.
The same data treatment is repeated four more
times for different masses, and the terminal velocity is
calculated for each. The final stage is to plot vt against
苶, in order to see whether the expected R = bv2
兹m
model for air resistance is borne out by the data. Results of a typical run are shown in Fig. 5, with a
straight-line fit that is reasonable.
Care has to be taken during the balloon drops: air
conditioning must be switched off, doors and windows shut, and student movement restricted in order
to prevent air currents that cause a drift in their path.
Because the surfaces of the ball and balloon are
curved, 2 to 3% inaccuracy in the heights measured is
179
2.4
2.5
2.3
2.2
2.1
1.5
distance (m)
velocity (m/s)
2
1
2.0
1.9
1.8
1.7
0.5
y = 2.5342x - 0.9791
1.6
0
1.5
0
0.4
0.8
1.2
1.4
0.95
time (s)
1.05
1.15
1.25
1.35
time (s)
Fig. 3. Velocity variation for a loaded balloon weighing
18.67 g, showing the terminal velocity region.
Fig. 4. Least-squares fit for raw y-vs-t data in the terminal velocity region given in Fig. 3.
2.6
expected, and an additional inaccuracy for the balloon
of the same order, due to the inevitable small drifts.
Lastly, the sonic ranger cannot cope with a sampling
rate much higher than quoted above, desirable as this
might be for more data points.
terminal velocity (m/s)
2.5
Closing Remarks
Combining the theoretical part with the laboratory
session seems to provide a more in-depth experience
and greater understanding so that we strongly recommend it. Although the experiment is MBL based, it
provides scope for care in arranging optimal conditions, taking repeated readings and finding standard
deviations, requiring care and circumspection in data
analysis and the testing of models. Its use of balls induces a pleasant atmosphere so that students carry it
out with gusto.
Acknowledgment
Many thanks to Oded Yaron for technical assistance.
References
1. M.S. Greenwood, C. Hanna, and J.W. Milton, “Air resistance acting on a sphere,” Phys. Teach. 24, 153–159
(March 1986).
2. C. Brueningsen et al., “Modeling air drag,” Phys. Teach.
32, 439–441 (Oct. 1994).
3. V. Pagonis et al., “Effects of air resistance,” Phys. Teach.
35, 364–368 (Sept. 1997).
4. S. Herbert and T. Toepker, “Terminal velocity,” Phys.
Teach. 37, 96–97 (Feb. 1999).
5. R. Young, “Improving the data analysis for falling coffee
180
2.4
2.3
y = 0.4597x + 0.557
2.2
2.1
2.0
1.9
3
3.5
4
4.5
square root of mass (g 1/2)
Fig. 5. Least-squares fit to terminal velocity vt as a func苶.
tion of 兹m
filters,” Phys. Teach. 39, 398 (Oct. 2001).
N.E. Derby, R.G. Fuller, and P.W. Gronseth, “The
ubiquitous coffee filter,” Phys. Teach. 35, 168–171
(March 1997).
7. K. Takahashi and D. Thompson, “Measuring air resistance in a computerized laboratory,” Am. J. Phys. 67,
709–711 (Aug. 1999).
PACS codes: 01.50P, 06.30Gv, 46.01A
6.
Paul Gluck has a Ph.D. in theoretical physics and statistical physics, from the University of London. His research
in that area was classical liquids, phase transitions, and
transport theory. He has held various positions at London
& Birmingham University UK, Northeastern University,
Boston, and Bar-Ilan University, Israel. For the past 10
years, he has been occupied with the training of physics
teachers in the greater Jerusalem area.
Israel Academy of Science and Arts, Jerusalem,
Israel, 96408; gluckpaul@hotmail.com
THE PHYSICS TEACHER ◆ Vol. 41, March 2003