Proper Inflation of a Basketball
Josephine Corcoran and Ralph Rackstraw
March 10, 2011
Summary
International basketball rules define the correct inflation of a ball in terms of its rebound properties when dropped. This report describes an experimental procedure to determine how correct
inflation can be achieved using a hand pump. A basketball’s rebound height was measured as a
function of the number of pump strokes used to inflate it. For the pump used in the experiment,
the ball pressure developed by 12 strokes was found to guarantee rebound to 60% of the drop
height, as required. A formula has been derived to determine the correct number of strokes for
different hand pumps. The effects that the ambient air temperature and the material from which
the ball is made might have on the accuracy of our results are discussed.
1
Introduction
Basketball is a game that relies on the skill of dribbling — that is, walking or running while
bouncing the basketball on the floor. To play the game most effectively, players need to be
able to rely on the ball rebounding off the floor as they move with it around the court. The
‘springiness’ of the ball also affects shots that rebound off the backboard. The amount that
the ball rebounds depends on how much air is used to inflate it. A ball with too little air is flat
and difficult to dribble. A ball with too much air is too lively and more difficult to control when
dribbling and shooting. Inflating a basketball with the correct amount of air is important to being
able to play the game well.
According to international basketball rules [1], a basketball is properly inflated when it is:
“. . . inflated to an air pressure such that, when it is dropped onto the playing floor from a
height of approximately 1,800 mm measured from the bottom of the ball, it will rebound to a
height of between 1,200 mm and 1,400 mm, measured to the top of the ball.”
Accounting for the 240 mm diameter of the ball, this means that a properly inflated ball rebounds
to 62 ± 6% of the height from which it is dropped. Unfortunately, using this standard alone to
determine when a ball is properly inflated implies a significant amount of trial and error: pump
some air into the ball, remove the pump, drop the ball and test its rebounding, pump more air
into the ball, and so forth.
The objective of this experiment was to develop a method for predicting, from the size of a given
hand-pump, the number of strokes needed to correctly inflate a basketball.
1
2
Materials and Methods
The basic setup used for this experiment is shown in Fig. 1. A 2-metre steel rule was used to
set a uniform height, h0 = 1.8 m, from which the basketball was dropped and to measure the
height h of its rebound. As Fig. 1 shows, h0 was measured from the floor to the bottom of the
ball, while h was measured from the floor to the top of the ball, in accordance with international
rules [1]. A video camera to record the motion of the basketball. By examining the video
recording frame by frame, the ball could be captured at the top of its rebound.
h0 = 1.8 m
basketball
video camera
h
2m
steel rule
Figure 1: To determine the rebound height of a basketball, a video camera was used to record
its motion with a metre rule in the background.
The video camera used was an 8-mm format Sony model CCD-F301. The resolution of the
image in the ‘stop-action’ frame was the critical factor in determining the precision of our height
measurements. In order for the marks on the metre rule to be seen in the frame, every centimetre marking was darkened with a permanent marker. The precision of our individual height
measurements was ±10 mm.
The basketball used (Wilson ‘Zone Buster’ model P1350) was made of moulded rubber. The
diameter of the ball was measured by holding it flush to the wall with a book (a convenient right
angle) and then measuring the distance from the edge of the book to the wall. The ball had a
diameter D = 240 mm, as shown in Fig. 2. The wall thickness of the ball was estimated to be
2.5 mm, and was therefore considered to be negligible compared to the ball diameter. Before
beginning the experiment, an inflating needle was inserted into the basketball to equalise its
internal pressure with that of the surrounding air.
The pump used to inflate the basketball was a common hand pump. This pump was disassembled to measure the inside diameter of the cylinder, d = 34 mm, and then reassembled to
measure the length of its stroke, ∆z = 380 mm. These measurements allowed the volume of
air pumped into the ball with each stroke to be calculated. The experiment was conducted indoors, using the room thermostat thermometer to measure the temperature of the surrounding
◦
air, T = 21 C. The ball was dropped onto a hardwood floor.
To complete the experiment the basketball was inflated, counting the number N of strokes used.
The ball was then dropped from a height of 1.8 m and its rebound height h recorded. Each
measurement was repeated five times. The ball was then inflated some more, counting the
additional number of strokes ∆N used, and measurement of the ball’s rebound was repeated
with this new total number of strokes N. This process was repeated until the ball rebounded to
more than 62% of its initial height.
2
ball
diameter, D
change in
length, Δz
cylinder
diameter, d
Figure 2: The outside diameter of the basketball used was D = 240 mm. The inside diameter
of the pump cylinder was d = 34 mm and the length of its stroke was ∆z = 380 mm.
3
Results
The specific results from this experiment allow us to determine the proper number of strokes of
our hand pump needed to properly inflate the basketball, and to calculate the proper number of
strokes to use with any pump.
Some typical data describing the rebounding of the basketball with increasing inflation are
shown in Table 1. These data show that there is a small amount of variation in the rebound
height from one trial to another, even for identical conditions. This is most probably due to
variations in the initial height from which the ball was dropped. For each set of trials, the
individual rebound heights were averaged to represent the mean rebound of the ball at that
inflation. The uncertainty in each of these averages was estimated as the standard deviation of
the results from five trials.
1.6
1.4
rebound height, h [m]
1.2
polynomial fit
1.0
0.8
0.6
0.4
0.2
0.0
0
5
10
15
20
25
number of pump strokes, N
Figure 3: Rebound height h as a function of the number N of pump strokes used to inflate the
basketball.
3
Figure 3 shows the variation of rebound height h as a function of the number of strokes of the
pump used to inflate the basketball. The ‘trendline’ feature in Microsoft Excel® was used to fit
the data to a polynomial function:
h = 0.64 + 0.093N − 0.0040N 2 + 0.000067N 3
(1)
where N is the number of strokes of the pump and h is measured in metres. These data show
that the proper number of strokes to inflate the basketball is N = 12.
Table 1: Typical data describing the rebounding of the basketball. These data were taken for
the case of 15 strokes having been used to inflate the ball.
Trial 1
1.31
4
Rebound height h [m]
Trial 2 Trial 3 Trial 4
1.34
1.34
1.34
Trial 5
1.36
Average h [m]
Uncertainty σh [m]
1.34
0.02
Analysis
It is important to recognise that the number of strokes needed to properly inflate the basketball
is valid only for the particular pump used. The important quantity is the pressure of the air that
is in the ball. Fortunately, the ideal gas law can be used to determine the pressure in the ball
from our measurements. The ideal gas law states [2] that:
P V = nkT
(2)
where P is the pressure of a given quantity of gas, n is the number of molecules in this quantity,
V is the volume the gas occupies, T is the absolute temperature of the gas, and k is Boltzmann’s
−23
constant (1.38 × 10
J/K [2]).
When a pump is used to inflate a basketball a fixed number of air molecules, n, initially at
room temperature and pressure, is squeezed into a smaller volume. Although the process of
compressing the gas heats it, the gas quickly loses this heat and returns to room temperature.
Since n and T are constants, Eq. (2) can be rewritten as:
P1 V1 = P2 V2
(3)
where the subscripts ‘1’ and ‘2’ refer to the initial and final conditions of the gas. Here, the initial
pressure of the gas is atmospheric pressure at room temperature, Patm . The initial volume of
the gas is the sum of the volume of the basketball, Vball , and N times the volume of the pump,
Vpump . Recall that N is the number of strokes of the pump used to inflate the ball. The process
is described graphically in Fig. 4.
The relative increase in the pressure of the ball, using Eq. (3), is just:
Vball + NVpump
P2
=
.
Patm
Vball
(4)
4
before: larger volume, lower pressure
after: smaller volume,
higher pressure
+
+
⋯+⋯
1
2
N−1
+
N
Figure 4: When a pump is used to inflate a basketball, a fixed amount of air is compressed into
a smaller volume.
Because the dimensions of the ball and the pump are known, these volumes can be calculated.
The volume of the pump is the product of its cross-sectional area and its stroke:
Vpump
πd 2 ∆z
=
= 3.45 × 10−4 m3
4
(5)
and the volume of the ball is approximately that of a sphere of its outside diameter:
3
Vball =
πD
= 7.24 × 10−3 m3 .
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(6)
Substituting Eqs. (5) and (6) into (4) shows that the increase in pressure that makes the ball
bounce properly is P2 /Patm = 1.57. With this knowledge, (4) can be used to derive a formula
that is useful in general for the number of strokes to properly inflate the ball, namely:
N = 0.57
Vball
(7)
Vpump
where Eqs. (5) and (6) give the volumes of the pump and the ball, respectively.
5
Discussion
These results were obtained at a fixed temperature. The ideal gas law (Eq. (2)) shows that
as the temperature increases, the pressure in the basketball will go up. Conversely, as the
temperature goes down, the pressure will decrease. This was observed at first hand when the
ball was brought in from a cold garage: it was noticeably flat.
Our results are therefore not valid for different temperatures. One way to address this problem
would be to account for it in the analysis, using the ideal gas law. This would make Eq. (7) more
complicated mathematically, but more generally useful. Another method would be to measure
data at different temperature conditions (e.g. outside during hot and cold weather).
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This investigation did not account for the effect of the material of the ball on its rebounding. The
material of the ball has some elastic stiffness, and a leather basketball would bounce slightly
differently to a rubber one. Our results (see Fig. 3) show that this is a more important effect
when there is very little air in the ball. Even a little air dramatically modifies rebounding of the
ball, as shown by the initially large slope in the best-fit line. The experiment could be improved
by repeating it with other basketballs of different materials to verify this assertion.
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Conclusions
The optimum number of strokes of a hand pump needed to properly inflate a basketball have
been determined. For the pump and basketball investigated it was found that 12 pump strokes
inflated the ball so that it rebounded to within 62 ± 6% of its original height when dropped.
Inflating the ball raises the pressure of the air inside it and gives it additional elasticity. On
this basis, a method was devised for determining the correct number of strokes necessary to
inflate a basketball using any hand pump, if the volume of the air in a single pump stroke can
◦
be determined. The results are valid for a temperature of 21 C. Extending the results to other
temperatures would require additional analysis, measurement, or both. Repeating these tests
with basketballs made of other materials would confirm our assertion that the material of the
ball has little effect on its rebounding once it is fully inflated.
Appendices
A
Error analysis
The first appendix.
B
About this document
The original version of this document was written in 1999 by the named authors to serve as
course material at James Madison University, Virginia USA under Professor C. Lon Enloe. It
has been edited, with permission, by Patrick Leevers (Mechanical Engineering Department,
Imperial College London) to adapt it to UK English spelling and Imperial College style.
References
[1] Fédération Internationale de Basketball. Official Basketball Rules 2010: Basketball Equipment, 17 April 2010.
[2] Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers. Brooks/Cole
CENGAGE Learning, 8th edition, 2010.
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