Football Physics

The controversy surrounding the Patriots deflated footballs in the AFC Championship game on Jan. 18, 2015 has made some headlines in the news, and raised some physics questions. There has been talk of the ideal gas law, guage pressure vs. absolute pressure and relative temperature vs. absolute temperature. So, I take this as an opportunity to explain some of what all this means if you have followed it or have an interest in this. Lets begin with the NFL rule book stating:

“The ball shall be made up of an inflated (12 1/2 to 13 1/2 pounds) urethane bladder enclosed in a pebble grained, leather case (natural tan color) without corrugations of any kind. It shall have the form of a prolate spheroid and the size and weight shall be: long axis, 11 to 11 1/4 inches; long circumference, 28 to 28 1/2 inches; short circumference, 21 to 21 1/4 inches; weight, 14 to 15 ounces.”

By “pounds” it is assumed this means “pounds per square inch” or psi. However, this term refers to a guage pressure, so the actual pressure is really guage pressure + atmospheric pressure. To understand this remember that the pressure measured with a guage, such as you would with a tire, indicates pressure relative to the atmospheric pressure. Scientists often speak of atmospheres of pressure or atm and define Standard Temperature and Pressure for sea level at 273.15 K as 1 atm = 101325 Pa, or 14.69595 psi. (Note: Pa is another pressure unit called the Pascal and K is a temperature unit called Kelvin). So, consider 1 atm of pressure inside some container, that is, there is the same pressure inside the container as outside of it. Taking a gauge will not show 1 atm, but zero atm, since the pressure in the container is just the same as the pressure outside. This means that relative to the outside there is pressure in the container and it is in balance or equilibrium. This is the meaning of gauge pressure. Now, absolute pressure is technically more accurate when speaking of pressure as it is the force that some gas is applying to the container surface area, by virtue of the fact that on the order of 1E23 molecules are bouncing around off each other and the container wall because of thermal stimulation (heat), a form of kinetic energy.

Heat is not the same thing as temperature, though they are related to each other. Heat is a form of energy that flows from a hotter substance to a colder one, which have higher and lower temperatures, respectively. So, there must be a temperature difference for heat to flow. Consider our container again, and it has a certain amount of heat associated with it, which can be probed by measuring the temperature. Another container made of a different substance may have more heat associated with it but still measure the same temperature, because the second container has more mass. Anyway, that’s the concept, and wanted to get that out of the was to talk about temperature specifically. Temperature scales we commonly use everyday to speak about the weather are measurements relative to some reference value. In the Celsius scale, for example, the reference value is the freezing point of water, or 0 °C. Fahrenheit uses 32 °F as the freezing point for peculiar scientific historical reasons. All measurements are made relative to these reference values, for example in speaking of above or below freezing. The Celsius scale and the Fahrenheit scales are relative temperature scales and can have both positive and negative numbers. This is not so with an Absolute temperature scale, which has only have positive numbers. The Kelvin scale and the Rankine scale are absolute temperature scales. The Rankine scale, in which the degree intervals are equal to those of the Fahrenheit scale and in which 0° Rankine (R) equals −459.7° Fahrenheit. The Kelvin scale, in which the degree intervals are equal to those of the Celsius scale and in which absolute zero is 0 degrees Kelvin and the triple point of water has the value of approximately 273 degrees Kelvin (K). The triple point is the temperature and pressure where the three phases of a substance (solid, liquid and gas) are in equilibrium. The triple point of water, 273.16 K at a pressure of 611.2 Pa, is chosen basis of Kelvin definition.

I explained all this (more than you the reader maybe wanted to know) because it is important to understand the difference between relative measurements and absolute measurements. This is important in scientific discussions describing values of things like pressure and temperature. So, we now have a better feel for things like gauge pressure vs absolute pressure and relative temperature versus absolute temperature. This allows discussion of the Ideal Gas Law that has been in the news. A good description of an ‘ideal gas’ is as follows: “An ideal gas is defined as one in which all collisions between atoms or molecules are perfectly elastic and in which there are no intermolecular attractive forces. One can visualize it as a collection of perfectly hard spheres which collide but which otherwise do not interact with each other.” The Ideal Gas Law is characterized by three variables: absolute pressure (P), volume (V), and absolute temperature (T) and written as:

$PV = nRT$

where n = number of moles of the gas, with each mole (abbreviated as mol) containing 6.02214129 × 10²³ atoms or molecules, known as Avogadro’s number. R is the gas constant (known as universal or ideal gas constant) universal having a value of 8.314 J/K-mol or 10.731 ft³ -psi -lb/R-mol. As will be seen, we won’t need these constants, but just note them for reference. The utility of this equation is that one can hold any variable constant and see how the others change. In the case we want to examine, our container is a football, and we can tentatively take the volume to be constant and see how the pressure changes with a known temperature change.

Before proceeding, let’s review the scenario that transpired in Foxborough, MA on the afternoon and early evening of Jan 18, 2015 before and during the AFC championship game between the New England Patriots and the Indianapolis Colts. The timeline of events can be found here – Timeline: Key Deflategate events probed in Wells Report. Key events needed for calculations are that footballs were checked at 3:45 pm and found to be at or above the minimum 12.5 psi gauge pressure, though they were not recorded. At 8:28 pm during halftime, the footballs are retested and found to be below psi specifications. The exact measurements of the under inflated footballs can be found here – Finally, the halftime PSI numbers are known. A couple of additional pieces of information are needed: (1) the temperature at which the footballs were and the atmospheric pressure in Foxborough at that time and (2) the field temperature and atmospheric pressure when the footballs were taken off the field. Data is available about the weather conditions and can be found here – Foxborough Weather Conditions (Jan. 18, 2015). This is actually data from Norwood, MA about 20 miles away from Foxborough. Only a guess can be made of the locker room temperature (say 70° F or 294.26 K) before the game, but the atmospheric pressure was approximately 29.9 Hg (or 14.686 psi) from the weather data. During halftime the temperature was approximately 50° F (or 283.15 K) and the atmospheric pressure at 29.6 Hg (or 14.538 psi). We have the needed data now for some calculations on all the footballs, but let’s play with the ideal gas law, and assume a constant volume for the football at time 1 (3:45 pm) and time 2 (8:35 pm), that is V₁ = V₂ and

$P{_1}V = nRT{_1}$

$P{_2}V = nRT{_2}$

Combining these equations, V, n and R cancel out and we are left with:

$P{_2} = {{T_2}\over{T_1}}P{_1}$

I ran the numbers, correcting gauge pressure to absolute pressure from the weather data and using the Kelvin scale temperatures, for both alternate referee measurements (Piroleau and Blakeman). Initial minimum pressure was assumed to be 12.5 psi and 13.0 psi for the Patriots and Colts footballs, respectibvely, at T =294.26 K and atmospheric pressure of 14.686 psi (with absolute pressure being 27.186 psi), while final temperature was 283.15 K with an atmospheric pressure of 14.538 psi. The results for the final absolute pressure on the 11 Patriots footballs and 4 colts footballs are:

Patriots Footballs, based on initial pressure of 12.5 psi

Colts Footballs, based on initial pressure of 13.0 psi

Taking the average of the two alternate referees measurements, gas law results and difference between the them, results in the following:

Patriots: Initial pressure = 27.186, average halftime measured pressure = 25.836, gas theory pressure = 26.10, average ∆P = -0.324
Colts: Initial pressure = 27.686, average halftime measured pressure = 27.069, gas theory pressure = 26.64, average ∆P = 0.429

This means that the actual alternate referee measurements and the gas law agree to within 0.324 psi with the actual measurement for the Patriots footballs, being slightly lower than what the gas law would predict, while they agree to within 0.429 psi for the Colts footballs, being slightly higher than what the gas law would predict . Said in another way this is only about 0.32 to 0.43 in 27, or approximately 1.2 to 1.5% difference in measurement and theory. The total average pressure drop, gas law aside, for Patriots footballs is 1.35 psi and for Colts footballs is 0.617. Both teams show an average drop in pressure, so something happened to both teams footballs that caused them to measure lower pressure. With that said, it is also curious that Prioleau’s measurements are consistently higher than Blakeman’s measurements for Patriots footballs, while Prioleau’s measurements are consistently lower than Blakeman’s measurements for Colts footballs. I don’t understand this unless they switched gauges between measurement of Patriots and Colts footballs. This appears to be the case. There are a lot of unknowns here: Initial pressures of the footballs before the game were never recorded, the initial temperature in each locker room is not known, the time between when the footballs were taken off the field and when they were measured is not precisely known, and the football pressures at games end were presumably not measured. In addition, only 4 Colts footballs were measured because referees ran out of time according to the Wells Report, implying the Colts footballs were measured after the Patriots footballs, which may have given them more time to warm up. What was the time gap from officials going from the Patriots to Colts locker room? All we really know is that halftime was around 13.5 minutes, where measurements and reinflation took place.

Based on this analysis the conclusion would be (from a scientific point of view) that the footballs were not tampered with and pressure differences are partly explained by the Ideal Gas Law. Hooray for physics! The footballs were re-inflated at halftime, but it doesn’t see that anybody bothered to measure them again at the end of the game. Nevertheless, the Wells Report seems to reject the Patriots explanation using physics. The scientific analysis of “Exponent”, the consulting firm used in the Wells report, seems thorough. However, the Wells Report may have cherry picked what they wanted from the scientific report by Exponent to phrase what they wanted to say. A key statement in the Wells Report is: “Exponent concluded that, within the range of likely game conditions and circumstances studied, they could identify no set of credible environmental or physical factors that completely accounts for the Patriots halftime measurements or for the additional loss in air pressure exhibited by the Patriots game balls, as compared to the loss in air.” True, but both teams footballs lost pressure when measured at halftime, and the Patriots footballs measured 0.733 psi lower in lost pressure than the Colts footballs, according to my analysis. The Wells Report makes this to be ~ 0.7 psi. Interestingly, the difference in pressure of the footballs explained by the Ideal Gas Law -0.324 for the patriots and 0.429 for the Colts, an absolute difference of 0.753 psi. The point is that both teams have a pressure discrepancy that has to be explained by something. Instead, the Wells Report states, “According to our scientific consultants, however, the reduction in pressure of the Patriots game balls cannot be explained completely by basic scientific principles, such as the Ideal Gas Law, based on the circumstances and conditions likely to have been present on the day of the AFC Championship Game.” So, what about the reduction in pressure of the Colts footballs? What is that explained by? This is not thorough, unbiased science as presented in the Wells report. Somebody should scrutinize the Wells Report more, as it’s full of assumptions and goes so far as to say science does not explain the Patriots footballs pressure drop. If true then by the same token, what has caused the pressure drop in Colts footballs, and science must not be able to explain that either?

Well, the NFL punishment has been doled out and it seems more about Patriots lack of cooperation in the investigation now, or more specifically, Tom Brady’s participation. I can’t say I blame him for not cooperating in today’s hypersensitive society where every little detail is scrutinized and people are presumed guilty until proven innocent. Maybe the Patriots real flaw is a culture of trying to gain a competitive edge without actually breaking any rules. It’s a grey line on morality, but part of sports, past and present. Some of the ways players try to get an edge up seem based more on psychology or physiology than physics. Athletes will be athletes and rely on brawn more than brains most of the time. Coaches or managers on the other hand often know more than they admit to, but it’s like protecting the commander in chief and players do that as they should. Maybe the Patriots did tamper with the footballs, but maybe they didn’t – it’s a stretch at best (without complete data) to conclude they did. To single out Tom Brady just seems unfair to me. He seems an honest guy and others say that of him too. His character is at stake here and I hope he come out on top in challenging the Wells Report!

Enough said, and here are some facts about footballs:

The NFL rule says it must be a prolate spheroid. It has a volume V = ${4\over 3}{\pi}{a^2}c$ , where a is half the length of the long axis and b is half the length of the short axis, called the major and minor axes, respectively. Since 1959, the inch has been defined and internationally accepted as being equivalent to 25.4mm, so with 1″ = 2.54 cm, an NFL regulation football is 27.94 to 28.575 cm along the long axis (giving b = 13.97 to 14.2875 cm) and 53.34 to 53.975 cm around the short circumference, which requires using the formula for circumference = $2{\pi}r$ (giving a = 8.489 to 8.59 cm). Using these numbers in the formula for an oblate spheroid to be V = 4216.95 to 4416.02 cm³ (or 257.33 to 269.48 square inches).

A football weight is 14 to 15 ounces. With 1 ounce = 28.349 gm, a football is 396.886 to 425.235 gm. It is generally assumed that the air in a fully inflated football accounts for only about 10 grams of its mass. Is this true? Assuming a gauge pressure of 13 psi (89632 Pa) at standard temperature and pressure (T = 273.15 K and P = 101325 Pa = 14.69595 psi) gives the absolute pressure of the football to be and knowing the volume of a football in addition to the molecular weight of O = 15.9994 and molecular weight of N = 14.0067, we can figure it out. By weight, dry air contains 23.2% O₂ and 75.47% N₂ by weight, which accounts for 98.67% of the weight of air. The actual major constituents of air are shown below:

Gas	Ratio compared to Dry Air (%)		Molecular Mass – M – (g/mol)	Chemical Symbol	Boiling Point
Gas	By volume	By weight	Molecular Mass – M – (g/mol)	Chemical Symbol	(K)	(^oC)
Oxygen	20.95	23.20	32.00	O₂	90.2	-182.95
Nitrogen	78.09	75.47	28.02	N₂	77.4	-195.79
Carbon Dioxide	0.03	0.046	44.01	CO₂	194.7	-78.5
Hydrogen	0.00005	~ 0	2.02	H₂	20.3	-252.87
Argon	0.933	1.28	39.94	Ar	84.2	-186
Neon	0.0018	0.0012	20.18	Ne	27.2	-246
Helium	0.0005	0.00007	4.00	He	4.2	-269
Krypton	0.0001	0.0003	83.8	Kr	119.8	-153.4
Xenon	9 10^-6	0.00004	131.29	Xe	165.1	-108.1

So, we can be a little more exact and include carbon dioxide (CO₂) and argon (Ar) to account for 99.99% of air composition by weight. Adding the numbers scaled by weight fraction gives:

Molecular weight of air = (0.7547 x 28.02) + (.2320 x 32) + (.0128 x 39.94) + (.00046 x 44.01) = 29.1 g/mol

The volume of a football we know is 4216.95 to 4416.02 cm³ , so take V = 4316.5 cm³ as the average. Using the Ideal Gas Law we can calculate the number of moles of air in a football at standard temperature and pressure as: n = PV/RT = [(190957 Pa) x (0.0043165 m³)]/[(8.314 J/K-mol) x (273.15 K)] = 0.362 moles. Each mole has Avogadro’s number of 6.02214129 × 10²³ molecules. With the molecular weight of air as 29.1 g/mol, we find that a football has and incredible 2.18 × 10²³ air molecules (more that 2 billion billion) that add to approximately 10.5 gm. So, indeed it is true, and air only accounts for ~ 2.5% of its mass. We may conclude from this that the NFL source who reportedly told Kravitz (Bob Kravitz at WTHR in Indiana) that “officials took a ball out of play at one point and weighed it”, and would investigate the deflation of footballs by the Patriot’s, is sort of half-baked. Kravitz broke the story and posed it as possible cheating. That’s how these things start and take on a life of their own. Anyway, that’s all I have to say or share on this matter, and a wild one it is for sure. Here are some other links I can offer:

Patriots football pressure links for further reading:

1.) But What about the Ideal Gas Law?
2.) Can physics explain the deflation of NFL footballs (aka #DeflateGate)?
3.) Deflation Experiments Show Patriots May Have a Point After All
4.) Clearing Up the Air on DeflateGate
5.) Deflate-Gate: Could Weather have played a role in deflated balls?
6.) Science! One Way To Deflate a Football Without Even Touching It
7.) How to reduce the amount of air in a football without letting any air out
8.) Football Physics: The Science of Deflate Gate
9.) So the Patriots really were playing small ball

Physics links for further reading:

1.) Ideal Gas Law – Hyperphysics
2.) Gas law calculators – WebQC
3.) Humid Air and the Ideal Gas Law – The Engineering Toolbox
4.) Heat and Heat Vs. Temperature – Online Physics Tutorials
5.) Calculate mass of air in a tyre from pressure – Physics StackExchange

Cogito Ergo