Science Is All Like, ‘Ooh Ooh We Can Explain Deflate-gate With Our Number Magic!’
At the core of Deflate-gate is the question of whether the pressure inside 11 of the Patriots' 12 game balls was purposefully under-inflated, or simply a function of nature exacting its toll on three-dimensional objects containing a gas.
There are a few things we know about footballs and the physics of air pressure.
1) Footballs are permeable, meaning they lose air over the course of a game.
2) Footballs with less air pressure are easier to grip.
3) Footballs with less air pressure have less mass, and thus do not travel as far.
4) Temperature effects the pressure inside a football.
Assuming the referees did, in fact, measure psi prior to kickoff of Sunday's AFC Championship game, all 12 balls must have started out with more than 12.5 pounds per square inch. If those balls were switched out or manually deflated by equipment managers -- or at least 11 of them -- then none of this science stuff matters because the Patriots would've been cheating by defrauding the officials' measurements. But if they weren't, there's only one reasonable explanation: the balls naturally deflated.
Considering that any tampering accusations can only be proven with video evidence (or an open and shut scientific proof that 11 footballs cannot lose two psi over the course of a half), let's look into the possibility that the weather at Foxborough could've decreased the pressure inside the under-inflated balls at the point in which they were re-measured at halftime.
One assessment of Deflate-gate concludes that balls measured in 70-degree room to be 12.5 psi two hours prior to the start of the game could decrease by as much .4 psi by kickoff.
[WUSA9] Let's assume that each ball was inflated to the minimum pressure required to meet the NFL rules regarding proper inflation: 12.5 psi. We convert psi (English) to pascals (Metric), which comes out to 86,184.5 Pa and assume room temperature (68ºF/20ºC) which converts to 293.15 K (Kelvin, the Metric equivalent). We now have,
86,184.5 Pa / 293.15 K = p2 / T2.
We're down to two variables. But we also know the temperature on the field at the start of the game was reported as 51ºF/10.6ºC (283.15 K). Plug it in...
86,184.5 Pa / 293.15 K = p2 / 283.15 K
Neat! Look, we're left with a solvable equation with one variable, p2, which is the pressure of the air inside the ball at game time! Let's solve this riddle...
Isolate the lone variable:
(86,184.5 Pa / 293.15 K) * 283.15 K = p2
83,244.6 Pa = p2 ---> 12.1 psi
83,244.6 Pa is 12.1 psi, so, according to our calculations, the balls could have been under-inflated by 0.4 psi on the field. This makes sense given the very first equation, which shows that a decrease in temperature would force a decrease in pressure, assuming the same volume of air in the football.
Admittedly, I don't know if I could feel the difference between a ball that has a pressure of 12.5 psi and a ball at 12.1 psi, but by the letter of the law and according to our calculations, the balls could have been under-inflated after having cooled down to the temperature on the field of play.
The question then becomes why this hasn't been an issue of this magnitude before, especially in games played at colder weather stadiums like Lambeau Field or Soldier Field? No one complained about the ball when the Giants squared off against the Packers at Lambeau in January of 2008, when the game time temperature was -1ºF. Granted, the Giants won, but plugging in the same numbers with a -1ºF/-18.3ºC value for the outdoor temperature, T2, we'd have a ball that was down to 10.9 psi.
A science teacher on Reddit concluded that, given the rainy conditions, temperature and barometric pressure, the pressure inside the footballs could've decreased by as much as .7 -- which is still far less than the Chris Mortensen's report that the balls were short by 2 psi.
[Reddit] Given the conditions of the game, a ball which meets specifications in the locker room could easily lose enough pressure to be considered under-inflated. Some math:
Guy-Lussac's Law describes the relationship between the pressure of a confined ideal gas and its temperature. For the sake of argument, we will assume that the football is a rigid enough container (unless a ball is massively deflated, it's volume won't change). The relationship is (P1/T1) = (P2/T2), where P is the pressure and T is the temperature in Kelvins.
The balls are inflated to between 12.5 and 13.5 psi at a temperature of 70 degrees Farenheit (294.1 K). Let's assume an average ball has a pressure of 13 psi. Since these are initial values, we will call them P1 and T1.
The game time temperature was 49 degrees F (278 K). We are attempting to solve for the new pressure at this temperature, P2. We plug everything into the equation and get (13/294.1) = (P2/278). At the game time temperature, the balls would have a pressure of 12.3 psi, below league specifications.
*Furthermore, given that it was raining all day, the air in the stadium was saturated with water vapor. At 70 degrees, water has a vapor pressure of 0.38 psi. The total pressure of the ball is equal to the pressure of the air inside the ball and the vaporized water in the ball. At 49 degrees, the vapor pressure of water is 0.13 psi. Up to 0.25 additional psi can be lost if the balls were inflated by either the team or the refs prior to the game. Granted, it's unlikely that anyone would inflate balls from 0, but it easily could cost another couple hundredths of a psi in pressure.
For a ball that barely meets specifications (12.5 psi), it's pressure would drop to 11.8 psi during the game... enough to be considered massively underinflated.
NPR's All Things Considered had two physicists on -- one of which was a professor of the science of sports -- to explain the potential effect weather may have played on the pressure inside the ball.
"You just can't imagine the ball being under-inflated for a significant portion of the game and the referees not noticing" John Eric Goff of Lynchburg College said.
Sure, that doesn't sound all that scientific, but it's the greater point that these very smart people who know how nature works aren't totally convinced foul play was necessarily involved.
All in all, it's hard to conclude the impact of the environment on the pressure inside a football without knowing the exact conditions at the time in which the balls were measured, both before the game and after a Colts' player picked a Tom Brady pass and notified the referees that the ball felt soft. That being said, it's very possible that the pressure inside the balls was affected by the weather -- but to the extent that is being reported (a decrease of two psi), that seems like a bit of a stretch.
The Weather Channel's Mike Bettes, a meteorologist, seems to agree with that last sentiment.
— Mike Bettes (@mikebettes) January 21, 2015
And hey -- wouldn't weather have affected the Colts' footballs too? Were they under-inflated?
MORE DEFLATE-GATE COVERAGE:
- NFL Finds 11 of Patriots' 12 footballs were deflated
- Twitter reacts to Chris Mortensen's Deflate-gate report
- Graham Gano accidentally explains how Pats' balls could've been under-inflated
- Colts' cornerback asks to replay AFC Championship game
- Tom Brady laughs off cheating accusations
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