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26 January 2015

Can physics explain what's wrong with Tom Brady's balls?

We're going to find out. Picking up on the speculation offered by a Boston sportswriter, who argued that weather alone was responsible for half of the underpressurization found in 11 of the 12 balls that the New England Patriots' offense brought with them to the 2015 AFC Championship game, we thought we'd check their math.

It's a good thing that we did, because they botched it. Badly.

We know that because of the physics that applies here, the combined gas law, which defines the relationship that exists between the internal pressure, volume and temperature of an enclosed vessel like a football will be a constant. That means that we can nearly perfectly predict how much the internal pressure, or inflation, of a football might change when the temperature of the ball changes from one point in time to another, using the following formula:

We're going to do that math using the information we know about the situation. We'll assume that the footballs were inflated at a temperature of 68 degrees Fahrenheit, which is consistent with the interior thermostat setting recommended by the U.S. Department of Energy for optimal personal comfort and low energy consumption in the winter, the gametime temperature of 51 degrees Fahrenheit, the minimum acceptable inflated pressure specified by the NFL's rules of 12.5 pounds per square inch, and the approximate internal volume of the football of 258.5 cubic inches, which we'll assume is unchanged from the time when the footballs were filled and checked to when they were discovered to have deflated by roughly 2.0 pounds per square inch.

Enough talk - let's do some math using the tool we built just for that purpose!

Update 30 January 2015: We've added the standard sea level atmospheric pressure to the NFL's pressure gauge readings in the tool below, as per the New York Times' article proclaiming that the weather was responsible for the deflation of Tom Brady's balls. Judge for yourself just how well that explains the situation.

Football Inflation Data Before the Game
Input Data Values
Initial Measured Internal Pressure [psi]
Initial Volume [cubic inches]
Initial Temperature [degrees Fahrenheit]
Football Inflation Data at Halftime
Internal Pressure at Halftime [psi]
Volume at Halftime [cubic inches]
Temperature at Halftime [degrees Fahrenheit]

How Much of Football Deflation Is Explained by Weather?
Calculated Results Values
Expected Internal Pressure of Footballs at Halftime
Actual Internal Pressure of Footballs at Halftime
Percentage of Change in Pressure Explained by Change in Temperature

So we find that, at best, the change in temperature in going from a room inside the stadium where the internal pressures of the footballs the New England Patriots brought to the game were inspected to on the field during the game can only, at best, explain 20% of the change in pressure that was measured.

We were curious to find out what of temperature would have to be held in order to produce the observed amount of deflation, and used trial and error in our tool above to find that Tom Brady's balls would have had to have been inflated and inspected at a temperature of 148.4 degrees Fahrenheit to account for a decrease in internal pressure from 12.5 psi to 10.5 psi.

That, of course, assumes that the volume of the bladder inside the football didn't change size. Going back to our initial temperature range, using trial and error in our tool once again, we found that it would have to expand to 297.85 cubic inches, or 115% of its initial volume, to account for the observed change in pressure.

That is something that might perhaps seem plausible, but a reality check indicates that would involve the footballs growing to be larger than the maximum dimensional specifications mandated by the NFL. It would have been incredibly easy for even the NFL's officials to determine that something was greatly amiss with the Patriots' footballs if that were the case.

Instead, the physics involved are such that it is clear that Tom Brady's balls achieved their deflated state through other means.

And if nothing else, we have perhaps demonstrated that one cannot trust either the scientific or mathematical claims made by a hack sportswriter from Boston.

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