One of the more farcical elements of a typical NFL game is when the referees call for a measurement to determine whether a play has gained enough yards to earn a first down. The position of the ball on the field is determined by referees several yards from the play trying to guess the precise position of the ball when the play ended. Then they bring out a set of markers connected by a ten-yard chain, and see whether the ball has covered the necessary ground. It's not infrequent for the ball to come up short by mere inches.
Of course, given the inherent imprecision involved in the "guess where the ball was when the guy carrying it got tackled" step, this is kind of ridiculous. And, of course, the approximation of the tackle position is followed by the bit where the referee marking the location of the tackle will toss the ball halfway across the field to another official who will attempt to match the marked position from a significant distance away. Then the back marker of the first-down chains will be aligned with the spot of the ball by another official who is standing on the sidelines. None of these positioning steps are anywhere near good enough to justify a measurement that turns on a few inches, and yet, whole games can turn on this pretense of precision.
This being 2015, of course, it's reasonable to ask whether you could find a better way of doing this using technology. Is there a high-tech way to improve the determination of the ball's position on the field in a way that doesn't use wild guesses as a basis for false precision?
This is a fairly difficult problem, because of all the other things that are on the field. If it were simply a matter of moving the ball a particular distance, you could imagine using lasers or video cameras to determine the location. A football game involves two teams of 11 players out on the field as well as the ball, though, and that makes it nearly impossible to get a clear line of sight to find the position of the ball (as seen time and again when officials try to use instant replay to determine the correct call).
We could, however, make use of the same sort of technology that enables smartphone navigation, namely the Global Positioning System (GPS). Which, as it turns out, involves quite a bit of physics.
Locating Schenectady, New York, by a method analogous to GPS. We're a three-hour drive from Boston,... [+]
GPS is based on a constellation of satellites in orbit around the Earth, each broadcasting a coded signal giving the time according to that particular satellite. A GPS receiver picks up several of these signals-- you need at least four for good results-- and determines the position by comparing the times signals from the different satellites.
This might seem like it has nothing to do with navigation, but in fact, it's not that unusual-- we frequently talk about distance in terms of travel time. We can, for example, give someone a good idea of the location of Union college, where I teach, simply by specifying travel time to other landmarks. We're about a three-hour drive from Boston, for example, which places us on the green circle in the figure above. That circle goes through Schenectady, New York, but also places in Main, New Hampshire, Vermont, and Connecticut, so it's not sufficient information to really nail things down. Giving the additional information that we're a three-hour drive from New York City, though, puts Schenectady at a point on the blue circle, and that narrows our location down to one of the two points where the blue and green circles cross-- either Schenectady, or a point in the North Atlantic Ocean. Common sense can narrow that down, or you can remove the need for thought by adding the final bit of information that we're four hours' drive from Montreal, on the purple circle, and there is exactly one point on the surface of the Earth where all of these cross.
GPS works the same way. The difference between time signals from the different satellites tells your GPS receiver how long the radio waves from each satellite spent getting to the receiver. This, in turn, tells you how far away from the satellite you are, because relativity tells us that light travels at a constant speed, so the time divided by the speed gives the distance. And the orbits of the satellites are very well known, so knowing the distance from each of three satellites specifies your location as a single point on the surface of the Earth.
Of course, the speed of light is very, very fast-- 299,792,458m/s, or very nearly one foot per nanosecond in American units-- so the timing accuracy needs to be very good for GPS to work correctly. This is accomplished using quantum mechanics-- each of the GPS satellites contains an atomic clock (actually multiple clocks, for redundant backup), that marks the passage of time according to the oscillation of microwaves set to the characteristic frequency of the atoms inside the clocks. Every cesium atom in the universe moves between two particular states by absorbing and emitting light of a single frequency determined by quantum physics, and we define the second as 9,192,631,770 oscillations of that light. The atomic clocks in the GPS satellites serve as a reference to ensure that the microwaves are at exactly the right frequency, and thus provide the necessary timing precision for navigation.
But there's even more physics involved in this, because the GPS satellites are orbiting 20,000 km above the Earth, which means they tick slightly faster than a clock on the ground, thanks to General Relativity. And the satellites are moving at high speed in their orbits, which causes them to tick a tiny bit slower thanks to Special Relativity. The combination of these means that a GPS clock would gain 34 microseconds (0.000034 seconds) per day, if the scientists and engineers who built the system didn't understand and correct for the effects of relativity. That doesn't sound like much, but it would be an error of around 11km per day, so the fact that your smartphone can locate you on the Earth to within a few meters using GPS is tangible proof that the time-bending effects of relativity are real, and properly understood by physicists and engineers.
GPS alone isn't sufficient to remove the need for football's silly little measurement dance with the chains-- a good commercial GPS unit is only accurate to a few meters, far too large an error for determining whether a play has resulted in a first down-- but you could imagine adapting the basic idea to find the position of the ball. Mounting a small radio transmitter on the ball, say, and placing receivers at the corners of the field to pick up that signal would let you determine the ball's position using the same time-to-distance conversion that GPS does. If you know how long it took radio waves from the ball to reach each corner of the field, you can determine the distance from the ball to that corner, and putting those signals together will determine the exact position of the ball.
The timing requirements here would be pretty stringent, though. If you want the position of the ball to within an inch, you need to know the travel time to within about 0.08 nanoseconds (0.00000000008s), which is a bit of a challenge. And that's before you deal with issues of filtering out reflections off different part of the stadium, possible delays from passing through large masses of players, and technical issues about powering the ball-mounted transmitter. To say nothing of the possibility that such a system might be meddled with by coaches like the New England Patriots' Bill Belichick. (And whether Belichick would have the engineering acumen to fiddle with GPS signals or not, you know that fans and coaches every other team would believe that he had...)
Those are problems that could be worked out with a bit of work-- physicists and electrical engineers are pretty ingenious. But until the NFL decides to start sinking some of its billions into research grants for atomic clocks and signal processing, we're probably stuck with the dingbat kabuki of "bring out the chains."
