Over at Grantland, Bill Barnwell offers some unorthodox suggestions for replacing the kickoff in NFL games, which has apparently been floated as a way to improve player safety. Appropriately enough, the suggestion apparently came from Giants owner John Mara, which makes perfect sense giving that the Giants haven’t had a decent kick returner since Dave Meggett twenty years ago, and their kick coverage team has lost them multiple games by giving up touchdowns to the other team.
Anyway, one of Barnwell’s suggestions invoked physics, in a way that struck me as puzzling:
Idea 3: The receiving team’s returner is handed the ball on his own goal line. His blockers must be positioned on the 20-yard line. The “kicking” team’s players are positioned on the opposition’s 40-yard line. Once the whistle blows, it’s a traditional kick return.
Advantages: Everyone on the field will still be colliding, but because the kick-coverage team will have been running for 20 yards as opposed to 45, there won’t be anywhere near as much momentum in those collisions. That should produce fewer injuries.
Now, if you know anything about introductory physics, you know that momentum is mass times velocity (for speeds much less than the speed of light). This doesn’t have any direct relationship to the distance somebody has run to get to that point, unless they’re accelerating the whole time. But it seems awfully unlikely to me that any of the whack jobs covering kicks in the NFL are actually speeding up appreciably between 20 and 45 yards– they probably hit their top speed well before that.
Ah, but is there a way to use our knowledge of introductory physics to test this idea? That is, can we estimate the distance over which an NFL player is likely to be accelerating? Well, I would hardly be posting this if I didn’t have a model for this sort of thing, so let’s have a run at it (heh).
This may seem like a ridiculously complicated problem, but in the fine tradition of physicists everywhere, we can drastically simplify the scenario to make the math work nicely. Let’s assume that our runner starts from rest, and accelerates up to his maximum speed, then covers the rest of the ground at that maximum speed. We can follow the same basic procedure used during the scientific commuting posts I did earlier, and find an expression for the time required to cover some distance in terms of the distance, the maximum speed of the runner (vmax), and the runner’s acceleration (a):
(There’s an extra factor of two in this equation that wasn’t in the scientific commuting posts because the runner in this case doesn’t need to slow down to a stop– he hits the ball carrier at maximum speed.)
So, what are vmax and a? Beats me, but if we had a list of times taken to run various distances, we could figure it out. And, hey, we do have such a list: the list of track world records. Yeah, fine, those are set by people who train to be the absolute fastest runners in the world, which isn’t quite the same thing as training to play football at the highest level. But NFL players are world-class athletes, right? And anyway, this will at least let us get an upper limit on vmax and a, which in turn will let us determine whether that extra 25 yards would make any difference.
So, let’s take the world records for the flat-out sprint events, ans see what they give up. From outdoor track, we can use the 100m and 200m dashes (the 400m is starting to add some extra endurance factor into the works, as you can see from the fact that the 4×100 relay record is significantly faster than the individual 400m record), and from indoor track the 50m and 60m dash. And, just for fun, we can throw in the fastest recorded 40-yard dash from the NFL combine, which appears to be 4.24 s (that’s 36m, give or take). If we plot all of these on a graph, we get the following:
The solid line here is a fit to the data, and you can see it does pretty well (an R-squared of 0.9987, if you want to be quantitative). The 40-yard-dash time is right in line with the others– leaving it out doesn’t change anything significantly.
From the equation above, we expect this to be a straight line, so finding a nice straight line on the graph is encouraging. The fit routine spits out values for the slope and intercept of this line, which are 0.0910 ± 0.0019 s2/m and 0.87 ± 0.20 s, respectively. By looking at the equation above, you can work out that vmax=10.99m/s and a=6.35m/s2 (do this for homework, if you like, and send it to Rhett Allain for grading…).
Do those values make sense? They’re within a reasonable-seeming range. If you look at cars and use the 0 to 60 times to estimate their acceleration, a typical car has an acceleration of about half this value (though it can sustain that far longer), and a top sports car has an acceleration of about twice this value. That top speed would correspond to a 100-meter-dash time of 9.09s, significantly below the world record, but not absurdly so. So these aren’t ridiculous numbers, at least.
So, given that max speed and acceleration, how much ground would a runner cover while accelerating? Well, trivial math tells you it would take a bit more than 1.7 s to reach top speed, during which time they would cover 9.51m, or 10.4 yards in NFL units. So, if you threw Usain Bolt out on the field, he would be moving at his maximum speed well before the 20-yard limit Barnwell suggests, let alone the 45-yard range of the current kickoff.
Of course, NFL kick coverage guys aren’t Usain Bolt, and they’re out there with a whole bunch of padding and whatnot, so they won’t accelerate quite as fast. But for a player to be accelerating for the full 20 yards, you’d need to drop the acceleration by almost half, to 3.3 m/s2 (assuming the same top speed, for simplicity). That seems a little too big to be realistic– I might believe a reduction to 5m/s2, but not dropping it by half.
So, would shortening the distance players have to run actually make kickoffs any safer? Not by reducing their momentum, it wouldn’t. You might get a reduction in the violence of the typical collision because the restricted field would make it easier for blockers to get in the way and slow things down. You’d also have less room for the guy who comes flying in completely unseen until the collision, which is where the biggest hits come from, but that’s a spatial effect, not a question of momentum. It’s a completely different branch of physics.
(Bonus Question: Using the simple model presented above, explain why this defensive track-and-field site is full of crap when they try to explain the apparent difference between NFL combine times and world-class track times. Leave your answer in the comments, or send a 300-word essay on the topic to Frank Noschese for grading…)
(Somebody has probably put a radar gun on a real sprinter and measured how their velocity changes over the course of the race, but I’m too lazy to look that up. It wouldn’t surprise me if there was something more like an asymptotic approach to a slightly higher top speed, so they accelerate quickly at the start, but more slowly afterwards, but keep speeding up for a longer time than the simple constant acceleration model would suggest. That doesn’t make a good introductory physics example, though…)