The Physics of Sprints and Kickoff Safety

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 4x100 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 &plusmn 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...)

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But -- the proper measure is ~10 yards, not twenty, as the line of blockers is 20 yards from the coverage team, and one must assume they're not going to just stand around. That said, they'd be very near the end of their acceleration, anyway.

I think the logic is that the big and dangerous hits come from guys who are unblocked for whatever reason, and those would have a 20-yard run (as would the ball carrier starting from the goal line). But that's a guess.

It's pretty obvious to runners and bicyclists that they accelerate faster than cars from a stop. But it never has been clear to me whether runners or bicyclists are the top in acceleration from a stop.

I just watched a video of Usain Bolt win the 100-meter dash in Berlin in 2009:

I used marks along the track's surface to measure his progress, and counted frames in the video to measure time. It's clear that he doesn't reach his top speed until at least 30, perhaps 40, meters into the race.

Watch for yourself and see.

By Michael Richmond (not verified) on 17 Apr 2012 #permalink

Russell @3: My guess is that the runner, and to a lesser extent the bicyclist, start accelerating sooner than the car does. There is a minimum amount of time needed to react to a signal, which is ~0.1 seconds (in foot races, this fact is used to detect false starts). A runner does not need any more time than that to start accelerating. A cyclist needs to move his foot from the ground to the pedal; if he's good (and has toe clips on his pedals) he can pedal with his other foot while moving the first from the ground to the pedal. A motorist has to move his foot from the brake to the accelerator pedal (as well as shift into gear and release the clutch if he is driving a manual transmission) and then wait a finite amount of time for the engine to respond to the additional fuel supply.

Once they start accelerating, the car is of course faster. But the delay in response time will be enough to tip the odds in favor of the runner at very short distances (less than ~50 m).

By Eric Lund (not verified) on 17 Apr 2012 #permalink

Can I post links? There's a PDF here with a table of Usain Bolt's split times for every 10m, indicating he reached top speed at ~77m.

I had always understood that most sprinters hit max at ~30m, straighten up and finish the race at constant speed, but world-class sprinters are those who straighten up ... and continue to accelerate.

By Lurker #753 (not verified) on 17 Apr 2012 #permalink

A quick web check of reports of the measured actual peak velocity position for sprints over 100 m turns up figures of around 35 m for beginning sprinters and around 70 m for the likes of Usain Bolt. They all decelerate over the last 20 m of the race. As you'd expect, the highest acceleration occurs earlier, but even beginners are accelerating near peak values to 20 m, and the elite are still burning rubber out to 50 m. NFL players are good athletes and likely fall at least in the middle of this range. Even in ideal circumstances they will reach their top velocity later than you have proposed, but there are other factors. NFL players do not launch as well as sprinters do for reasons of equipment, technique and intention and likely reach their maximum velocity relatively later than equivalent athletes in a pure sprint situation. The position of the proponents of this rule change is likely to be more correct than you've estimated.

Eric, the greater acceleration of the runner and bicyclist is not just reaction time. They can wait until they see the car has started moving, and still jump ahead. For a short distance. Cars just don't accelerate that quickly. Their forte is that they can accelerate longer and maintain a higher speed.

Cars can accelerate quickly, it's just that very few drivers actually try to max out their acceleration. A driver in a performance car doing a jackrabbit start has his acceleration capped by the coefficient of static friction for his tires, which on a 4WD vehicle will allow 9-10 m/s^2, and lets you manage 6 m/s^2 on a 2WD vehicle as long as the wheel loading is reasonable. For reference, 6.35m/s^2 is a 0-60 time of 4.22s, and most cars will have their acceleration drop off sharply when they have to shift into second gear, so it's fair to say that any car with a 0-60 time of less than 6-7s will have no real trouble keeping up with even an olympic athlete if the driver really tries.

"The winner is the runner who slows down the least"

A great sprinter's comments on the 100m.

Sorry I can't remember who he was;-)

I read the Grantland article and thought the same thing. Most of the acceleration occurs right away. Halving the distance that the players run before colliding would likely not change the collision momentum much. In fact, I wouldn't be surprised if the players are running faster after 20yds than 40, since they will have had to sprint 20yrds less. This seems to be potentially supported by the Bolt data, when you account for the ways in which NFL players are not Usain Bolt - they are probably slowing down much earlier than 80m. I can't understand why you don't just change the rules so players don't block and tackle with their heads! In what other sport is there so much contact with the head? Boxing and Hockey come to mind, which are hardly good models to look at for reducing concussion. It seems to me that scrimmage line blocking should be shoulder to shoulder (like a rugby scrum) and tackles in the field should be below the waist. Get rid of the helmets and see how many players still want to block with their heads!

By John Evens (not verified) on 18 Apr 2012 #permalink

As said above by several people, players are unlikely to maintain maximum speed over the last 25 yards, so maybe it would be safer to kick off from the 10 yard line, so players would be travelling slower when they collide.

The big difference between 40 yard times and 100m times is that in the 40 yard dash, the timer reacts to the person starting, whereas, in the 100m, it is the runner who reacts to the gun.

Cars react slower because they don't have the low speed torque of a bicyclist. Runners are even more advantaged. (We were made to run, not bike). They simply have to push straight off rather than having to conform to the bicycles crank. Greater force over short distance.

Using Orzel's calculation of an acceleration of 6.35m/s, we get a 0 to 60mph time for a runner of 4.2s, which is better than all but race cars and dragsters. I imagine the peak acceleration over the first 2-3 m is even higher. However, the force of human muscle contraction drops off quickly with speed so we humans reach near top speed quickly.

So, whatever the initial advantages for the runner and then the cyclist they are reversed quickly

Another relevant fact is that, for kickoffs, the players on the kicking team start running from 10 yards back, so they have gained some speed before the play starts.