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The Numbers Behind the Hail Mary

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The Numbers Behind the Hail Mary

By now you have no doubt seen the ridiculous game-winning play from the Green Bay Packers and Detroit Lions game last Thursday. If you haven't seen the play, hell, even if you have seen it, take another look - or two:

Give Richard Rodgers a lot of credit for getting in perfect position to catch that pass and then actually executing and making the play. Richard Rodgers deserves credit, but the headliner on the play was that throw and for good reason. In the midst of a career of greatness, Aaron Rodgers made potentially the most memorable play in that career.

The thing that made the play so great was not just that he got the ball all the way to the end zone, it is how high he threw it to give his teammates a chance to get in position to catch it. 

Lets take a look at the numbers behind the throw.

It is a little tough to see, but in the picture above, Aaron Rodgers (circled) has his front foot at the Green Bay 36-yard line. However the ball is actually released around the 35-yard line, so for the sake of this piece, we are going to say he released it at the 35-yard line. 

Richard Rodgers catches the ball about three yards deep into the end zone while the ball was roughly nine feet in the air. 

Aaron Rodgers releases the ball at a arm angle of 61 degrees, which means the release point of the ball is going to be at about eight feet from the ground.

After Rodgers releases it, the ball is in the air for a truly impressive 4.15 seconds.

Putting together what we know now, the ball travels 68 yards down the field, was in the air for 4.15 seconds, it was released about eight feet off the ground, and the ball was nine feet in the air when it was caught.

When you factor in most punters kick the ball about 13 yards behind the line of scrimmage, that would make Rodgers' throw a 55-yard punt with 4.15 seconds of hang time, aka a pretty nice punt.

Plugging what we know now into equations for constant acceleration for projectiles we can figure out the velocity that Rodgers throws the ball at and how high the ball goes in the air.


Using those equations you find that the ball travels at 33.5 mph in the x-direction (downfield) and 45.7 mph in the y-direction (vertically). This means that Rodgers throws the ball at 57 miles per hour. I think baseball tends to skew people's idea of how fast a thrown football is. While 57 miles per hour would be a very slow pitch in baseball, the same cannot be said for a football because the mechanics are not the same, in large part to the flick of the wrist. To put that into perspective, is a NFL scouting service and they track throwing speeds at the NFL combine, the fastest they have ever recorded is 60 miles per hour.

With a flight time of 4.15 seconds and only a foot difference in height from the release point to the catch point, the ball reaches its peak height in 2.08 seconds. That means that the height of the ball at its peak is 78 feet high. The NFL rulebook calls out that stadiums have at least 85 feet of clearance above the field of play. While Rodgers did not hit that 85 foot mark on this throw, he came close, as can be seen below by the ball almost hitting the rafters at Ford Field.


That gif was taken from this fan's instagram account if you want a better view of it. If you freeze the gif while the ball is at its highest point, you can see just how close it came to hitting the rafters at Ford Field.

If a player wants to throw a ball the maximum distance they can, assuming air resistance is ignored, they would throw the ball into the air at a 45 degree angle. Obviously Rodgers threw this one a little higher to give his receivers time to get under it. Rodgers ended up throwing this pass at an angle of 53 degrees. If Rodgers would have thrown this same pass at 45 degrees instead of 53 degrees it would have flown out of the back of the end zone and either hit the wall or fly into the first couple rows of the stands. At 45 degrees the ball would have traveled an extra 14 yards for a total of 82 yards downfield. 

Using the below equation we can find the arc length of the flight of the ball and figure out just how far the ball traveled during its flight from Rodgers to Rodgers.

While the ball may have traveled 68 yards down the field, due to the height that Rodgers put on the throw, altogether the ball traveled an astonishing 105.5 total yards only to fall in front of the defense and into Richard Rodgers' outstretched arms for a game-winning and unforgettable touchdown. Well done Rodgerses.

To sum it all up, the ball was thrown at 57 miles per hour at an angle of 53 degrees and flew a total of 105.5 total yards and 68 yards down the field for a game-winning score as time was expired.

One more time:

Go Pack Go.

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Fan friendly comments only: off Comments (80) This filter will hide comments which have ratio of 5 to 1 down-vote to up-vote.

Nick Perry's picture

Awesome Mike, but I'll just stick to saying it was the most beautiful pass I've seen since the 3rd and 10 pass to Jennings in SB 45! : )

MikeReuter's picture

I'm not going to argue with that. Rodgers to Rodgers vs the Cowboys last year was pretty great too.

Nick Perry's picture

Good one!!

RCPackerFan's picture

Great Article!
So the only question I have is why are we having Masthay punting, when we can have Rodgers throwing it. :)

That was possibly the best ending to a game that I have seen.

Mojo's picture

That's the first thing I thought of when seeing it on replay. Unless they're trying to pooch inside the 20, any punter would have been delighted with 68 yards in the air and 4.15 hang-time.

Considering how much more power their is in the average leg versus arm, this was an amazing throw. With over 105 yards of total distance, hope ARod didn't hurt himself. Plus the throw is designed to go right down the numbers, which is exactly where he put It. And near perfect depth into the endzone.

hobbes's picture

I think the big catch is that Rodgers needs a running start to throw the ball the way he did. Aussie Rules Football (kinda a weird mix of American football and Rugby) has "punting" with running starts and apparently the record is 116 yards (rumored but in the stat books, the ball is bigger so that probably factors in positively as well). That would mean you could coffin kick a punt standing at the back end of your own endzone.

Docgperez's picture

great job ! I was waiting for sports science to do this.

RCPackerFan's picture

sports science did do this? I was wondering if they were going to. I hadn't seen it though.

Dan Stodola's picture

Nice enough article, except you forgot to mention one very important thing! R. Rodgers didn't do his JOB! You know the same way Brandon Bostic didn't do his JOB. He was supposed to be in the end zone boxing out so Adams could be they guy who makes the catch.

Isnt that all you guys cared about when Bostic didn't do his job? I'm very sure everyone bitched at Bostic for weeks about doing his Job. Now when R. Rodgers makes the catch, instead of doing his job you have all forgotten that the VAST majority of you don't seem to care.

See, I for one didn't mind that Bostic tried to make a play, Charles Woodson did exactly that MANY, many times. Only thing is that if you try to make a play offscript, you have to actually MAKE the play. But I'm very sure the only thing almost EVERYONE else said to Bostic, was DO YOU JOB. Just do your job.

Hate to say it again, but R. Rodgers didn't do his JOB! But go ahead and ignore that fact and let the dislikes come. Just remember how all of you are now being hypocrites for ignoring that one VERY important FACT, even tho R. Rodgers himself admitted it!

D.D. Driver's picture

Meh. He called an audible in the ♫ Heat of the Moment♫. That's part of his job, too. It's like a running back that bounced a run outside that was supposed to go up the gut.

Dan Stodola's picture

R. Rodgers didn't call an audible. And A Rodgers didn't either. THe lateral play was out the window, used it on the previous play. R. Rodgers admitted his JOB was to box out for Adams. And he didn't, which is exactly what EVERYONE (and their mother) complained about w/ Bostic. R. Rodgers had plenty of time to get in position to block out, but he didn't. His job was to block out, period! At least thats all anyone cared about when Bostic F'd up. Now everyone is ignoring that fact.

I'm very happy it worked out too, but at least I'm not being hypocritical and ignoring recent past when doing so.

D.D. Driver's picture

I was using audible metaphorically....

Bottom line: players routine go off script depending of the situation. But if you are going to bounce an inside dive outside the tackles, you better no lose five yards. If you are going to try to catch an onside's kick, you had better catch it.

Dan Stodola's picture

Which is whst I said, but not the point.

hobbes's picture

Supposedly they were in the wrong formation, meaning they were all winging it. If the play call and formation matched, then Rodgers should not have gone for the ball (like Bostick shouldn't have gone for the ball). But if the play has obviously broken down then he's doing the what gives his team the best chance to win, which is to go for the ball.

Dan Stodola's picture

Not in the wrong formation. However they hadn't run that play w/ that personnel group before. Specifically R. Rodgers usually wouln't have been on the field. But, R. Rodgers DID know his job assignment when the playcall came in. So even tho he knew his assignment he didn't preform it.

Like I mentioned, I didn't have an issue w/ Bostic going for the ball, but if you do, you must make the play. However, that isn't what ANYONE else said. It was do you JOB, do your damn Job, not anything about making a play if you go for the ball.

hobbes's picture

From ESPN (

"In the Green Bay Packers’ playbook, you won’t find the words “Hail Mary” anywhere. But the play they used to shock the Detroit Lions on Thursday night does have a name: Two Jet Rebound Pass. “We think we’re going to go catch it like a rebound,” Packers coach Mike McCarthy said Friday. “So it’s called a rebound pass.” However, because the Packers had different personnel on the field for the final play of Thursday night's 27-23 win at the Detroit Lions, it actually turned into Scat Two Rebound Pass."

I agree somewhat with your sentiment, if you decide to not follow the play, you better come up with the ball, which is what Bostick (not Bostic, who was an inside linebacker for the bears/patriots) didn't do. On the other hand, following the playbook when the play has broken down (wrong personnel group and wrong formation until the motion) is not the smart thing to do nor is it how the Packers are taught. Receivers are taught to break off their routes and just get open because Rodgers can get the ball to you even if you are not where you are supposed to be.

I think it's a far less egregious error if the play breaks down, Rodgers goes for the ball and drops it versus if the play is run properly, Rodgers breaks his assignment and then drops the ball. One is a guy not being a teammate, the other is a guy trying to optimize his team's chance of winning in unknown circumstances.

PaulRosik's picture

The play may have called for him to box out, but it all depends on where the throw falls and int his case it was Adams who blocked for Rodgers to go after the ball with a totally clear path and no one in front or in back of him except packer players.

paxbak's picture

Onside kick is a routine play by the receiving team. If everyone does their job, the receiving team recovers the kick. A hail mary is different in that it is a one in a million play. Anyone that has a shot at catching it should ignore their job and catch it thus maximizing the chances of a positive outcome. Bostic was not involved in a hail mary - he should have done his job and we would have won the Super Bowl last year

Idiot Fan's picture

Actually, I kind of see some similarities between the two plays - in both cases you want the bulk of your guys to block out the other team while one guy gets a clean shot at catching the ball. It's one of those plays that is a lot more coordinated that we often suppose.

Dan Stodola's picture

They are very similar plays. And each player is supposed to do his job. R. Rodgers didn't and neither did Bostic. Everyone had a problem w Bostic but don't Rodgers and now they're being hypocritical about it.

Oppy's picture

A detail that should probably be included here; Richard Rodgers has been singled out as having "perhaps the best hands on the team" since his rookie camp- something Bostic could never claim.

Also worthy of noting, Aaron Rodgers was throwing practice bombs specifically to Richard Rodgers during warm ups. He also literally caught every pass thrown at him all game long.

Lastly, Richard Rodgers was in ideal position to make the play. All defenders within striking distance were behind him by roughly a yard, and there was already a wall formed.

Bostic was in perfect position to do his job- a job that still needed doing- and even though Jordy Nelson has infinitely better hands than Bostic, and there wasn't a wall to keep the opposing team off the ball... Bostic went for it anyways.

The difference is Bostic was reckless and used poor judgment. Richard Rodgers was in position, the situation was right, he had practiced the ultra-high ball catch, and his hand talent and performance that night made his decision to break from the script a rational, and ultimately a good one.

Dan Stodola's picture

You only say reckless and poor judgment since Bostic didnt catch it. If he had you would say great play. Rodgers could have gotten in position to box out if he had hustled a little. Either way he didn't perform his job on the play. So your pissed at Bostic and call him out, but your fine w Rodgers even tho he didn't do his job either.

Oppy's picture

Please, don't put words in my mouth, or emotions on my sleeve.

I regularly voice my displeasure with playcalls that succeed, or my approval of ones that fail.

Just pointing out it's not as black and white as you're painting it. Sorry to throw observation and logic into your tirade against fans.

Dan Stodola's picture

No not black and white, but If you try to make the play, a la Bostic you better damn well make it. Great that Rodgers made the catch, but the fact remains he could have easily done he job assigned but didn't.

Pumpguy1988's picture

What is the reason you box out? You're trying to get the ball. So in reality RR did his job

dobber's picture

"But I'm very sure the only thing almost EVERYONE else said to Bostic, was DO YOU JOB. Just do your job."

I'm reminded of a scene from the movie Major League where Wesley Snipes makes a fancy basket catch in the outfield and comes to the dugout where Lou Brown slaps him on the back and says "Nice catch...don't ever F***KIN' do it again!"

If you make the play, all might be forgiven, but you're probably going to go to the woodshed for what you should have been doing, anyway.

Evan's picture

Good lord...who gives a shit.

Dan Stodola's picture

Thanks for making my point! LMFAO

Pumpguy1988's picture

What is the point of boxing out anyhow? To get the ball. So what RR did was his job

Packatron's picture

Bostic was a third string nobody.
Richard Rodgers is a high draft pick and a starting NFL tight end.

Big difference there.

Mojo's picture

Don't know if RR was even fast enough to get into a blockout position. He just meandered in their at the end. So he did the next best thing possible - make the catch (and actually he did end up screening players out - his own).

thebeast431's picture

The way I saw it was the ball was undert-hrown. If it was 3-4 deeper in the endzone and RR was already there he would of blocked out. Facts are it was undert-hrown, Rodgers was to slow to play blocker and had to improvise.
With Bostick, nothing changed. The onside kick was standard. Now if the kick was a bullet right at him, then yes, its ok to go offscript.

4thand1's picture

No one honestly believes he wasn't supposed to catch the dam ball if he had the chance. Sure he was supposed to block out but probably told himself,"I'm not leaving this up to Adams, no fucking way". No one can be sure exactly where it will end in the 1st place. It would have bounced off of Bostic's helmet. Danny , congratulations, you pull off a cow.

Oppy's picture

under thrown?


TommyG's picture

All of us? I never was upset with bostic. As a matter of fact the only part of that bostic play that upset me was that he dropped the ball. He had a great shot at catching it and just didn't. Rodgers had the best shot at catching this one and did.

4thand1's picture

Yes, if botchkick catches it, everyone is happy. He might still on the team not getting the playbook.

Thegreatreynoldo's picture

It is true that RR was supposed to block out but did not do so. However, I did not see any Detroit player that needed blocking out, so I think RR did the right thing in going up for the catch. It would be like a pulling guard whose responsibility is to block the OLB, but the OLB slipped or dropped into coverage. Assuming the OLB is not there to block, I sure hope the OG would find someone else to block.

ray nichkee's picture

If botchkick catcthes it he's still bostick and the superbowl is possibly rewritten.

Stroh, are you going out of your way to be a complete douchebag?

D.D. Driver's picture

This is fantastic. That is all.

zoellner25's picture

I was waiting for an ESPN sport science on this play, but this is even better. throwing a ball 78 feet into the air, that is insane!

croatpackfan's picture

I already said that was the best throw I ever seen.
Now, can we move on to the next Sunday game. Enough about past...

WinUSA's picture

Mike..a very impressive article chock full of algorithms and calculus formulas that I, who never made it out of General Math don't have a clue if the are correct or not...but hey, the explanations in between was crystal clear to even the ignorants like myself. Well done.

Given all that....the play was a play for the ages and I am so happy that I gutted it out to see it.....

I would also like to say that previous play, the lst Rodgers to Rodgers lateral play, though didn't end as planned was pretty darn impressive in its own right. The laterals were done to perfection...... Lady luck for once shined down on us to give us the opportunity to have the Motown Miracle happen.

Now let's get Dallas!!!!!

Mojo's picture

Right, the two laterals were well executed, although I don't think ARod was supposed to get the last one.

porupack's picture

Impressive mike. Thanks.

Some other things about the play was QB Rogers' elusiveness fully on display, or that throw never happens.

How do 3 passrushers get through 5 Olinemen that easily?

MikeReuter's picture

That's a fair question and certainly not an ideal scenario. At the end of the day, you probably want Rodgers scrambling out to build up his momentum, but not under those circumstances.

Idiot Fan's picture

I almost wonder if it's part of the plan to let the rushers come (hopefully slowly) up the middle. For a play like that, you need time to let the receivers get downfield (and A LOT of time for RR to get there...), and you also need space to step up into the throw. It seems like the ideal is to have the QB roll out wide to accomplish both of those things, but if there are defenders there, that can be a problem too. If you can sort-of corral the defenders into the middle then you can allow the QB to get outside and have room out there. That's a total guess, but it wouldn't shock me if something like that was part of the play.

4thand1's picture

That play was what execution can do.

dobber's picture

I still love, as I watch that clip and the ball is coming down, to see the blue jerseys all fade to the back of the group and the white jerseys come forward. Someone was going to catch that ball, and it's a wonder the Packer receivers didn't get in each others way and knock the ball down...might have happened if RRodg. hadn't "drifted" in and made that catch.

MikeReuter's picture

I wondered that too, it looks like Davante was in position to catch it... but there were so many players, there could have been some "friendly fire" knocking the ball down. Looked like Adams being there actually held off the defenders for Richard Rodgers.

Dan Stodola's picture

If Adams hadn't caught it and the ball is knocked down somewhat, Cobb was in perfect position for any rebound. He was in position to do his job on the play. Glad Rodgers caught it but he didnt technically do his job.

Cobb and Adams both did their jobs on the play. Pretty much everyone but R. Rodgers did.

4thand1's picture

RR had no one to box out because the lolions played it so poorly. There should have been at least 2 lolion guys at the goal line. So he did the only he could do, HE CAUGHT THE FUCKER! RR did his job, sorry but I think you're wrong this time DS.

PaulRosik's picture

That's the key there 4thand1. He was totally free and clear and no one was around him. I am sure he would have boxed out if there was someone within 10 yards of him but having a totally free run he could jog into place and do his best vertical jump just like they do at the combine and then make a grab for the ball.

4thand1's picture

The biggest part of the "hail Mary" was the lolions totally screwing it up. They were playing for another lateral, lmfao!

Dan Stodola's picture

No one to block out becuz he didn't run downfield like he's supposed to. Probably 8 sec from snap to ball arriving in the end zone. He's a professional athlete, he should have easily run 65 yds in 8 sec. If he had hustled he would have been in positon to block out for Adams.

Idiot Fan's picture

"Someone was going to catch that ball"

Or at least, it was going to hit one of our guys in the hands... :)

dobber's picture

Too true for words...

Jersey Al's picture

At least you stopped short of going to differential equations, which drove me crazy in college. Great stuff, Mike!!!

MikeReuter's picture

The only thing I remember about differential equations is that Carlos Gomez hit for the cycle while I was in class one of the nights. Which is about all I need to know.

WKUPackFan's picture

Differential equations? Dang Jersey Al, I apologize. Anyone who can do those is smarter than I am (at least in that area).

Pack88's picture

After almost vomiting when Bostic failed to box out the Seattle defender in NFC championship game - on this play Rodgers job might have been to box out for Adams, however when the ball entered the "catch window" Rodgers is still at the 10 yard line and every other key player, Lion or Packer, is in the end zone . The trajectory of the ball more resembled a punt than a pass so that as it fell and Adams sped ahead of his defender Rodgers and Adams were both in front of the Lions defender in such a manner that 82 could not have boxed him out without offensive interference being called. The debate about doing your job will go on and on but look at the tape- there is no way for Rodgers to box out Adams defender- 17 is in front of that defender . Don't wan't to start the instincts vs assignment debate but slow the tape down and look at the position

Tundraboy's picture

Good point. I agree.

Dan Stodola's picture

If Rodgers had run a little faster he would have been in position to do his job. He had about 8 sec from snap before the ball go to the end zone. Hope he can at least run 65 yd in 8 sec.

4thand1's picture

He was trying to locate the ball. What if the throw doesn't make it the end zone? I'm sure they take every scenario into account.

Tundraboy's picture

It was a perfect throw given the situation. Rodgers was as close to bending over backwards without falling that one could possibly be. As he said he needed to to arc it high to give his recievers time to get down the field. Not the most beautiful throw but the most amazing I have ever seen. As for Bostic I was mad because he didn't catch the ball.

Steve Cheez's picture

I get a chubby every time I watch that.

Ibleedgreenmore's picture

Saw earlier the throw and catch set two more records, amazing how many records we seem to break.

PaulRosik's picture

As a math and science teacher awesome stuff! The only thing I would add is a comparison to an "average" bomb to see how much this one was just launched. Maybe Wisconsin high schools can use the Rodgers Hail Mary as a prop or even a project to teach the laws of motion (uvast equations) when they get to them in physics. Might help a few more people pay attention that day.

4thand1's picture

I'd of been screwed.

bart's picture

Nice article Mike.

Big Moe's picture

Good stuff, reminds me of an equation from school using a Tiger Woods drive (pre-falling apart).

Go Pack!!! Now Pound the cowboys.

4thand1's picture

Wow, queens actually should have won that one. But it's still a loss. Now Packers gotta take care of business Sunday.

LAS VEGAS-TOM's picture

4thand1, What is the Health of the OL? LVT

4thand1's picture


LAS VEGAS-TOM's picture

4thand1, Are any Game Time decisions. It wouldn't surprise me at all if MM gave them another weeks rest. Like I said, MM will have this team as Healthy as he can get them by the playoffs. He may want to hold some people out, without announcing it. He can't legally do that. Did Bulaga, Sitton, Bak, practice this week? LVT

4thand1's picture

They all listed as limited. Lindsley was the only non participant. Ty Montgomery practiced again. So Lindsley was the only one who sat out.

ray nichkee's picture

I forget what day it actually is but in a regular week Friday is a walk through and Saturday is a light practice. If you don't practice Friday you dont play Sunday. There was an article I read a while back about the change up. Feel free to correct me if I'm wrong.

4thand1's picture

Thursday was a padded practice, and I think you're right about Friday. I'm haven't heard how Ty Montgomery did in the padded practice. hopefully it went good. just watched an Eddie lacy interview, he says the best it yet to come. He and MM had a one on one and he's ready to go.

LAS VEGAS-TOM's picture

4thand1, I could be wrong but I believe that's a GB rule. I think each team has to report to the league by Wednesday their injury report & the status of players. Maybe the Friday thing is now a league rule. There are game time decisions, & teams do get around the rules by pulling players out early. There is also a lot of disinformation put out by all the teams. You can't believe every thing you hear. I could be way out in Left field, but I think the #1 thing on MM's mind is the health of his team. See if there aren't a few surprises this week. LVT

Rich Beckman's picture

Kudos to, I think, Tretter, for having the awareness to stop short of the line of scrimmage just prior to the the throw.

sheppercheeser's picture

In an interview, RR said that the play is prolly going to cost him an "MA", which I have to assume means Missed Assignment. I have no doubt that if RR had just "done his job", Davante would NOT have come down with this, or any catch (lately) for that matter.

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