Calculating the Ridiculous Physics of Disney’s Hercules

By Anthony Zonfrelli and Dmitri Ilushin

This post can also be viewed on Deadspin here.

Disney’s Hercules: the classic tale of the clumsy boy blessed with superhuman strength who fights to become a hero to rejoin his family on Mount Olympus. With the help of his goat-man trainer, Philoctetes, Hercules learns to use his incredible gift to “Go the Distance,” performing all kinds of outrageous physical feats.

In order to give Hercules his due —  and to judge Disney’s ability to follow the rules of the known universe — we looked into the physics behind these feats to better appreciate how otherworldly his strength really was. I don’t care if I’m watching a movie with shape-shifters and talking tornados -– I want to see some consistency.

We see Hercules’ first truly impressive feat of strength during his teenage years, when he is seen sprinting down a barren road with a cart loaded with an incredible amount of hay bound for the marketplace. Eager to help out, Hercules lifts the entire haystack, but his father tells him to wait by the cart while he haggles with the local hay-mongers. A chastened Hercules then drops the haystack back onto the cart, causing the cart to launch his father’s donkey, Penelope, into the sky. With a few assumptions and basic kinematic equations, we can dig a little deeper into Penelope’s wild ride through the clouds.


Poor Penelope was in the air for 7.7 seconds of screen time. With this measurement alone, we can deduce that she reached a maximum height of 73 meters, meaning she went on a ride higher than the eighth-tallest roller coaster in the world without so much as a seatbelt. Furthermore, we estimate that the g-force required to catch Penelope — a task Hercules managed with little to no stress — exceeds 19 g. Keep in mind that anything over 25 is likely to result in serious injury or death to a human (and probably a donkey). At Penelope’s speed, the force of the catch is equivalent to her crashing the cart into a telephone pole at 84 mph.


To propel an average-sized donkey that high, the mass of the haystacks must have totaled 12,864 kg (28,360 lbs), which didn’t seem to faze Hercules, who managed its weight with one arm. What’s less clear is how Hercules’ father expected Penelope to lug over 14 tons of hay into town. Donkey rights have come a long way.


We fast-forward past a strenuous Danny DeVito training program, when Hercules takes his first pass at being a hero. Two boys are trapped in a rock quarry, begging someone to “call IX-I-I.” Hercules sees his big chance and hefts the enormous boulder off of the boys, presses it up over his head, and saves the day.

From a screenshot, we estimate that the boulder has a radius of approximately five times Hercules’s height (6’5”, according to legend). Using this measurement and Wolfram Alpha’s estimate of the average density of rock, we find that the gargantuan boulder weighs nearly 23 million pounds. In non-rock terms, this is equivalent to the weight of 575 Greyhound Buses or nearly 2400 African elephants. Had this task been conducted in an official competitive setting, Hercules would have shoulder pressed 44,000 times the world-record weight.


Soon after achieving hero status, Herc engages in that other favored pursuit of teenage males: chasing tail. After wooing the beautiful Megara, Hercules apparently gets a boost of testosterone from falling in love and goes straight to the arena to lift weights with his flying horse pal, Pegasus. In a fit of youthful exuberance, Hercules jumps off the high bar and deep into the night sky, achieving an absurd hang time of 30.1 seconds — long enough for evil demons Pain and Panic to capture Pegasus by shape-shifting into a sexy Pegaslut (complete with tramp stamp) and luring him into their clutches.

Using the same method we used to calculate Penelope’s height, and taking into account the standard height of a high bar, we find that Hercules took off with an incredible velocity of 147.6 m/s (330 mph) and reached a towering height of 1,113.7 meters (3,654 feet). Let’s put that into perspective – the Empire State Building is 1,250 feet tall, the Sears Tower is 1,450 feet tall, and the tallest building in the world, the Burj Khalifa in Dubai, U.A.E., is 2,717 feet tall.


To figure out what kind of view our hero enjoyed from that height, we’ll need a lesson in geometry. We only need two pieces of information: Hercules’s maximum height and the radius of the Earth. Knowing that the line of Herc’s vision will be tangent to Earth’s surface, we can make a right triangle with this point on the horizon, Herc’s viewpoint, and the Earth’s very core. Taking the inverse cosine of the ratio of R/(R+h), we can figure out angle x. Then using tan(x), we can find d.


This calculation shows that Herc’s spectacular view is over 74 miles in any direction, nearly twice as far as one’s view from the top of the Eiffel Tower. Assuming his training area is located in the center of Thebes, he can see well past all of Athens and into the Aegean Sea.

The evil Hades greets Hercules upon his return to Earth, and in exchange for Megara’s promised safety and Pegasus’s release, Hercules agrees to give up his strength for 24 hours while Hades unleashes the Titans in an attempt to overthrow Zeus and take control of Mount Olympus. The deal is confirmed with a dramatic handshake that sucks the power right from Hercules’s body. But there’s only one problem with it – they shake with their left hands. In what universe is that deal legal and binding? Hercules might as well have just crossed his fingers behind his back if this is what constituted a legitimate, godly contract in Ancient Greece.


After a brief interlude involving a tender love scene and the recovery of his strength, Hercules chases down Hades and joins the gods in the war against the Titans where he performs his final, absolutely ridiculous feat of strength. Hercules grabs the tornado Titan, Stratos, and swings him around until he has sucked up all of the bad guys, and, in super slow motion, hurls them all into outer space. Where they explode. This feat exhibits incalculable strength, and the explosion in space breaks all kinds of laws of physics. I cannot even begin to understand, let alone explain, how it all transpired.


The rest is (ancient) history. Hercules takes a dip in the underworld, reviving Meg, and restoring order to Mount Olympus. Take it away, Hades.

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TEBOW TEBOW TEBOW: Is Tim Tebow Capable of Adding Value to New England’s Offense?

By Anthony Zonfrelli

It’s “Tebow Time” in New England. The Patriots’ signing of the former Broncos star and Jets backup has catalyzed yet another mass analysis of the poor guy. All of New England is now wondering, “What in the world could Tim Tebow possibly add to New England’s powerhouse of an offense?”

The statistic posted in various places by ESPN suggests that Tebow actually hurt the Jets’ offense last year, averaging 3.8 yards per play when he was on the field versus 4.7 yards per play when he sat.  But this doesn’t quite encapsulate the Tim Tebow experience. For one, it includes plays in which Tebow simply handed the ball off, where he likely had little to no effect on the play. What the New Englanders really need to know is how well Tim Tebow did in situations where he will be relevant.

It’s common knowledge by now that Tebow is probably best suited in short yardage situations, using his size, athleticism, and childlike enthusiasm to get those tough first downs. It is highly unlikely that the Patriots will ever try their luck with Tebow on 1st or 2nd down when they have an elite quarterback like Brady at their disposal, let alone permit him to chuck the football. Instead, the Patriots are probably only looking to use him as a pounding running back when they would have otherwise risked Brady’s health by sneaking him up the middle. So how well did Tebow do in these situations?

Using play-by-play data from Advanced NFL Stats, I compiled all rushing plays by the Jets on 3rd or 4th down with less than 5 yards to go. Rather than simply comparing Tebow plays versus non-Tebow plays, I looked specifically at plays where Tebow made an impact – only plays where Tebow actually ran with the ball. I found that, when rushing the ball on 3rd or 4th down with less than 5 yards to go, Tebow averaged 2.86 yards per play, while the rest of the Jets averaged only 2.00 yards per play.  Furthermore, Tebow converted every first down when he had to go less than 5 yards to attain it; his only shortcoming resulted from a single 3rd and 5 attempt.

In the situations that mirror those in which New England expects to see him, Tebow provided nearly an additional yard of value for his former team. The Patriots can use Tebow’s dependability in these 3rd and short situations, granting them the privilege of not having to risk their starting quarterback. Brady will no longer have to grind his way through the trenches with the obedient Tebow at the ready. That said, Brady is certainly no slouch when it comes to short yardage situations; as Bill Barnwell noted on Twitter, Brady is 18-for-19 in converting 4th and 1 sneaks since 2002.

Still, it appears as though the Patriots may have found yet another valuable role player to add to their roster. Either that, or Belichick is just using Tebow for information on the Jets (we wouldn’t put it past him) and will dump him by the end of training camp. Once the season comes, we’ll see if Belichick deems Tebow’s athleticism and divine touch worth the media circus that will likely follow him to New England.

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YouTube Athlete Hall of Fame, Pt. 1

HSAC would like to express its solidarity with the students on campus currently slaving away at their studies. We now present a brief diversion for your Reading Period consumption: the first inaugural class of the YouTube Athlete Hall of Fame, dedicated to those athletes who honed their craft not to win games, but to produce incredible highlights for posterity to view for eternity.

Andrew Mooney: For God’s sake, it’s a highlight reel of 100 dunks from the same dude, and they’re all just awesome. Look at the form on that windmill. It can’t be duplicated: the speed, the full arm extension, the perfectly round arc it traces in the air. Vince Carter was put on this Earth to stuff round things into slightly larger round holes and send them bouncing all over hardwood surfaces, the perfect combination of grace, power, and style. And, I mean, Frederic Weis (9:34)…it’s a testament to French pacifism that they didn’t immediately initiate a retaliatory missile strike for that assault. That’s one of their citizens Carter murdered.

Anthony Zonfrelli: Shawn Kemp may have been the baddest dog ever to play in the NBA. You can feel the primal satisfaction each time he throws down a monster jam or swats a shot into oblivion. The disrespect he shows defenders when embarrassing them with a posterization makes the aggressive, hard-core rap backing track all too appropriate. The only thing rivaling the fury with which he throws down each dunk is the humiliation he makes the defense feel afterward (0:52).

Nick Jaroszewicz: Here’s one for the rafters. High school football highlight tapes don’t get much better than Sam McGuffie, who took YouTube by storm during its “infancy” in 2007. With 3.15 million views today, many people have seen McGuffie elude, knock over, carry, and occasionally hurdle would-be tacklers. After struggling at Michigan, McGuffie found a place at Rice, where he showed a few flashes of his greatness, and was recently picked up by the Raiders, whose insatiable desire for speed and flash seems to persist, even with the passing of Al Davis.

Andrew Mooney: When you re-watch videos like these, you can understand why Texans fans rent their garments when their team passed on Reggie Bush for Mario Williams with the No. 1 pick. He’s a video game character. I remember watching that full-speed hurdle at 0:22 live and laughing gleefully that the underworld had spat up its running back demon to afflict the hated Irish of Notre Dame. After treating the likes of Fresno State and UCLA as rudely as he did, he must have thought he’d rush for 4,500 yards a year in the NFL. He should have five Heismans.

Anthony Zonfrelli: Some entertaining highlight reels are only possible because of “Big Fish-Small Pond” Syndrome, a classic affliction of the high school athlete video mashup. While this tape may be another one of those culprits, this fish doesn’t even fit in the pond. In this video, Seventh Woods, listed at 6’2″, 170 pounds, is shown dominating his high school league…at the age of 14. Though it may not be as “impressive” to see somebody that large continuously dunk over his short white peers, who doesn’t like to see armpit blocks (0:32), MJ-like free-throw tip slams (0:33), and behind-the-backboard shots reminiscent of Larry Bird (0:57)? Let’s hope there’s an Eighth.

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The United States of Football

By Andrew Mooney

In 2013, it seems safe to say that by almost any measure—viewership, ticket sales, merchandising—America considers football to be its national pastime. Gone are the days when youngsters snuck into the Polo Grounds or Fenway Park to catch the matinee performance of their favorite ballplayers; now is the era of Friday Night Lights andMadden.

Of course, only the thinnest slice of those kids who grow up dreaming of gridiron glory actually achieve it at the highest professional level; over a million U.S. teenagers play high school football, and only 1,696 roster spots exist in the NFL. Yet despite this miniscule success rate, nearly two thousand successful cases is certainly a large enough sample on which to perform some analysis. In this case, I’ll look at how that NFL talent is distributed across the country.

I examined the birth states of all active NFL players from the records kept on and combined this with state population estimates from 2012 U.S. Census data. There were quite a few players that PFR listed as having “Unknown” birthplaces, but I assumed for the purposes of this post that the distribution of this group’s birthplaces was similar to the data I was able to gather. Armed with this dataset, I determined the most prolific football talent-producing states, both on an absolute and per capita basis.

The first state north of the Mason-Dixon line to appear on the per capita list is Nebraska, at number eight. Though the state contains less than two million people, 14 Nebraskans are on NFL rosters at present.

It’s not surprising that the South, the holy land of football, leads the way in NFL players per capita with seven of the top ten states. But the top individual states aren’t the ones you might expect. The triangle of Louisiana, Mississippi, and Alabama comprise three of the top seven states, all of which lie ahead of traditional football sanctuaries Texas and Florida.


Of course, these latter two are penalized by this method of measurement for their sheer size. In absolute numbers, Texas and Florida rank second and third in number of active pros with 130 and 107, respectively. The only other state to surmount the century mark for active players is the behemoth of this list, California, which boasts 143 current NFL players. Though it has by far the largest population of any state in the country (nearly 50 percent larger than second place Texas), California still comes in around the middle of the pack (18th) in the per capita rankings, a testament to the football prowess of the Golden State.

Bringing up the rear is New England. Despite a near-dynastic pro franchise in close proximity, the Northeast produces almost no native NFL players of its own. Maine, New Hampshire, and Rhode Island have a combined two players currently in the NFL. Massachusetts, the home state of the Patriots, has a slightly less embarrassing total of seven active players, and Connecticut, a state of a little over three million people, has a respectable 11. Still, just two in a million people from all of Patriots territory have reached the highest level of the sport.


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At Which Positions Have NFL Rookies Had the Most Success?

By Alex Koenig

The NFL draft is an exercise in patience. Unlike the NBA, it is rare for highly touted NFL rookies to make significant impacts at key positions. Sure, a player like Cam Newton will come around every now and then, but by and large draft picks are made under the reasonable assumption that the transition from the college to the pro game takes a while.

Nevertheless, talking heads and coaches alike regularly claim that players can “step in right away and make an impact.” Often these statements are made because of a players’ supernatural athletic ability or because he’s run a pro-style offense for the last five years – both pretty reasonable assumptions. But it got us thinking: is there any basis for these claims? Are there any patterns in terms of what types of players at what positions are able to step in and regularly prove effective?

It’s important to clarify how we measure quality in a rookie season. If we just look at the recipients of AP Offensive and Defensive Player of the Year awards, we get a jaded perception. Since the Offensive Rookie of the Year award was created in 1967, it’s been given out to six quarterbacks, eight wide receivers and 30 running backs. It’s possible that no offensive linemen have ever been worthy of the award but unlikely. On the defensive side, six cornerbacks have been so honored, six defensive tackles, eight defensive ends, 21 linebackers, and just two safeties.

But that’s just compared to other rookies; what about compared to the rest of the league? Another thing to look at is Pro Bowl appearances. Since these are allotted by position, and not limited to rookies, it can be assumed that – if rookie impact is indeed the same across positions – there will be a more even distribution of success.

Since 2000, 29 rookies have played in the Pro Bowl. Only six positions had multiple representatives in this time period. Running backs, which make up 68 percent of all Offensive Rookie of the Year awards, only had two – Chris Johnson in 2008 and Adrian Peterson in 2007. Meanwhile, the offensive line had four representatives, including three at the all-important position of left tackle. Perhaps the award system should be revised to reflect actual impact, but that’s a debate for a different time.

On the defensive end, linebackers had seven representatives (four on the inside, three on the inside), perhaps justifying their disproportionate number of Defensive ROY awards and solidifying themselves as the most readily impactful position on the defense. The most commonly occurring position for rookies on Pro Bowl rosters, however, is return man, with rookies notching eight of the possible 24 spots since 2000.

But these awards just represent the best the league has to offer each year. A better statistic for measuring cumulative impact – independent of season – is Approximate Value (AV), from Pro Football Reference. The methodology is explained here, but AV is a way of trying to equalize players’ contributions across positions.

First, let’s limit the sample size to first and second rounders. No matter what coaches and media members say, no one really expects an immediate impact from the later rounds.

Since the AFL-NFL merger in 1970, the rookie season that measured best in terms of AV was cornerback/returnman Patrick Peterson of the Arizona Cardinals in 2011. Peterson’s four punt return touchdowns and two interceptions gave him an AV of 21. For reference, the highest AV total Peyton Manning ever accumulated was 21 in 2004.

The distribution of the top 100 rookie seasons since 1970 in terms of AV is as follows:




Running Back


Edgerrin James (21) – 1999



Lawrence Taylor (17) – 1981

Wide Receiver


Randy Moss (17) – 1998



Cam Newton (19) – 2011



Patrick Peterson (25) – 2011



Ronnie Lott (18) – 1981

Left Tackle


Ryan Clady (13) – 2008

Defensive End


Jevon Kearse (16) – 1999

Tight End


Charle Young (13) – 1973

Defensive Tackle


Ndamukong Suh (15) – 2010

What is it about running backs and linebackers that make them so much more effective as rookies than players at other positions?

There’s probably a pretty simple answer. Both positions are very much reliant on instincts and reaction time. Unlike a quarterback, an offensive lineman, or a wide receiver, the playbook does not expand a whole lot for a running back in transitioning to the NFL. Certainly adjustments need to be made – new blocking assignments, an increased role catching passes out of the backfield, etc.  – but by and large running backs are relying on good blocking up front, an ability to quickly assess a situation and react to it, and their own physical gifts. Yes, the defense will be better prepared and conditioned than most they will face in college, but key attributes for running backs like speed and agility tend to peak early on in their careers.

Similarly, the linebacker position relies on an ability to react and hit the hole. The factors that contribute to making that hole may be more complex, but the reaction time is still the same. Whether it’s figuring out a blocking scheme or adjusting to play action, linebackers are forced to rely on their instincts more than almost any other position, and while experience will certainly fine tune those instincts, for the elite young linebacker, those instincts are already there.

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Lakers Resurgence: New Team or Regression to the Mean?

By Julian Ryan

The Lakers have risen from the grave. Seemingly destined to hand over their lottery pick to the Suns for much of the season, the team has turned their season around and are now furiously battling it out with the Jazz for the final playoff spot in the West. (Editor’s Note: This article was written before Kobe’s Achilles injury, hence the more optimistic tone — they should still make the playoffs, but their outlook is pretty bleak once they are there without their star player.)

LA started out 17-25 – a far cry from the expected output of the superstar filled team– but have since rallied, going 25-12. Kobe Bryant has rolled back the clock with a series of stellar performances, including a season high 47 points against Portland on Wednesday, earning himself Western Conference Player of the Month in February for his efforts. The turnaround is all the more remarkable given the injuries sustained by starters Metta World Peace and Steve Nash.

Is this improvement in form entirely a result of the team of all-stars finally coming together and clicking as a unit? If the Lakers had objectively improved in the second half of the season, one would expect that their average point differential would be improving. Point differential data can help us strip out some of the luck of winning and losing close games and is thus a good indicator over time of true performance as a complement to simple win-loss percentage.

The graph below charts the Lakers’ winning percentage and point differential over the course of the season. The blue line shows changes in winning percentage by game and corresponds to the y-axis on the right, while the red line shows changes in point differential by game and corresponds to the y-axis on the left.

Screen shot 2013-04-15 at 2.26.30 AM

Looking at the Lakers’ average point differential (+/-) at each point of the season, there does not seem to be a discernible increase during their recent run. Their winning percentage has been increasing while their +/- declines. It appears that to some extent, the Lakers’ luck has turned and they are simply regressing to the mean. They were 3-7 in their first ten games decided by five points or less, and since then have been 9-2 in such fixtures. Their increasing winning percentage is more likely a return to the true quality of the team this year, its perception not distorted by close losses, than an increase in the quality of the team.

All of this does not take away from the achievements of the team. The big three of Bryant, Howard and Gasol have all improved their game and deserve credit for dragging the Lakers back into contention. However, the data would seem to suggest that regression to the mean through better luck in close games may also be at play.

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Rethinking the Greatest Performances in Masters History

By Andrew Mooney

Cue Jim Nantz and the tinkling piano: the Masters is officially under way. The field of 93 will attempt to duplicate the legendary green jacket-winning performances of years past, including a “who’s who” of golfing royalty: Tiger Woods, Jack Nickalus, Arnold Palmer, Phil Mickelson, Seve Ballesteros.

Though the competitors always play the same course, Augusta National has undergone many significant changes since it hosted the first Masters tournament in 1934. As a result, it’s difficult to compare performances at the Masters across the years. Simply tabulating raw scores is not the most accurate way to do it; in different years, the field faces different temperatures, winds, and moisture, not to mention the alterations to the course itself and the equipment in players’ bags.

In a piece written in 2011, Grantland’s Bill Barnwell proposed another method for evaluating golf scores using a statistical measure called a Z-score. Barnwell’s argument was that a golfer’s actual raw score was less important than how that score compared to the rest of the field.

Take two victories by Golfer A and Golfer B, each of whom shoot 11-under-par, while the runners-up each shoot 6-under-par. They look equal, but the rest of the field’s performance matters. Let’s say the third-place finisher in Golfer A’s tournament shoots 5-under-par, but the third-placed duffer in Golfer B’s tourney shoots 1-over-par. Player B has clearly outperformed the rest of the field to a greater level than Player A, but raw margin of victory fails to capture that detail.

A Z-score measures the number of standard deviations a particular observation is from the mean of those observations. When applied to golf scores, a Z-score can tell us how different a player’s score was from the average, in addition to incorporating the range of those scores. Barnwell applied this method to the four major tournaments and recorded the top-20 performances since 1960, three of which came at the Masters, which you can read here.

As the first round kicks off today, I decided to examine the greatest performances in the history of the Masters more in-depth. I gathered the 72-hole scores for every Masters tournament, pulling the data from all the way back to 1934. I then converted each year’s raw scores into Z-scores so I would have a uniform standard for comparison across years. Since a low score in golf is good, a more negative Z-score reflects a better performance. The 20 best scores are in the table below.


Many people point to Tiger Woods’ 1997 Masters as the most impressive 72 holes Augusta has ever seen, significant not only for his otherworldly display of golf, but for the fact that it came at age 21, it was his first major win, and it was the first time a non-white player had won the tournament.

Though Woods’ score of 270 remains a tournament record, his performance was slightly less superior relative to the rest of the field than Jack Nicklaus’ 271 in 1965. Woods defeated second-place Tom Kite by 12 strokes in 1997, three more than Nicklaus bested Arnold Palmer by. But only eight other players were under par in the 1965 tournament, compared to 15 in 1997. Nicklaus’ lower Z-score suggests that his total of 17-under came in a more difficult overall course environment and thus exhibited more dominance than Tiger’s 18-under.

A couple of other takeaways from this list:

  • Woods, Nicklaus, Raymond Floyd, and Nick Faldo are the only golfers to appear on this list more than once.
  • In 2005, Chris DiMarco submitted one of the best-ever Masters performances and finished second, falling to Woods in a playoff, much to the relief of the Nike marketing department.
  • Doug Ford tied with Tommy Aaron for the worst to-par score on this list at five-under, but in Ford’s case, only two other players in the tournament finished under par.
  • In gathering this data, I was also able to uncover the worst performances in Masters history (for players that made the 36-hole cut). The most shameful 72-hole score at Augusta came in 1940, when Chick Evans posted a 43-over, shooting 82-84-86-79 for a total of 331 (Z-score: 3.627). In second-to-last place is the aforementioned Tommy Aaron’s 2000 Masters, which came 27 years after he donned the green jacket. A score of 25-over can be forgiven for a man of 63, but it still yielded a ghastly Z-score of 3.346.
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